Quantum Physics full article

<h1 id="firstHeading" class="firstHeading" lang="en" >Quantum mechanics</h1>

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<div class="hatnote" >For a more accessible and less technical introduction to this topic, see <a href="https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics" >Introduction to quantum mechanics</a>.</div>

Solution to <a href="https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation" >Schrödinger's equation</a> for the hydrogen atom at different energy levels. The brighter areas represent a higher probability of finding an <a href="https://en.wikipedia.org/wiki/Electron" title="Electron" >electron</a></div>

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<th ><strong class="selflink" >Quantum mechanics</strong></th>

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<div ><a href="https://en.wikipedia.org/wiki/Uncertainty_principle" title="Uncertainty principle" >Uncertainty principle</a></div>

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<li ><a href="https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics" >Introduction</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics" >Glossary</a></li>

<li ><a href="https://en.wikipedia.org/wiki/History_of_quantum_mechanics" title="History of quantum mechanics" >History</a></li>

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<div class="NavHead" >Fundamentals<a class="NavToggle" id="NavToggle2" href="https://en.wikipedia.org/wiki/Quantum_mechanics#" >[show]</a></div>

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<div class="NavHead" >Experiments<a class="NavToggle" id="NavToggle3" href="https://en.wikipedia.org/wiki/Quantum_mechanics#" >[show]</a></div>

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<div class="NavHead" >Formulations<a class="NavToggle" id="NavToggle4" href="https://en.wikipedia.org/wiki/Quantum_mechanics#" >[show]</a></div>

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<div class="NavHead" ><a href="https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics" >Interpretations</a><a class="NavToggle" id="NavToggle6" href="https://en.wikipedia.org/wiki/Quantum_mechanics#" >[show]</a></div>

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<div class="NavHead" >Scientists<a class="NavToggle" id="NavToggle8" href="https://en.wikipedia.org/wiki/Quantum_mechanics#" >[show]</a></div>

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<p ><b>Quantum mechanics</b> (<b>QM</b>; also known as <b>quantum physics</b>, or <b>quantum theory</b>), including quantum field theory, is a fundamental branch of <a href="https://en.wikipedia.org/wiki/Physics" title="Physics" >physics</a>concerned with processes involving, for example, <a href="https://en.wikipedia.org/wiki/Atom" title="Atom" >atoms</a> and <a href="https://en.wikipedia.org/wiki/Photons" title="Photons" class="mw-redirect" >photons</a>. In such processes, said to be quantized, the <a href="https://en.wikipedia.org/wiki/Action_(physics)" title="Action (physics)" >action</a> has been observed to be only in integer multiples of the <a href="https://en.wikipedia.org/wiki/Planck_constant" title="Planck constant" >Planck constant</a>, a physical quantity that is exceedingly, indeed perhaps ultimately, small. This is utterly inexplicable in <a href="https://en.wikipedia.org/wiki/Classical_physics" title="Classical physics" >classical physics</a>.

Quantum mechanics <a href="https://en.wikipedia.org/wiki/History_of_quantum_mechanics" title="History of quantum mechanics" >gradually arose</a> from <a href="https://en.wikipedia.org/wiki/Max_Planck" title="Max Planck" >Max Planck</a>'s solution in 1900 to the<a href="https://en.wikipedia.org/wiki/Black-body_radiation" title="Black-body radiation" >black-body radiation</a> problem (reported 1859) and <a href="https://en.wikipedia.org/wiki/Albert_Einstein" title="Albert Einstein" >Albert Einstein</a>'s 1905 paper which offered a quantum-based theory to explain the <a href="https://en.wikipedia.org/wiki/Photoelectric_effect" title="Photoelectric effect" >photoelectric effect</a> (reported 1887). <a href="https://en.wikipedia.org/wiki/Old_quantum_theory" title="Old quantum theory" >Early quantum theory</a> was significantly reformulated in the mid-1920s.

The <a href="https://en.wikipedia.org/wiki/Mathematical_formulations_of_quantum_mechanics" title="Mathematical formulations of quantum mechanics" class="mw-redirect" >mathematical formulations of quantum mechanics</a> are abstract. A mathematical function, the <a href="https://en.wikipedia.org/wiki/Wave_function" title="Wave function" >wave function</a>, provides information about the <a href="https://en.wikipedia.org/wiki/Probability_amplitude" title="Probability amplitude" >probability amplitude</a> of position, momentum, and other physical properties of a particle.

Important applications of quantum mechanical theory include <a href="https://en.wikipedia.org/wiki/Superconducting_magnet" title="Superconducting magnet" >superconducting magnets</a>, <a href="https://en.wikipedia.org/wiki/LED" title="LED" class="mw-redirect" >LEDs</a> and the <a href="https://en.wikipedia.org/wiki/Laser" title="Laser" >laser</a>, the <a href="https://en.wikipedia.org/wiki/Transistor" title="Transistor" >transistor</a> and <a href="https://en.wikipedia.org/wiki/Semiconductor" title="Semiconductor" >semiconductors</a> such as the<a href="https://en.wikipedia.org/wiki/Microprocessor" title="Microprocessor" >microprocessor</a>, <a href="https://en.wikipedia.org/wiki/Medical_imaging" title="Medical imaging" >medical and research imaging</a> such as <a href="https://en.wikipedia.org/wiki/MRI" title="MRI" class="mw-redirect" >MRI</a> and <a href="https://en.wikipedia.org/wiki/Electron_microscope" title="Electron microscope" >electron microscopy</a>, and explanations for many biological and physical phenomena.

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<h2 >Contents</h2>

  [<a href="https://en.wikipedia.org/wiki/Quantum_mechanics#" id="togglelink" >hide</a>] </div>

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<li class="toclevel-1 tocsection-1" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#History" >1 History</a></li>

<li class="toclevel-1 tocsection-2" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Mathematical_formulations" >2 Mathematical formulations</a></li>

<li class="toclevel-1 tocsection-3" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Mathematically_equivalent_formulations_of_quantum_mechanics" >3 Mathematically equivalent formulations of quantum mechanics</a></li>

<li class="toclevel-1 tocsection-4" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Interactions_with_other_scientific_theories" >4 Interactions with other scientific theories</a>

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<li class="toclevel-2 tocsection-5" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Quantum_mechanics_and_classical_physics" >4.1 Quantum mechanics and classical physics</a></li>

<li class="toclevel-2 tocsection-6" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Copenhagen_interpretation_of_quantum_versus_classical_kinematics" >4.2 Copenhagen interpretation of quantum versus classical kinematics</a></li>

<li class="toclevel-2 tocsection-7" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Relativity_and_quantum_mechanics" >4.3 Relativity and quantum mechanics</a></li>

<li class="toclevel-2 tocsection-8" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Attempts_at_a_unified_field_theory" >4.4 Attempts at a unified field theory</a></li>

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<li class="toclevel-1 tocsection-9" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Philosophical_implications" >5 Philosophical implications</a></li>

<li class="toclevel-1 tocsection-10" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Applications" >6 Applications</a></li>

<li class="toclevel-1 tocsection-11" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Examples" >7 Examples</a>

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<li class="toclevel-2 tocsection-12" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Free_particle" >7.1 Free particle</a></li>

<li class="toclevel-2 tocsection-13" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Step_potential" >7.2 Step potential</a></li>

<li class="toclevel-2 tocsection-14" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Rectangular_potential_barrier" >7.3 Rectangular potential barrier</a></li>

<li class="toclevel-2 tocsection-15" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Particle_in_a_box" >7.4 Particle in a box</a></li>

<li class="toclevel-2 tocsection-16" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Finite_potential_well" >7.5 Finite potential well</a></li>

<li class="toclevel-2 tocsection-17" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Harmonic_oscillator" >7.6 Harmonic oscillator</a></li>

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<li class="toclevel-1 tocsection-19" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Notes" >9 Notes</a></li>

<li class="toclevel-1 tocsection-20" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#References" >10 References</a></li>

<li class="toclevel-1 tocsection-21" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#Further_reading" >11 Further reading</a></li>

<li class="toclevel-1 tocsection-22" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#External_links" >12 External links</a></li>

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<h2 >History[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=1" title="Edit section: History" >edit</a>]</h2>

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<th >Modern physics</th>

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<div ><a href="https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation" >Schrödinger equation</a></div>

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<td ><a href="https://en.wikipedia.org/wiki/History_of_physics#The_emergence_of_a_new_physics_circa_1900" title="History of physics" >History of modern physics</a></td>

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<div class="NavHead" >Founders<a class="NavToggle" id="NavToggle9" href="https://en.wikipedia.org/wiki/Quantum_mechanics#" >[show]</a></div>

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<div class="NavHead" >Concepts<a class="NavToggle" id="NavToggle10" href="https://en.wikipedia.org/wiki/Quantum_mechanics#" >[show]</a></div>

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<div class="NavHead" >Branches<a class="NavToggle" id="NavToggle11" href="https://en.wikipedia.org/wiki/Quantum_mechanics#" >[show]</a></div>

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<div class="NavHead" >Scientists<a class="NavToggle" id="NavToggle12" href="https://en.wikipedia.org/wiki/Quantum_mechanics#" >[show]</a></div>

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<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/History_of_quantum_mechanics" title="History of quantum mechanics" >History of quantum mechanics</a></div>

<p >Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when<a href="https://en.wikipedia.org/wiki/Scientist" title="Scientist" >scientists</a> such as <a href="https://en.wikipedia.org/wiki/Robert_Hooke" title="Robert Hooke" >Robert Hooke</a>, <a href="https://en.wikipedia.org/wiki/Christiaan_Huygens" title="Christiaan Huygens" >Christiaan Huygens</a> and <a href="https://en.wikipedia.org/wiki/Leonhard_Euler" title="Leonhard Euler" >Leonhard Euler</a> proposed a wave theory of light based on experimental observations.<sup id="cite_ref-Born.26_Wolf_1-0" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Born.26_Wolf-1" >[1]</a></sup> In 1803, <a href="https://en.wikipedia.org/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)" >Thomas Young</a>, an English <a href="https://en.wikipedia.org/wiki/Polymath" title="Polymath" >polymath</a>, performed the famous <a href="https://en.wikipedia.org/wiki/Young%27s_interference_experiment" title="Young's interference experiment" >double-slit experiment</a> that he later described in a paper entitled <i>On the nature of light and colours</i>. This experiment played a major role in the general acceptance of the <a href="https://en.wikipedia.org/wiki/Wave_theory_of_light" title="Wave theory of light" class="mw-redirect" >wave theory of light</a>.

In 1838, <a href="https://en.wikipedia.org/wiki/Michael_Faraday" title="Michael Faraday" >Michael Faraday</a> discovered <a href="https://en.wikipedia.org/wiki/Cathode_rays" title="Cathode rays" class="mw-redirect" >cathode rays</a>. These studies were followed by the 1859 statement of the <a href="https://en.wikipedia.org/wiki/Black-body_radiation" title="Black-body radiation" >black-body radiation</a> problem by <a href="https://en.wikipedia.org/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff" >Gustav Kirchhoff</a>, the 1877 suggestion by <a href="https://en.wikipedia.org/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann" >Ludwig Boltzmann</a> that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of <a href="https://en.wikipedia.org/wiki/Max_Planck" title="Max Planck" >Max Planck</a>.<sup id="cite_ref-2" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-2" >[2]</a></sup> Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy elements) precisely matched the observed patterns of black-body radiation.

In 1896, <a href="https://en.wikipedia.org/wiki/Wilhelm_Wien" title="Wilhelm Wien" >Wilhelm Wien</a> empirically determined a distribution law of black-body radiation,<sup id="cite_ref-3" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-3" >[3]</a></sup> known as <a href="https://en.wikipedia.org/wiki/Wien_approximation" title="Wien approximation" >Wien's law</a> in his honor. Ludwig Boltzmann independently arrived at this result by considerations of <a href="https://en.wikipedia.org/wiki/Maxwell%27s_equations" title="Maxwell's equations" >Maxwell's equations</a>. However, it was valid only at high frequencies and underestimated the radiance at low frequencies. Later, Planck corrected this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called <a href="https://en.wikipedia.org/wiki/Planck%27s_law" title="Planck's law" >Planck's law</a>, which led to the development of quantum mechanics.

Following <a href="https://en.wikipedia.org/wiki/Max_Planck" title="Max Planck" >Max Planck</a>'s solution in 1900 to the <a href="https://en.wikipedia.org/wiki/Black-body_radiation" title="Black-body radiation" >black-body radiation</a> problem (reported 1859), <a href="https://en.wikipedia.org/wiki/Albert_Einstein" title="Albert Einstein" >Albert Einstein</a> offered a quantum-based theory to explain the <a href="https://en.wikipedia.org/wiki/Photoelectric_effect" title="Photoelectric effect" >photoelectric effect</a> (1905, reported 1887). Around 1900-1910, the <a href="https://en.wikipedia.org/wiki/Atomic_theory" title="Atomic theory" >atomic theory</a> and the <a href="https://en.wikipedia.org/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light" >corpuscular theory of light</a><sup id="cite_ref-4" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-4" >[4]</a></sup> first came to be widely accepted as scientific fact; these latter theories can be viewed as quantum theories of <a href="https://en.wikipedia.org/wiki/Matter" title="Matter" >matter</a>and <a href="https://en.wikipedia.org/wiki/Electromagnetic_radiation" title="Electromagnetic radiation" >electromagnetic radiation</a>, respectively.

Among the first to study quantum phenomena in nature were <a href="https://en.wikipedia.org/wiki/Arthur_Compton" title="Arthur Compton" >Arthur Compton</a>, <a href="https://en.wikipedia.org/wiki/C.V.Raman" title="C.V. Raman" class="mw-redirect" >C.V. Raman</a>, and <a href="https://en.wikipedia.org/wiki/Pieter_Zeeman" title="Pieter Zeeman" >Pieter Zeeman</a>, each of whom has a quantum effect named after him. <a href="https://en.wikipedia.org/wiki/Robert_A.Millikan" title="Robert A. Millikan" class="mw-redirect" >Robert A. Millikan</a> studied the <a href="https://en.wikipedia.org/wiki/Photoelectric_effect" title="Photoelectric effect" >photoelectric effect</a> experimentally, and <a href="https://en.wikipedia.org/wiki/Albert_Einstein" title="Albert Einstein" >Albert Einstein</a> developed a theory for it. At the same time, <a href="https://en.wikipedia.org/wiki/Niels_Bohr" title="Niels Bohr" >Niels Bohr</a> developed his theory of the atomic structure, which was later confirmed by the experiments of <a href="https://en.wikipedia.org/wiki/Henry_Moseley" title="Henry Moseley" >Henry Moseley</a>. In 1913, <a href="https://en.wikipedia.org/wiki/Peter_Debye" title="Peter Debye" >Peter Debye</a> extended Niels Bohr's theory of atomic structure, introducing <a href="https://en.wikipedia.org/wiki/Elliptical_orbit" title="Elliptical orbit" class="mw-redirect" >elliptical orbits</a>, a concept also introduced by <a href="https://en.wikipedia.org/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld" >Arnold Sommerfeld</a>.<sup id="cite_ref-5" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-5" >[5]</a></sup> This phase is known as <a href="https://en.wikipedia.org/wiki/Old_quantum_theory" title="Old quantum theory" >old quantum theory</a>.

According to Planck, each energy element (<i>E</i>) is proportional to its <a href="https://en.wikipedia.org/wiki/Frequency" title="Frequency" >frequency</a> (<i>ν</i>):</p>

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<a href="https://en.wikipedia.org/wiki/Max_Planck" title="Max Planck" >Max Planck</a> is considered the father of the quantum theory.</div>

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<p >where <i>h</i> is <a href="https://en.wikipedia.org/wiki/Planck_constant" title="Planck constant" >Planck's constant</a>.

Planck cautiously insisted that this was simply an aspect of the <i>processes</i> of absorption and emission of radiation and had nothing to do with the <i>physical reality</i> of the radiation itself.<sup id="cite_ref-6" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-6" >[6]</a></sup> In fact, he considered his <a href="https://en.wikipedia.org/wiki/Quantum_hypothesis" title="Quantum hypothesis" class="mw-redirect" >quantum hypothesis</a> a mathematical trick to get the right answer rather than a sizable discovery.<sup id="cite_ref-Kragh_7-0" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Kragh-7" >[7]</a></sup> However, in 1905 <a href="https://en.wikipedia.org/wiki/Albert_Einstein" title="Albert Einstein" >Albert Einstein</a> interpreted Planck's quantum hypothesis <a href="https://en.wikipedia.org/wiki/Local_realism" title="Local realism" class="mw-redirect" >realistically</a>and used it to explain the <a href="https://en.wikipedia.org/wiki/Photoelectric_effect" title="Photoelectric effect" >photoelectric effect</a>, in which shining light on certain materials can eject electrons from the material. He won the 1921 Nobel Prize in Physics for this work.

Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle (later called the <a href="https://en.wikipedia.org/wiki/Photon" title="Photon" >photon</a>), with a discrete quantum of energy that was dependent on its frequency.<sup id="cite_ref-8" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-8" >[8]</a></sup></p>

The 1927 <a href="https://en.wikipedia.org/wiki/Solvay_Conference" title="Solvay Conference" >Solvay Conference</a> in <a href="https://en.wikipedia.org/wiki/Brussels" title="Brussels" >Brussels</a>.</div>

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<p >The foundations of quantum mechanics were established during the first half of the 20th century by<a href="https://en.wikipedia.org/wiki/Max_Planck" title="Max Planck" >Max Planck</a>, <a href="https://en.wikipedia.org/wiki/Niels_Bohr" title="Niels Bohr" >Niels Bohr</a>, <a href="https://en.wikipedia.org/wiki/Werner_Heisenberg" title="Werner Heisenberg" >Werner Heisenberg</a>, <a href="https://en.wikipedia.org/wiki/Louis_de_Broglie" title="Louis de Broglie" >Louis de Broglie</a>, <a href="https://en.wikipedia.org/wiki/Arthur_Compton" title="Arthur Compton" >Arthur Compton</a>, <a href="https://en.wikipedia.org/wiki/Albert_Einstein" title="Albert Einstein" >Albert Einstein</a>, <a href="https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger" >Erwin Schrödinger</a>, <a href="https://en.wikipedia.org/wiki/Max_Born" title="Max Born" >Max Born</a>, <a href="https://en.wikipedia.org/wiki/John_von_Neumann" title="John von Neumann" >John von Neumann</a>, <a href="https://en.wikipedia.org/wiki/Paul_Dirac" title="Paul Dirac" >Paul Dirac</a>, <a href="https://en.wikipedia.org/wiki/Enrico_Fermi" title="Enrico Fermi" >Enrico Fermi</a>, <a href="https://en.wikipedia.org/wiki/Wolfgang_Pauli" title="Wolfgang Pauli" >Wolfgang Pauli</a>, <a href="https://en.wikipedia.org/wiki/Max_von_Laue" title="Max von Laue" >Max von Laue</a>, <a href="https://en.wikipedia.org/wiki/Freeman_Dyson" title="Freeman Dyson" >Freeman Dyson</a>, <a href="https://en.wikipedia.org/wiki/David_Hilbert" title="David Hilbert" >David Hilbert</a>,<a href="https://en.wikipedia.org/wiki/Wilhelm_Wien" title="Wilhelm Wien" >Wilhelm Wien</a>, <a href="https://en.wikipedia.org/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose" >Satyendra Nath Bose</a>, <a href="https://en.wikipedia.org/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld" >Arnold Sommerfeld</a>, and <a href="https://en.wikipedia.org/wiki/Category:Quantum_physicists" title="Category:Quantum physicists" >others</a>. The <a href="https://en.wikipedia.org/wiki/Copenhagen_interpretation" title="Copenhagen interpretation" >Copenhagen interpretation</a> of <a href="https://en.wikipedia.org/wiki/Niels_Bohr" title="Niels Bohr" >Niels Bohr</a> became widely accepted.

In the mid-1920s, developments in quantum mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory. Out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called <a href="https://en.wikipedia.org/wiki/Photon" title="Photon" >photons</a> (1926). From Einstein's simple postulation was born a flurry of debating, theorizing, and testing. Thus, the entire field of <a href="https://en.wikipedia.org/wiki/Quantum_physics" title="Quantum physics" class="mw-redirect" >quantum physics</a> emerged, leading to its wider acceptance at the Fifth <a href="https://en.wikipedia.org/wiki/Solvay_Conference" title="Solvay Conference" >Solvay Conference</a> in 1927.

It was found that <a href="https://en.wikipedia.org/wiki/Subatomic_particles" title="Subatomic particles" class="mw-redirect" >subatomic particles</a> and electromagnetic waves are neither simply particle nor wave but have certain properties of each. This originated the concept of <a href="https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality" >wave–particle duality</a>.

By 1930, quantum mechanics had been further unified and formalized by the work of <a href="https://en.wikipedia.org/wiki/David_Hilbert" title="David Hilbert" >David Hilbert</a>, <a href="https://en.wikipedia.org/wiki/Paul_Dirac" title="Paul Dirac" >Paul Dirac</a> and <a href="https://en.wikipedia.org/wiki/John_von_Neumann" title="John von Neumann" >John von Neumann</a><sup id="cite_ref-9" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-9" >[9]</a></sup> with greater emphasis on <a href="https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics" >measurement</a>, the statistical nature of our knowledge of reality, and <a href="https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics" >philosophical speculation about the 'observer'</a>. It has since permeated many disciplines including <a href="https://en.wikipedia.org/wiki/Quantum_chemistry" title="Quantum chemistry" >quantum chemistry</a>, <a href="https://en.wikipedia.org/wiki/Quantum_electronics" title="Quantum electronics" class="mw-redirect" >quantum electronics</a>, <a href="https://en.wikipedia.org/wiki/Quantum_optics" title="Quantum optics" >quantum optics</a>, and <a href="https://en.wikipedia.org/wiki/Quantum_information_science" title="Quantum information science" >quantum information science</a>. Its speculative modern developments include <a href="https://en.wikipedia.org/wiki/String_theory" title="String theory" >string theory</a> and <a href="https://en.wikipedia.org/wiki/Quantum_gravity" title="Quantum gravity" >quantum gravity</a> theories. It also provides a useful framework for many features of the modern <a href="https://en.wikipedia.org/wiki/Periodic_table" title="Periodic table" >periodic table of elements</a>, and describes the behaviors of <a href="https://en.wikipedia.org/wiki/Atoms" title="Atoms" class="mw-redirect" >atoms</a>during <a href="https://en.wikipedia.org/wiki/Chemical_bond" title="Chemical bond" >chemical bonding</a> and the flow of <a href="https://en.wikipedia.org/wiki/Electron" title="Electron" >electrons</a> in <a href="https://en.wikipedia.org/wiki/Computer" title="Computer" >computer</a> <a href="https://en.wikipedia.org/wiki/Semiconductor" title="Semiconductor" >semiconductors</a>, and therefore plays a crucial role in many modern technologies.

While quantum mechanics traditionally described the world of the very small, it is also needed to explain certain recently investigated<a href="https://en.wikipedia.org/wiki/Macroscopic" title="Macroscopic" class="mw-redirect" >macroscopic</a> systems such as <a href="https://en.wikipedia.org/wiki/Superconductivity" title="Superconductivity" >superconductors</a>, <a href="https://en.wikipedia.org/wiki/Superfluid" title="Superfluid" class="mw-redirect" >superfluids</a>, and large organic molecules.<sup id="cite_ref-10" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-10" >[10]</a></sup>

The word <i>quantum</i> derives from the <a href="https://en.wikipedia.org/wiki/Latin_language" title="Latin language" class="mw-redirect" >Latin</a>, meaning "how great" or "how much".<sup id="cite_ref-11" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-11" >[11]</a></sup> In quantum mechanics, it refers to a discrete unit assigned to certain <a href="https://en.wikipedia.org/wiki/Physical_quantity" title="Physical quantity" >physical quantities</a> such as the <a href="https://en.wikipedia.org/wiki/Energy" title="Energy" >energy</a> of an <a href="https://en.wikipedia.org/wiki/Atom" title="Atom" >atom</a> at rest (see Figure 1). The discovery that particles are discrete packets of energy with wave-like properties led to the branch of physics dealing with atomic and subatomic systems which is today called quantum mechanics. It underlies the <a href="https://en.wikipedia.org/wiki/Mathematical" title="Mathematical" class="mw-redirect" >mathematical</a> framework of many fields of <a href="https://en.wikipedia.org/wiki/Physics" title="Physics" >physics</a> and <a href="https://en.wikipedia.org/wiki/Chemistry" title="Chemistry" >chemistry</a>, including <a href="https://en.wikipedia.org/wiki/Condensed_matter_physics" title="Condensed matter physics" >condensed matter physics</a>, <a href="https://en.wikipedia.org/wiki/Solid-state_physics" title="Solid-state physics" >solid-state physics</a>, <a href="https://en.wikipedia.org/wiki/Atomic_physics" title="Atomic physics" >atomic physics</a>, <a href="https://en.wikipedia.org/wiki/Molecular_physics" title="Molecular physics" >molecular physics</a>, <a href="https://en.wikipedia.org/wiki/Computational_physics" title="Computational physics" >computational physics</a>, <a href="https://en.wikipedia.org/wiki/Computational_chemistry" title="Computational chemistry" >computational chemistry</a>, <a href="https://en.wikipedia.org/wiki/Quantum_chemistry" title="Quantum chemistry" >quantum chemistry</a>, <a href="https://en.wikipedia.org/wiki/Particle_physics" title="Particle physics" >particle physics</a>, <a href="https://en.wikipedia.org/wiki/Nuclear_chemistry" title="Nuclear chemistry" >nuclear chemistry</a>, and <a href="https://en.wikipedia.org/wiki/Nuclear_physics" title="Nuclear physics" >nuclear physics</a>.<sup id="cite_ref-12" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-12" >[12]</a></sup> Some fundamental aspects of the theory are still actively studied.<sup id="cite_ref-13" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-13" >[13]</a></sup>

Quantum mechanics is essential to understanding the behavior of systems at <a href="https://en.wikipedia.org/wiki/Atom" title="Atom" >atomic</a> length scales and smaller. If the physical nature of an atom was solely described by <a href="https://en.wikipedia.org/wiki/Classical_mechanics" title="Classical mechanics" >classical mechanics</a>, electrons would not <i>orbit</i> the nucleus, since orbiting electrons emit radiation (due to <a href="https://en.wikipedia.org/wiki/Circular_motion" title="Circular motion" >circular motion</a>) and would eventually collide with the nucleus due to this loss of energy. This framework was unable to explain the stability of atoms. Instead, electrons remain in an uncertain, non-deterministic, <i>smeared</i>, <a href="https://en.wikipedia.org/wiki/Probability" title="Probability" >probabilistic</a> wave–particle <a href="https://en.wikipedia.org/wiki/Atomic_orbital" title="Atomic orbital" >orbital</a> about the nucleus, defying the traditional assumptions of classical mechanics and <a href="https://en.wikipedia.org/wiki/Electromagnetism" title="Electromagnetism" >electromagnetism</a>.<sup id="cite_ref-14" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-14" >[14]</a></sup>

Quantum mechanics was initially developed to provide a better explanation and description of the atom, especially the differences in the <a href="https://en.wikipedia.org/wiki/Spectrum" title="Spectrum" >spectra</a> of light emitted by different <a href="https://en.wikipedia.org/wiki/Isotope" title="Isotope" >isotopes</a> of the same <a href="https://en.wikipedia.org/wiki/Chemical_element" title="Chemical element" >chemical element</a>, as well as subatomic particles. In short, the quantum-mechanical atomic model has succeeded spectacularly in the realm where classical mechanics and electromagnetism falter.

Broadly speaking, quantum mechanics incorporates four classes of phenomena for which classical physics cannot account:</p>

<ul >

<li ><a href="https://en.wikipedia.org/wiki/Quantization_(physics)" title="Quantization (physics)" >quantization</a> of <a href="https://en.wikipedia.org/wiki/Canonical_conjugate_variables" title="Canonical conjugate variables" class="mw-redirect" >certain physical properties</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality" >wave–particle duality</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Uncertainty_principle" title="Uncertainty principle" >principle of uncertainty</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Quantum_entanglement" title="Quantum entanglement" >quantum entanglement</a></li>

</ul>

<h2 >Mathematical formulations[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=2" title="Edit section: Mathematical formulations" >edit</a>]</h2>

<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/Mathematical_formulations_of_quantum_mechanics" title="Mathematical formulations of quantum mechanics" class="mw-redirect" >Mathematical formulations of quantum mechanics</a></div>

<div class="hatnote" >See also: <a href="https://en.wikipedia.org/wiki/Quantum_logic" title="Quantum logic" >Quantum logic</a></div>

<p >In the mathematically rigorous formulation of quantum mechanics developed by <a href="https://en.wikipedia.org/wiki/Paul_Dirac" title="Paul Dirac" >Paul Dirac</a>,<sup id="cite_ref-15" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-15" >[15]</a></sup> <a href="https://en.wikipedia.org/wiki/David_Hilbert" title="David Hilbert" >David Hilbert</a>,<sup id="cite_ref-16" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-16" >[16]</a></sup> <a href="https://en.wikipedia.org/wiki/John_von_Neumann" title="John von Neumann" >John von Neumann</a>,<sup id="cite_ref-17" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-17" >[17]</a></sup> and <a href="https://en.wikipedia.org/wiki/Hermann_Weyl" title="Hermann Weyl" >Hermann Weyl</a>,<sup id="cite_ref-18" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-18" >[18]</a></sup> the possible states of a quantum mechanical system are represented by <a href="https://en.wikipedia.org/wiki/Unit_vector" title="Unit vector" >unit vectors</a> (called<i>state vectors</i>). Formally, these reside in a <a href="https://en.wikipedia.org/wiki/Complex_number" title="Complex number" >complex</a> <a href="https://en.wikipedia.org/wiki/Separable_space" title="Separable space" >separable</a> <a href="https://en.wikipedia.org/wiki/Hilbert_space" title="Hilbert space" >Hilbert space</a>—variously called the <a href="https://en.wikipedia.org/wiki/State_space(physics)" title="State space (physics)" ><i>state space</i></a> or the <i>associated Hilbert space</i> of the system—that is well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the <a href="https://en.wikipedia.org/wiki/Projective_space" title="Projective space" >projective space</a> of a Hilbert space, usually called the <a href="https://en.wikipedia.org/wiki/Complex_projective_space" title="Complex projective space" >complex projective space</a>. The exact nature of this Hilbert space is dependent on the system—for example, the state space for position and momentum states is the space of <a href="https://en.wikipedia.org/wiki/Square-integrable" title="Square-integrable" class="mw-redirect" >square-integrable</a>functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally <a href="https://en.wikipedia.org/wiki/Hermitian_adjoint" title="Hermitian adjoint" >Hermitian</a> (precisely: by a <a href="https://en.wikipedia.org/wiki/Self-adjoint_operator" title="Self-adjoint operator" >self-adjoint</a>) linear <a href="https://en.wikipedia.org/wiki/Operator(physics)" title="Operator (physics)" >operator</a> acting on the state space. Each <a href="https://en.wikipedia.org/wiki/Eigenstate" title="Eigenstate" class="mw-redirect" >eigenstate</a> of an observable corresponds to an <a href="https://en.wikipedia.org/wiki/Eigenvector" title="Eigenvector" class="mw-redirect" >eigenvector</a> of the operator, and the associated <a href="https://en.wikipedia.org/wiki/Eigenvalue" title="Eigenvalue" class="mw-redirect" >eigenvalue</a> corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can attain only those discrete eigenvalues.

In the formalism of quantum mechanics, the state of a system at a given time is described by a <a href="https://en.wikipedia.org/wiki/Complex_number" title="Complex number" >complex</a> <a href="https://en.wikipedia.org/wiki/Wave_function" title="Wave function" >wave function</a>, also referred to as state vector in a complex <a href="https://en.wikipedia.org/wiki/Vector_space" title="Vector space" >vector space</a>.<sup id="cite_ref-19" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-19" >[19]</a></sup> This abstract mathematical object allows for the calculation of <a href="https://en.wikipedia.org/wiki/Probability" title="Probability" >probabilities</a> of outcomes of concrete experiments. For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of <a href="https://en.wikipedia.org/wiki/Conjugate_variables" title="Conjugate variables" >conjugate variables</a>, such as position and momentum, with accuracy. For instance, electrons may be considered (to a certain probability) to be located somewhere within a given region of space, but with their exact positions unknown. Contours of constant probability, often referred to as "clouds", may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. Heisenberg's <a href="https://en.wikipedia.org/wiki/Uncertainty_principle" title="Uncertainty principle" >uncertainty principle</a> quantifies the inability to precisely locate the particle given its conjugate momentum.<sup id="cite_ref-20" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-20" >[20]</a></sup>

According to one interpretation, as the result of a measurement the wave function containing the probability information for a system collapses from a given initial state to a particular eigenstate. The possible results of a measurement are the eigenvalues of the operator representing the observable—which explains the choice of <i>Hermitian</i> operators, for which all the eigenvalues are real. The probability distribution of an observable in a given state can be found by computing the <a href="https://en.wikipedia.org/wiki/Spectral_theorem" title="Spectral theorem" >spectral decomposition</a> of the corresponding operator. Heisenberg's <a href="https://en.wikipedia.org/wiki/Uncertainty_principle" title="Uncertainty principle" >uncertainty principle</a> is represented by the statement that the operators corresponding to certain observables do not <a href="https://en.wikipedia.org/wiki/Commutator" title="Commutator" >commute</a>.

The <a href="https://en.wikipedia.org/wiki/Probability" title="Probability" >probabilistic</a> nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous <a href="https://en.wikipedia.org/wiki/Bohr-Einstein_debates" title="Bohr-Einstein debates" class="mw-redirect" >Bohr-Einstein debates</a>, in which the two scientists attempted to clarify these fundamental principles by way of <a href="https://en.wikipedia.org/wiki/Thought_experiment" title="Thought experiment" >thought experiments</a>. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer <a href="https://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics" title="Interpretation of quantum mechanics" class="mw-redirect" >interpretations of quantum mechanics</a> have been formulated that do away with the concept of "wavefunction collapse" (see, for example, the <a href="https://en.wikipedia.org/wiki/Relative_state_interpretation" title="Relative state interpretation" class="mw-redirect" >relative state interpretation</a>). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become <a href="https://en.wikipedia.org/wiki/Quantum_Entanglement" title="Quantum Entanglement" class="mw-redirect" >entangled</a>, so that the original quantum system ceases to exist as an independent entity. For details, see the article on <a href="https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics" >measurement in quantum mechanics</a>.<sup id="cite_ref-google215_21-0" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-google215-21" >[21]</a></sup>

Generally, quantum mechanics does not assign definite values. Instead, it makes a prediction using a <a href="https://en.wikipedia.org/wiki/Probability_distribution" title="Probability distribution" >probability distribution</a>; that is, it describes the probability of obtaining the possible outcomes from measuring an observable. Often these results are skewed by many causes, such as dense probability clouds. Probability clouds are approximate (but better than <a href="https://en.wikipedia.org/wiki/Bohr_model" title="Bohr model" >the Bohr model</a>) whereby electron location is given by a <a href="https://en.wikipedia.org/wiki/Probability_function" title="Probability function" class="mw-redirect" >probability function</a>, the <a href="https://en.wikipedia.org/wiki/Wave_function" title="Wave function" >wave function</a> <a href="https://en.wikipedia.org/wiki/Eigenvalue" title="Eigenvalue" class="mw-redirect" >eigenvalue</a>, such that the probability is the squared modulus of the<a href="https://en.wikipedia.org/wiki/Probability_amplitude" title="Probability amplitude" >complex amplitude</a>, or quantum state nuclear attraction.<sup id="cite_ref-22" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-22" >[22]</a></sup><sup id="cite_ref-23" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-23" >[23]</a></sup> Naturally, these probabilities will depend on the quantum state at the "instant" of the measurement. Hence, uncertainty is involved in the value. There are, however, certain states that are associated with a definite value of a particular observable. These are known as <a href="https://en.wikipedia.org/wiki/Eigenstate" title="Eigenstate" class="mw-redirect" >eigenstates</a> of the observable ("eigen" can be translated from<a href="https://en.wikipedia.org/wiki/German_language" title="German language" >German</a> as meaning "inherent" or "characteristic").<sup id="cite_ref-24" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-24" >[24]</a></sup>

In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate. Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum (since they are <a href="https://en.wikipedia.org/wiki/Conjugate_variables" title="Conjugate variables" >conjugate pairs</a>) or its energy and time (since they too are conjugate pairs); rather, it provides only a range of probabilities in which that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having <i><a href="https://en.wikipedia.org/wiki/Uncertainty_principle" title="Uncertainty principle" >uncertain</a></i> values and states having <i>definite</i> values (eigenstates). Usually, a system will not be in an <a href="https://en.wikipedia.org/wiki/Eigenstate" title="Eigenstate" class="mw-redirect" >eigenstate</a> of the observable (particle) we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or "generalized" eigenstate) of that observable. This process is known as <a href="https://en.wikipedia.org/wiki/Wavefunction_collapse" title="Wavefunction collapse" class="mw-redirect" >wavefunction collapse</a>, a controversial and much-debated process<sup id="cite_ref-25" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-25" >[25]</a></sup> that involves expanding the system under study to include the measurement device. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of the wavefunction collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a <a href="https://en.wikipedia.org/wiki/Wave_packet" title="Wave packet" >wave packet</a> centered around some mean position <i>x</i><sub >0</sub> (neither an eigenstate of position nor of momentum). When one measures the position of the particle, it is impossible to predict with certainty the result.<sup id="cite_ref-google215_21-1" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-google215-21" >[21]</a></sup> It is probable, but not certain, that it will be near <i>x</i><sub >0</sub>, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result <i>x</i>, the wave function collapses into a position eigenstate centered at <i>x</i>.<sup id="cite_ref-26" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-26" >[26]</a></sup>

The time evolution of a quantum state is described by the <a href="https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation" >Schrödinger equation</a>, in which the <a href="https://en.wikipedia.org/wiki/Hamiltonian(quantum_mechanics)" title="Hamiltonian (quantum mechanics)" >Hamiltonian</a> (the <a href="https://en.wikipedia.org/wiki/Operator(physics)" title="Operator (physics)" >operator</a>corresponding to the <a href="https://en.wikipedia.org/wiki/Total_energy" title="Total energy" class="mw-redirect" >total energy</a> of the system) generates the time evolution. The <a href="https://en.wikipedia.org/wiki/Time_evolution" title="Time evolution" >time evolution</a> of wave functions is <a href="https://en.wikipedia.org/wiki/Determinism" title="Determinism" >deterministic</a> in the sense that - given a wavefunction at an <i>initial</i> time - it makes a definite prediction of what the wavefunction will be at any <i>later</i>time.<sup id="cite_ref-27" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-27" >[27]</a></sup>

During a <a href="https://en.wikipedia.org/wiki/Quantum_measurement" title="Quantum measurement" class="mw-redirect" >measurement</a>, on the other hand, the change of the initial wavefunction into another, later wavefunction is not deterministic, it is unpredictable (i.e., <a href="https://en.wikipedia.org/wiki/Random" title="Random" class="mw-redirect" >random</a>). A time-evolution simulation can be seen here.<sup id="cite_ref-28" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-28" >[28]</a></sup><sup id="cite_ref-29" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-29" >[29]</a></sup>

Wave functions change as time progresses. The <a href="https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation" >Schrödinger equation</a> describes how wavefunctions change in time, playing a role similar to <a href="https://en.wikipedia.org/wiki/Newton%27s_second_law" title="Newton's second law" class="mw-redirect" >Newton's second law</a> in <a href="https://en.wikipedia.org/wiki/Classical_mechanics" title="Classical mechanics" >classical mechanics</a>. The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain with time. This also has the effect of turning a position eigenstate (which can be thought of as an infinitely sharp wave packet) into a broadened wave packet that no longer represents a (definite, certain) position eigenstate.<sup id="cite_ref-30" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-30" >[30]</a></sup></p>

Fig. 1: <a href="https://en.wikipedia.org/wiki/Probability_density_function" title="Probability density function" >Probability densities</a> corresponding to the wavefunctions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: <i>n</i> = 1, 2, 3, ...) and angular momenta (increasing across from left to right: <i>s</i>, <i>p</i>, <i>d</i>, ...). Brighter areas correspond to higher probability density in a position measurement. Such wavefunctions are directly comparable to<a href="https://en.wikipedia.org/wiki/Chladni%27s_figures" title="Chladni's figures" class="mw-redirect" >Chladni's figures</a> of <a href="https://en.wikipedia.org/wiki/Acoustics" title="Acoustics" >acoustic</a> modes of vibration in<a href="https://en.wikipedia.org/wiki/Classical_physics" title="Classical physics" >classical physics</a>, and are modes of oscillation as well, possessing a sharp <a href="https://en.wikipedia.org/wiki/Energy" title="Energy" >energy</a> and, thus, a definite <a href="https://en.wikipedia.org/wiki/Frequency" title="Frequency" >frequency</a>. The <a href="https://en.wikipedia.org/wiki/Angular_momentum" title="Angular momentum" >angular momentum</a> and energy are <a href="https://en.wikipedia.org/wiki/Quantization_(physics)" title="Quantization (physics)" >quantized</a>, and take <b>only</b> discrete values like those shown (as is the case for<a href="https://en.wikipedia.org/wiki/Resonant_frequency" title="Resonant frequency" class="mw-redirect" >resonant frequencies</a> in acoustics)</div>

</div>

<p >Some wave functions produce probability distributions that are constant, or independent of time—such as when in a <a href="https://en.wikipedia.org/wiki/Eigenstate#Schr.C3.B6dinger_equation" title="Eigenstate" class="mw-redirect" >stationary state</a> of constant energy, time vanishes in the absolute square of the wave function. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single <a href="https://en.wikipedia.org/wiki/Electron" title="Electron" >electron</a> in an unexcited <a href="https://en.wikipedia.org/wiki/Atom" title="Atom" >atom</a> is pictured classically as a particle moving in a circular trajectory around the <a href="https://en.wikipedia.org/wiki/Atomic_nucleus" title="Atomic nucleus" >atomic nucleus</a>, whereas in quantum mechanics it is described by a static, <a href="https://en.wikipedia.org/wiki/Spherical_coordinate_system" title="Spherical coordinate system" >spherically symmetric</a> wavefunction surrounding the nucleus (<a href="https://en.wikipedia.org/wiki/File:HAtomOrbitals.png" title="File:HAtomOrbitals.png" >Fig. 1</a>) (note, however, that only the lowest angular momentum states, labeled <i>s</i>, are spherically symmetric).<sup id="cite_ref-31" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-31" >[31]</a></sup>

The Schrödinger equation acts on the <i>entire</i> probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its <a href="https://en.wikipedia.org/wiki/Phase(waves)" title="Phase (waves)" >phase</a> encodes information about the <a href="https://en.wikipedia.org/wiki/Interference(wave_propagation)" title="Interference (wave propagation)" >interference</a>between quantum states. This gives rise to the "wave-like" behavior of quantum states. As it turns out, analytic solutions of the Schrödinger equation are available for only <a href="https://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions" title="List of quantum-mechanical systems with analytical solutions" >a very small number of relatively simple model Hamiltonians</a>, of which the<a href="https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator" >quantum harmonic oscillator</a>, the <a href="https://en.wikipedia.org/wiki/Particle_in_a_box" title="Particle in a box" >particle in a box</a>, the <a href="https://en.wikipedia.org/wiki/Hydrogen_molecular_ion" title="Hydrogen molecular ion" class="mw-redirect" >hydrogen molecular ion</a>, and the <a href="https://en.wikipedia.org/wiki/Hydrogen_atom" title="Hydrogen atom" >hydrogen atom</a> are the most important representatives. Even the <a href="https://en.wikipedia.org/wiki/Helium" title="Helium" >helium</a> atom—which contains just one more electron than does the hydrogen atom—has defied all attempts at a fully analytic treatment.

There exist several techniques for generating approximate solutions, however. In the important method known as <a href="https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)" >perturbation theory</a>, one uses the analytic result for a simple quantum mechanical model to generate a result for a more complicated model that is related to the simpler model by (for one example) the addition of a weak <a href="https://en.wikipedia.org/wiki/Potential_energy" title="Potential energy" >potential energy</a>. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces only weak (small) deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of <a href="https://en.wikipedia.org/wiki/Quantum_chaos" title="Quantum chaos" >quantum chaos</a>.</p>

<h2 >Mathematically equivalent formulations of quantum mechanics[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=3" title="Edit section: Mathematically equivalent formulations of quantum mechanics" >edit</a>]</h2>

<p >There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the "<a href="https://en.wikipedia.org/wiki/Transformation_theory_(quantum_mechanics)" title="Transformation theory (quantum mechanics)" >transformation theory</a>" proposed by <a href="https://en.wikipedia.org/wiki/Paul_Dirac" title="Paul Dirac" >Paul Dirac</a>, which unifies and generalizes the two earliest formulations of quantum mechanics - <a href="https://en.wikipedia.org/wiki/Matrix_mechanics" title="Matrix mechanics" >matrix mechanics</a> (invented by <a href="https://en.wikipedia.org/wiki/Werner_Heisenberg" title="Werner Heisenberg" >Werner Heisenberg</a>) and <a href="https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation" >wave mechanics</a> (invented by <a href="https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger" >Erwin Schrödinger</a>).<sup id="cite_ref-32" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-32" >[32]</a></sup>

Especially since <a href="https://en.wikipedia.org/wiki/Werner_Heisenberg" title="Werner Heisenberg" >Werner Heisenberg</a> was awarded the <a href="https://en.wikipedia.org/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics" >Nobel Prize in Physics</a> in 1932 for the creation of quantum mechanics, the role of <a href="https://en.wikipedia.org/wiki/Max_Born" title="Max Born" >Max Born</a> in the development of QM was overlooked until the 1954 Nobel award. The role is noted in a 2005 biography of Born, which recounts his role in the matrix formulation of quantum mechanics, and the use of probability amplitudes. Heisenberg himself acknowledges having learned matrices from Born, as published in a 1940 <i><a href="https://en.wikipedia.org/wiki/Festschrift" title="Festschrift" >festschrift</a></i> honoring <a href="https://en.wikipedia.org/wiki/Max_Planck" title="Max Planck" >Max Planck</a>.<sup id="cite_ref-33" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-33" >[33]</a></sup> In the matrix formulation, the <a href="https://en.wikipedia.org/wiki/Quantum_state" title="Quantum state" >instantaneous state of a quantum system</a> encodes the probabilities of its measurable properties, or "<a href="https://en.wikipedia.org/wiki/Observable" title="Observable" >observables</a>". Examples of observables include <a href="https://en.wikipedia.org/wiki/Energy" title="Energy" >energy</a>, <a href="https://en.wikipedia.org/wiki/Position_operator" title="Position operator" >position</a>, <a href="https://en.wikipedia.org/wiki/Momentum_operator" title="Momentum operator" >momentum</a>, and <a href="https://en.wikipedia.org/wiki/Angular_momentum" title="Angular momentum" >angular momentum</a>. Observables can be either <a href="https://en.wikipedia.org/wiki/Continuous_function" title="Continuous function" >continuous</a> (e.g., the position of a particle) or <a href="https://en.wikipedia.org/wiki/Discrete_mathematics" title="Discrete mathematics" >discrete</a> (e.g., the energy of an electron bound to a hydrogen atom).<sup id="cite_ref-34" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-34" >[34]</a></sup> An alternative formulation of quantum mechanics is <a href="https://en.wikipedia.org/wiki/Feynman" title="Feynman" class="mw-redirect" >Feynman</a>'s <a href="https://en.wikipedia.org/wiki/Path_integral_formulation" title="Path integral formulation" >path integral formulation</a>, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the<a href="https://en.wikipedia.org/wiki/Action_principle" title="Action principle" class="mw-redirect" >action principle</a> in classical mechanics.</p>

<h2 >Interactions with other scientific theories[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=4" title="Edit section: Interactions with other scientific theories" >edit</a>]</h2>

<p >The rules of quantum mechanics are fundamental. They assert that the state space of a system is a <a href="https://en.wikipedia.org/wiki/Hilbert_space" title="Hilbert space" >Hilbert space</a> and that observables of that system are <a href="https://en.wikipedia.org/wiki/Hermitian_operators" title="Hermitian operators" class="mw-redirect" >Hermitian operators</a> acting on that space—although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the <a href="https://en.wikipedia.org/wiki/Correspondence_principle" title="Correspondence principle" >correspondence principle</a>, which states that the predictions of quantum mechanics reduce to those of classical mechanics when a system moves to higher energies or, equivalently, larger quantum numbers, i.e. whereas a single particle exhibits a degree of randomness, in systems incorporating millions of particles averaging takes over and, at the high energy limit, the statistical probability of random behaviour approaches zero. In other words, classical mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the <i>classical</i> or <i>correspondence limit</i>. One can even start from an established classical model of a particular system, then attempt to guess the underlying quantum model that would give rise to the classical model in the correspondence limit.</p>

<caption ><a href="https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics" >List of unsolved problems in physics</a></caption>

<tbody>

<tr>

<td >

<div ><i>In the <a href="https://en.wikipedia.org/wiki/Correspondence_limit" title="Correspondence limit" class="mw-redirect" >correspondence limit</a> of<b>quantum mechanics</b>: Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "<a href="https://en.wikipedia.org/wiki/Superposition_principle" title="Superposition principle" >superposition</a>of states" and "<a href="https://en.wikipedia.org/wiki/Wavefunction_collapse" title="Wavefunction collapse" class="mw-redirect" >wavefunction collapse</a>", give rise to the reality we perceive?</i></div>

</td>

</tr>

</tbody></table>

<p >When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was <a href="https://en.wikipedia.org/wiki/Theory_of_relativity" title="Theory of relativity" >non-relativistic</a> <a href="https://en.wikipedia.org/wiki/Classical_mechanics" title="Classical mechanics" >classical mechanics</a>. For instance, the well-known model of the <a href="https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator" >quantum harmonic oscillator</a> uses an explicitly non-relativistic expression for the <a href="https://en.wikipedia.org/wiki/Kinetic_energy" title="Kinetic energy" >kinetic energy</a> of the oscillator, and is thus a quantum version of the<a href="https://en.wikipedia.org/wiki/Harmonic_oscillator" title="Harmonic oscillator" >classical harmonic oscillator</a>.

Early attempts to merge quantum mechanics with <a href="https://en.wikipedia.org/wiki/Special_relativity" title="Special relativity" >special relativity</a> involved the replacement of the Schrödinger equation with a covariant equation such as the <a href="https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation" >Klein–Gordon equation</a> or the <a href="https://en.wikipedia.org/wiki/Dirac_equation" title="Dirac equation" >Dirac equation</a>. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of <a href="https://en.wikipedia.org/wiki/Quantum_field_theory" title="Quantum field theory" >quantum field theory</a>, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, <a href="https://en.wikipedia.org/wiki/Quantum_electrodynamics" title="Quantum electrodynamics" >quantum electrodynamics</a>, provides a fully quantum description of the<a href="https://en.wikipedia.org/wiki/Electromagnetism" title="Electromagnetism" >electromagnetic interaction</a>. The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been employed since the inception of quantum mechanics, is to treat <a href="https://en.wikipedia.org/wiki/Electric_charge" title="Electric charge" >charged</a> particles as quantum mechanical objects being acted on by a classical <a href="https://en.wikipedia.org/wiki/Electromagnetic_field" title="Electromagnetic field" >electromagnetic field</a>. For example, the elementary quantum model of the <a href="https://en.wikipedia.org/wiki/Hydrogen_atom" title="Hydrogen atom" >hydrogen atom</a> describes the <a href="https://en.wikipedia.org/wiki/Electric_field" title="Electric field" >electric field</a> of the hydrogen atom using a classical <img src="/194714a8-edf2-5f9c-a019-a0c16a22a84d.png"/> <a href="https://en.wikipedia.org/wiki/Electric_potential" title="Electric potential" >Coulomb potential</a>. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of <a href="https://en.wikipedia.org/wiki/Photon" title="Photon" >photons</a> by<a href="https://en.wikipedia.org/wiki/Charged_particle" title="Charged particle" >charged particles</a>.

<a href="https://en.wikipedia.org/wiki/Field_(physics)" title="Field (physics)" >Quantum field</a> theories for the <a href="https://en.wikipedia.org/wiki/Strong_nuclear_force" title="Strong nuclear force" class="mw-redirect" >strong nuclear force</a> and the <a href="https://en.wikipedia.org/wiki/Weak_nuclear_force" title="Weak nuclear force" class="mw-redirect" >weak nuclear force</a> have also been developed. The quantum field theory of the strong nuclear force is called <a href="https://en.wikipedia.org/wiki/Quantum_chromodynamics" title="Quantum chromodynamics" >quantum chromodynamics</a>, and describes the interactions of subnuclear particles such as <a href="https://en.wikipedia.org/wiki/Quark" title="Quark" >quarks</a>and <a href="https://en.wikipedia.org/wiki/Gluon" title="Gluon" >gluons</a>. The weak nuclear force and the <a href="https://en.wikipedia.org/wiki/Electromagnetic_force" title="Electromagnetic force" class="mw-redirect" >electromagnetic force</a> were unified, in their quantized forms, into a single quantum field theory (known as <a href="https://en.wikipedia.org/wiki/Electroweak_theory" title="Electroweak theory" class="mw-redirect" >electroweak theory</a>), by the physicists <a href="https://en.wikipedia.org/wiki/Abdus_Salam" title="Abdus Salam" >Abdus Salam</a>, <a href="https://en.wikipedia.org/wiki/Sheldon_Glashow" title="Sheldon Glashow" class="mw-redirect" >Sheldon Glashow</a> and <a href="https://en.wikipedia.org/wiki/Steven_Weinberg" title="Steven Weinberg" >Steven Weinberg</a>. These three men shared the Nobel Prize in Physics in 1979 for this work.<sup id="cite_ref-35" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-35" >[35]</a></sup>

It has proven difficult to construct quantum models of <a href="https://en.wikipedia.org/wiki/Gravity" title="Gravity" >gravity</a>, the remaining <a href="https://en.wikipedia.org/wiki/Fundamental_force" title="Fundamental force" class="mw-redirect" >fundamental force</a>. Semi-classical approximations are workable, and have led to predictions such as <a href="https://en.wikipedia.org/wiki/Hawking_radiation" title="Hawking radiation" >Hawking radiation</a>. However, the formulation of a complete theory of <a href="https://en.wikipedia.org/wiki/Quantum_gravity" title="Quantum gravity" >quantum gravity</a> is hindered by apparent incompatibilities between <a href="https://en.wikipedia.org/wiki/General_relativity" title="General relativity" >general relativity</a> (the most accurate theory of gravity currently known) and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as <a href="https://en.wikipedia.org/wiki/String_theory" title="String theory" >string theory</a> are among the possible candidates for a future theory of quantum gravity.

Classical mechanics has also been extended into the <a href="https://en.wikipedia.org/wiki/Complex_domain" title="Complex domain" class="mw-redirect" >complex domain</a>, with complex classical mechanics exhibiting behaviors similar to quantum mechanics.<sup id="cite_ref-36" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-36" >[36]</a></sup></p>

<h3 >Quantum mechanics and classical physics[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=5" title="Edit section: Quantum mechanics and classical physics" >edit</a>]</h3>

<p >Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy.<sup id="cite_ref-37" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-37" >[37]</a></sup> According to the<a href="https://en.wikipedia.org/wiki/Correspondence_principle" title="Correspondence principle" >correspondence principle</a> between classical and quantum mechanics, all objects obey the laws of quantum mechanics, and classical mechanics is just an approximation for large systems of objects (or a statistical quantum mechanics of a large collection of particles).<sup id="cite_ref-Tipler_38-0" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Tipler-38" >[38]</a></sup> The laws of classical mechanics thus follow from the laws of quantum mechanics as a statistical average at the limit of large systems or large <a href="https://en.wikipedia.org/wiki/Quantum_number" title="Quantum number" >quantum numbers</a>.<sup id="cite_ref-39" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-39" >[39]</a></sup> However, <a href="https://en.wikipedia.org/wiki/Chaos_theory" title="Chaos theory" >chaotic systems</a> do not have good quantum numbers, and <a href="https://en.wikipedia.org/wiki/Quantum_chaos" title="Quantum chaos" >quantum chaos</a>studies the relationship between classical and quantum descriptions in these systems.

<a href="https://en.wikipedia.org/wiki/Quantum_coherence" title="Quantum coherence" class="mw-redirect" >Quantum coherence</a> is an essential difference between classical and quantum theories as illustrated by the <a href="https://en.wikipedia.org/wiki/EPR_paradox" title="EPR paradox" >Einstein–Podolsky–Rosen (EPR) paradox</a> — an attack on a certain philosophical interpretation of quantum mechanics by an appeal to <a href="https://en.wikipedia.org/wiki/Local_realism" title="Local realism" class="mw-redirect" >local realism</a>.<sup id="cite_ref-40" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-40" >[40]</a></sup><a href="https://en.wikipedia.org/wiki/Quantum_interference#Quantum_interference" title="Quantum interference" class="mw-redirect" >Quantum interference</a> involves adding together <i><a href="https://en.wikipedia.org/wiki/Probability_amplitude" title="Probability amplitude" >probability amplitudes</a></i>, whereas classical "waves" infer that there is an adding together of <i>intensities</i>. For microscopic bodies, the extension of the system is much smaller than the <a href="https://en.wikipedia.org/wiki/Coherence_length" title="Coherence length" >coherence length</a>, which gives rise to long-range entanglement and other nonlocal phenomena characteristic of quantum systems.<sup id="cite_ref-41" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-41" >[41]</a></sup> Quantum coherence is not typically evident at macroscopic scales, though an exception to this rule may occur at extremely low temperatures (i.e. approaching<a href="https://en.wikipedia.org/wiki/Absolute_zero" title="Absolute zero" >absolute zero</a>) at which quantum behavior may manifest itself macroscopically.<sup id="cite_ref-42" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-42" >[42]</a></sup> This is in accordance with the following observations:</p>

<ul >

<li >Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and <a href="https://en.wikipedia.org/wiki/Molecule" title="Molecule" >molecules</a> which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of <a href="https://en.wikipedia.org/wiki/Electric_charge" title="Electric charge" >electric charges</a> under the rules of quantum mechanics.<sup id="cite_ref-43" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-43" >[43]</a></sup></li>

<li >While the seemingly "exotic" behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with particles of extremely small size or velocities approaching the <a href="https://en.wikipedia.org/wiki/Speed_of_light" title="Speed of light" >speed of light</a>, the laws of classical, often considered "<a href="https://en.wikipedia.org/wiki/Isaac_Newton" title="Isaac Newton" >Newtonian</a>", physics remain accurate in predicting the behavior of the vast majority of "large" objects (on the order of the size of large molecules or bigger) at velocities much smaller than the <a href="https://en.wikipedia.org/wiki/Speed_of_light" title="Speed of light" >velocity of light</a>.<sup id="cite_ref-44" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-44" >[44]</a></sup></li>

</ul>

<h3 >Copenhagen interpretation of quantum versus classical kinematics[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=6" title="Edit section: Copenhagen interpretation of quantum versus classical kinematics" >edit</a>]</h3>

<p >A big difference between classical and quantum mechanics is that they use very different kinematic descriptions.<sup id="cite_ref-45" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-45" >[45]</a></sup>

In <a href="https://en.wikipedia.org/wiki/Niels_Bohr" title="Niels Bohr" >Niels Bohr</a>'s mature view, quantum mechanical phenomena are required to be experiments, with complete descriptions of all the devices for the system, preparative, intermediary, and finally measuring. The descriptions are in macroscopic terms, expressed in ordinary language, supplemented with the concepts of classical mechanics.<sup id="cite_ref-46" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-46" >[46]</a></sup><sup id="cite_ref-47" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-47" >[47]</a></sup><sup id="cite_ref-48" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-48" >[48]</a></sup><sup id="cite_ref-49" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-49" >[49]</a></sup> The initial condition and the final condition of the system are respectively described by values in a configuration space, for example a position space, or some equivalent space such as a momentum space. Quantum mechanics does not admit a completely precise description, in terms of both position and momentum, of an initial condition or "state" (in the classical sense of the word) that would support a precisely deterministic and causal prediction of a final condition.<sup id="cite_ref-Heisenberg_1927_50-0" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Heisenberg_1927-50" >[50]</a></sup><sup id="cite_ref-51" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-51" >[51]</a></sup> In this sense, advocated by Bohr in his mature writings, a quantum phenomenon is a process, a passage from initial to final condition, not an instantaneous "state" in the classical sense of that word.<sup id="cite_ref-52" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-52" >[52]</a></sup><sup id="cite_ref-53" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-53" >[53]</a></sup> Thus there are two kinds of processes in quantum mechanics: stationary and transitional. For a stationary process, the initial and final condition are the same. For a transition, they are different. Obviously by definition, if only the initial condition is given, the process is not determined.<sup id="cite_ref-Heisenberg_1927_50-1" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Heisenberg_1927-50" >[50]</a></sup> Given its initial condition, prediction of its final condition is possible, causally but only probabilistically, because the Schrödinger equation is deterministic for wave function evolution, but the wave function describes the system only probabilistically.<sup id="cite_ref-54" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-54" >[54]</a></sup><sup id="cite_ref-55" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-55" >[55]</a></sup>

For many experiments, it is possible to think of the initial and final conditions of the system as being a particle. In some cases it appears that there are potentially several spatially distinct pathways or trajectories by which a particle might pass from initial to final condition. It is an important feature of the quantum kinematic description that it does not permit a unique definite statement of which of those pathways is actually followed. Only the initial and final conditions are definite, and, as stated in the foregoing paragraph, they are defined only as precisely as allowed by the configuration space description or its equivalent. In every case for which a quantum kinematic description is needed, there is always a compelling reason for this restriction of kinematic precision. An example of such a reason is that for a particle to be experimentally found in a definite position, it must be held motionless; for it to be experimentally found to have a definite momentum, it must have free motion; these two are logically incompatible.<sup id="cite_ref-56" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-56" >[56]</a></sup><sup id="cite_ref-57" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-57" >[57]</a></sup>

Classical kinematics does not primarily demand experimental description of its phenomena. It allows completely precise description of an instantaneous state by a value in phase space, the Cartesian product of configuration and momentum spaces. This description simply assumes or imagines a state as a physically existing entity without concern about its experimental measurability. Such a description of an initial condition, together with Newton's laws of motion, allows a precise deterministic and causal prediction of a final condition, with a definite trajectory of passage. <a href="https://en.wikipedia.org/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics" >Hamiltonian</a> dynamics can be used for this. Classical kinematics also allows the description of a process analogous to the initial and final condition description used by quantum mechanics. <a href="https://en.wikipedia.org/wiki/Lagrangian_mechanics" title="Lagrangian mechanics" >Lagrangian mechanics</a>applies to this.<sup id="cite_ref-58" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-58" >[58]</a></sup> For processes that need account to be taken of actions of a small number of <a href="https://en.wikipedia.org/wiki/Planck%27s_constant" title="Planck's constant" class="mw-redirect" >Planck constants</a>, classical kinematics is not adequate; quantum mechanics is needed.</p>

<h3 >Relativity and quantum mechanics[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=7" title="Edit section: Relativity and quantum mechanics" >edit</a>]</h3>

<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics" >Relativistic quantum mechanics</a></div>

<p >Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being indisputably supported by rigorous and repeated <a href="https://en.wikipedia.org/wiki/Empirical_evidence" title="Empirical evidence" >empirical evidence</a>, and while they do not directly contradict each other theoretically (at least with regard to their primary claims), they have proven extremely difficult to incorporate into one consistent, cohesive model.<sup id="cite_ref-59" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-59" >[59]</a></sup>

Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly contributing to the field, he did not accept many of the more "philosophical consequences and interpretations" of quantum mechanics, such as the lack of deterministic<a href="https://en.wikipedia.org/wiki/Causality" title="Causality" >causality</a>. He is famously quoted as saying, in response to this aspect, "My God does not play with dice". He also had difficulty with the assertion that a single <a href="https://en.wikipedia.org/wiki/Subatomic_particle" title="Subatomic particle" >subatomic particle</a> can occupy numerous areas of space at one time. However, he was also the first to notice some of the apparently exotic consequences of <a href="https://en.wikipedia.org/wiki/Quantum_entanglement" title="Quantum entanglement" >entanglement</a>, and used them to formulate the <a href="https://en.wikipedia.org/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox" class="mw-redirect" >Einstein–Podolsky–Rosen paradox</a> in the hope of showing that quantum mechanics had unacceptable implications if taken as a complete description of physical reality. This was 1935, but in 1964 it was shown by John Bell (see <a href="https://en.wikipedia.org/wiki/Bell_inequality" title="Bell inequality" class="mw-redirect" >Bell inequality</a>) that - although Einstein was correct in identifying seemingly paradoxical implications of <a href="https://en.wikipedia.org/wiki/Quantum_nonlocality" title="Quantum nonlocality" >quantum mechanical nonlocality</a> - these implications could be experimentally tested. Alain Aspect's initial experiments in 1982, and many subsequent experiments since, have definitively verified quantum entanglement.

According to the paper of J. Bell and the <a href="https://en.wikipedia.org/wiki/Copenhagen_interpretation" title="Copenhagen interpretation" >Copenhagen interpretation</a>—the common interpretation of quantum mechanics by physicists since 1927 - and contrary to Einstein's ideas, quantum mechanics was <i>not</i>, at the same time a "realistic" theory and a "<a href="https://en.wikipedia.org/wiki/Principle_of_locality" title="Principle of locality" >local</a>" theory.

The <a href="https://en.wikipedia.org/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox" class="mw-redirect" >Einstein–Podolsky–Rosen paradox</a> shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner - although the two particles can be an arbitrary distance apart. However, this effect does not violate <a href="https://en.wikipedia.org/wiki/Causality" title="Causality" >causality</a>, since no transfer of information happens. Quantum entanglement forms the basis of<a href="https://en.wikipedia.org/wiki/Quantum_cryptography" title="Quantum cryptography" >quantum cryptography</a>, which is used in high-security commercial applications in banking and government.

Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of <a href="https://en.wikipedia.org/wiki/Quantum_gravity" title="Quantum gravity" >quantum gravity</a> is an important issue in<a href="https://en.wikipedia.org/wiki/Cosmology" title="Cosmology" >cosmology</a> and the search by physicists for an elegant "<a href="https://en.wikipedia.org/wiki/Theory_of_everything" title="Theory of everything" >Theory of Everything</a>" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th and 21st century physics. Many prominent physicists, including <a href="https://en.wikipedia.org/wiki/Stephen_Hawking" title="Stephen Hawking" >Stephen Hawking</a>, have labored for many years in the attempt to discover a theory underlying <i>everything</i>. This TOE would combine not only the different models of subatomic physics, but also derive the four <a href="https://en.wikipedia.org/wiki/Fundamental_interaction" title="Fundamental interaction" >fundamental forces of nature</a> - the <a href="https://en.wikipedia.org/wiki/Strong_interaction" title="Strong interaction" >strong force</a>, <a href="https://en.wikipedia.org/wiki/Electromagnetism" title="Electromagnetism" >electromagnetism</a>, the <a href="https://en.wikipedia.org/wiki/Weak_interaction" title="Weak interaction" >weak force</a>, and <a href="https://en.wikipedia.org/wiki/Gravity" title="Gravity" >gravity</a> - from a single force or phenomenon. While Stephen Hawking was initially a believer in the Theory of Everything, after considering <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_Incompleteness_Theorem" title="Gödel's Incompleteness Theorem" class="mw-redirect" >Gödel's Incompleteness Theorem</a>, he has concluded that one is not obtainable, and has stated so publicly in his lecture "Gödel and the End of Physics" (2002).<sup id="cite_ref-60" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-60" >[60]</a></sup></p>

<h3 >Attempts at a unified field theory[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=8" title="Edit section: Attempts at a unified field theory" >edit</a>]</h3>

<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/Grand_unified_theory" title="Grand unified theory" class="mw-redirect" >Grand unified theory</a></div>

<p >The quest to unify the <a href="https://en.wikipedia.org/wiki/Fundamental_force" title="Fundamental force" class="mw-redirect" >fundamental forces</a> through quantum mechanics is still ongoing. <a href="https://en.wikipedia.org/wiki/Quantum_electrodynamics" title="Quantum electrodynamics" >Quantum electrodynamics</a> (or "quantum electromagnetism"), which is currently (in the perturbative regime at least) the most accurately tested physical theory in competition with general relativity,<sup id="cite_ref-61" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-61" >[61]</a></sup><sup id="cite_ref-62" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-62" >[62]</a></sup><sup class="noprint Inline-Template" >[<i><a href="https://en.wikipedia.org/wiki/Wikipedia:Identifying_reliable_sources" title="Wikipedia:Identifying reliable sources" >unreliable source?</a></i>]</sup><sup >(blog)</sup> has been successfully merged with the weak nuclear force into the <a href="https://en.wikipedia.org/wiki/Electroweak_force" title="Electroweak force" class="mw-redirect" >electroweak force</a>and work is currently being done to merge the electroweak and strong force into the <a href="https://en.wikipedia.org/wiki/Electrostrong_force" title="Electrostrong force" class="mw-redirect" >electrostrong force</a>. Current predictions state that at around 10<sup >14</sup> GeV the three aforementioned forces are fused into a single unified field.<sup id="cite_ref-63" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-63" >[63]</a></sup> Beyond this "grand unification", it is speculated that it may be possible to merge gravity with the other three gauge symmetries, expected to occur at roughly 10<sup >19</sup> GeV. However — and while special relativity is parsimoniously incorporated into quantum electrodynamics — the expanded <a href="https://en.wikipedia.org/wiki/General_relativity" title="General relativity" >general relativity</a>, currently the best theory describing the gravitation force, has not been fully incorporated into quantum theory. One of those searching for a coherent TOE is <a href="https://en.wikipedia.org/wiki/Edward_Witten" title="Edward Witten" >Edward Witten</a>, a theoretical physicist who formulated the <a href="https://en.wikipedia.org/wiki/M-theory" title="M-theory" >M-theory</a>, which is an attempt at describing the supersymmetrical based <a href="https://en.wikipedia.org/wiki/String_theory" title="String theory" >string theory</a>. M-theory posits that our apparent 4-dimensional <a href="https://en.wikipedia.org/wiki/Spacetime" title="Spacetime" >spacetime</a> is, in reality, actually an 11-dimensional spacetime containing 10 spatial dimensions and 1 time dimension, although 7 of the spatial dimensions are - at lower energies - completely "compactified" (or infinitely curved) and not readily amenable to measurement or probing.

Another popular theory is <a href="https://en.wikipedia.org/wiki/Loop_quantum_gravity" title="Loop quantum gravity" >Loop quantum gravity</a> (LQG), a theory first proposed by <a href="https://en.wikipedia.org/wiki/Carlo_Rovelli" title="Carlo Rovelli" >Carlo Rovelli</a> that describes the quantum properties of gravity. It is also a theory of <a href="https://en.wikipedia.org/w/index.php?title=Quantum_space&action=edit&redlink=1" class="new" title="Quantum space (page does not exist)" >quantum space</a> and <a href="https://en.wikipedia.org/wiki/Chronon" title="Chronon" >quantum time</a>, because in general relativity the geometry of spacetime is a manifestation of <a href="https://en.wikipedia.org/wiki/Gravity" title="Gravity" >gravity</a>. LQG is an attempt to merge and adapt standard quantum mechanics and standard <a href="https://en.wikipedia.org/wiki/General_relativity" title="General relativity" >general relativity</a>. The main output of the theory is a physical picture of space where space is granular. The granularity is a direct consequence of the quantization. It has the same nature of the granularity of the photons in the quantum theory of electromagnetism or the discrete levels of the energy of the atoms. But here it is space itself which is discrete. More precisely, space can be viewed as an extremely fine fabric or network "woven" of finite loops. These networks of loops are called <a href="https://en.wikipedia.org/wiki/Spin_networks" title="Spin networks" class="mw-redirect" >spin networks</a>. The evolution of a spin network over time is called a spin foam. The predicted size of this structure is the <a href="https://en.wikipedia.org/wiki/Planck_length" title="Planck length" >Planck length</a>, which is approximately 1.616×10<sup >−35</sup> m. According to theory, there is no meaning to length shorter than this (cf. <a href="https://en.wikipedia.org/wiki/Planck_scale" title="Planck scale" >Planck scale</a> energy). Therefore, LQG predicts that not just matter, but also space itself, has an atomic structure.</p>

<h2 >Philosophical implications[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=9" title="Edit section: Philosophical implications" >edit</a>]</h2>

<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics" >Interpretations of quantum mechanics</a></div>

<p >Since its inception, the many <a href="https://en.wikipedia.org/wiki/Counter-intuitive" title="Counter-intuitive" class="mw-redirect" >counter-intuitive</a> aspects and results of quantum mechanics have provoked strong <a href="https://en.wikipedia.org/wiki/Philosophy" title="Philosophy" >philosophical</a> debates and many <a href="https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics" >interpretations</a>. Even fundamental issues, such as <a href="https://en.wikipedia.org/wiki/Max_Born" title="Max Born" >Max Born</a>'s basic <a href="https://en.wikipedia.org/wiki/Born_rule" title="Born rule" >rules</a> concerning <a href="https://en.wikipedia.org/wiki/Probability_amplitude" title="Probability amplitude" >probability amplitudes</a> and <a href="https://en.wikipedia.org/wiki/Probability_distribution" title="Probability distribution" >probability distributions</a>, took decades to be appreciated by society and many leading scientists. <a href="https://en.wikipedia.org/wiki/Richard_Feynman" title="Richard Feynman" >Richard Feynman</a> once said, "I think I can safely say that nobody understands quantum mechanics."<sup id="cite_ref-64" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-64" >[64]</a></sup> According to <a href="https://en.wikipedia.org/wiki/Steven_Weinberg" title="Steven Weinberg" >Steven Weinberg</a>, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."<sup id="cite_ref-65" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-65" >[65]</a></sup>

The <a href="https://en.wikipedia.org/wiki/Copenhagen_interpretation" title="Copenhagen interpretation" >Copenhagen interpretation</a> - due largely to the Danish theoretical physicist <a href="https://en.wikipedia.org/wiki/Niels_Bohr" title="Niels Bohr" >Niels Bohr</a> - remains the quantum mechanical formalism that is currently most widely accepted amongst physicists, some 75 years after its enunciation. According to this interpretation, the probabilistic nature of quantum mechanics is not a <i>temporary</i> feature which will eventually be replaced by a deterministic theory, but instead must be considered a <i>final</i> renunciation of the classical idea of "causality." It is also believed therein that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the <a href="https://en.wikipedia.org/wiki/Complementarity_(physics)" title="Complementarity (physics)" >conjugate</a> nature of evidence obtained under different experimental situations.

<a href="https://en.wikipedia.org/wiki/Albert_Einstein" title="Albert Einstein" >Albert Einstein</a>, himself one of the founders of quantum theory, <a href="https://en.wikipedia.org/wiki/Bohr-Einstein_debates" title="Bohr-Einstein debates" class="mw-redirect" >rejected</a> the quantum theoretical doctrine that the state of a system depends on the experimental arrangement for its measurement. He held that underlying quantum mechanics there should be a theory that thoroughly and directly expresses the rule against <a href="https://en.wikipedia.org/wiki/Action_at_a_distance" title="Action at a distance" >action at a distance</a>; in other words, he insisted on the <a href="https://en.wikipedia.org/wiki/Principle_of_locality" title="Principle of locality" >principle of locality</a>. He inferred that the present theory was incomplete, contrary to the Copenhagen doctrine that it is complete. He therefore produced a series of objections, the most famous of which has become known as the <a href="https://en.wikipedia.org/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox" class="mw-redirect" >Einstein–Podolsky–Rosen paradox</a>.

<a href="https://en.wikipedia.org/wiki/John_Stewart_Bell" title="John Stewart Bell" >John Bell</a> showed that this "EPR" paradox led to <a href="https://en.wikipedia.org/wiki/Bell%27s_theorem" title="Bell's theorem" >experimentally testable differences</a> between quantum mechanics and theories that rely on added hidden variables. <a href="https://en.wikipedia.org/wiki/Bell_test_experiments" title="Bell test experiments" >Experiments</a> have been performed confirming the accuracy of quantum mechanics, thereby demonstrating that quantum mechanics cannot be improved upon by addition of hidden variables.<sup id="cite_ref-66" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-66" >[66]</a></sup> The <i><a href="https://en.wikipedia.org/wiki/Bohr-Einstein_debates" title="Bohr-Einstein debates" class="mw-redirect" >Bohr-Einstein debates</a></i>provide a vibrant critique of the Copenhagen Interpretation from an <a href="https://en.wikipedia.org/wiki/Epistemological" title="Epistemological" class="mw-redirect" >epistemological</a> point of view.

The <a href="https://en.wikipedia.org/wiki/Everett_many-worlds_interpretation" title="Everett many-worlds interpretation" class="mw-redirect" >Everett many-worlds interpretation</a>, formulated in 1956, holds that <i>all</i> the possibilities described by quantum theory<i>simultaneously</i> occur in a <a href="https://en.wikipedia.org/wiki/Multiverse" title="Multiverse" >multiverse</a> composed of mostly independent parallel universes.<sup id="cite_ref-67" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-67" >[67]</a></sup> This is not accomplished by introducing some "new axiom" to quantum mechanics, but on the contrary, by <i>removing</i> the axiom of the collapse of the wave packet. <i>All</i> of the possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a <i>real</i>physical - not just formally mathematical, as in other interpretations - <a href="https://en.wikipedia.org/wiki/Quantum_superposition" title="Quantum superposition" >quantum superposition</a>. Such a superposition of consistent state combinations of different systems is called an <a href="https://en.wikipedia.org/wiki/Quantum_entanglement" title="Quantum entanglement" >entangled state</a>. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we can only observe the universe (i.e., the consistent state contribution to the aforementioned superposition) that we, as observers, inhabit. Everett's interpretation is perfectly consistent with <a href="https://en.wikipedia.org/wiki/John_Stewart_Bell" title="John Stewart Bell" >John Bell</a>'s experiments and makes them intuitively understandable. However, according to the theory of <a href="https://en.wikipedia.org/wiki/Quantum_decoherence" title="Quantum decoherence" >quantum decoherence</a>, these "parallel universes" will never be accessible to us. The inaccessibility can be understood as follows: once a measurement is done, the measured system becomes <a href="https://en.wikipedia.org/wiki/Quantum_entanglement" title="Quantum entanglement" >entangled</a> with <i>both</i> the physicist who measured it <i>and</i> a huge number of other particles, some of which are <a href="https://en.wikipedia.org/wiki/Photon" title="Photon" >photons</a> flying away at the <a href="https://en.wikipedia.org/wiki/Speed_of_light" title="Speed of light" >speed of light</a> towards the other end of the universe. In order to prove that the wave function did not collapse, one would have to bring <i>all</i> these particles back and measure them again, together with the system that was originally measured. Not only is this completely impractical, but even if one <i>could</i> theoretically do this, it would have to destroy any evidence that the original measurement took place (including the physicist's memory). In light of these <a href="https://en.wikipedia.org/wiki/Bell_test" title="Bell test" class="mw-redirect" >Bell tests</a>, Cramer (1986) formulated his<a href="https://en.wikipedia.org/wiki/Transactional_interpretation" title="Transactional interpretation" >transactional interpretation</a>.<sup id="cite_ref-Cramer_68-0" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Cramer-68" >[68]</a></sup> <a href="https://en.wikipedia.org/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics" >Relational quantum mechanics</a> appeared in the late 1990s as the modern derivative of the<a href="https://en.wikipedia.org/wiki/Copenhagen_Interpretation" title="Copenhagen Interpretation" class="mw-redirect" >Copenhagen Interpretation</a>.</p>

<h2 >Applications[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=10" title="Edit section: Applications" >edit</a>]</h2>

<p >Quantum mechanics has had enormous<sup id="cite_ref-69" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-69" >[69]</a></sup> success in explaining many of the features of our universe. Quantum mechanics is often the only tool available that can reveal the individual behaviors of the <a href="https://en.wikipedia.org/wiki/Subatomic_particle" title="Subatomic particle" >subatomic particles</a> that make up all forms of matter (<a href="https://en.wikipedia.org/wiki/Electron" title="Electron" >electrons</a>,<a href="https://en.wikipedia.org/wiki/Proton" title="Proton" >protons</a>, <a href="https://en.wikipedia.org/wiki/Neutron" title="Neutron" >neutrons</a>, <a href="https://en.wikipedia.org/wiki/Photon" title="Photon" >photons</a>, and others). Quantum mechanics has strongly influenced <a href="https://en.wikipedia.org/wiki/String_theory" title="String theory" >string theories</a>, candidates for a <a href="https://en.wikipedia.org/wiki/Theory_of_everything" title="Theory of everything" >Theory of Everything</a> (see <a href="https://en.wikipedia.org/wiki/Reductionism" title="Reductionism" >reductionism</a>).

Quantum mechanics is also critically important for understanding how individual atoms combine covalently to form <a href="https://en.wikipedia.org/wiki/Molecules" title="Molecules" class="mw-redirect" >molecules</a>. The application of quantum mechanics to <a href="https://en.wikipedia.org/wiki/Chemistry" title="Chemistry" >chemistry</a> is known as <a href="https://en.wikipedia.org/wiki/Quantum_chemistry" title="Quantum chemistry" >quantum chemistry</a>. Relativistic quantum mechanics can, in principle, mathematically describe most of chemistry. Quantum mechanics can also provide quantitative insight into <a href="https://en.wikipedia.org/wiki/Ionic_bond" title="Ionic bond" class="mw-redirect" >ionic</a> and <a href="https://en.wikipedia.org/wiki/Covalent_bonding" title="Covalent bonding" class="mw-redirect" >covalent bonding</a>processes by explicitly showing which molecules are energetically favorable to which others and the magnitudes of the energies involved.<sup id="cite_ref-70" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-70" >[70]</a></sup> Furthermore, most of the calculations performed in modern <a href="https://en.wikipedia.org/wiki/Computational_chemistry" title="Computational chemistry" >computational chemistry</a> rely on quantum mechanics.</p>

A working mechanism of a resonant tunneling diode device, based on the phenomenon of quantum tunneling through potential barriers</div>

</div>

<p >A great deal of modern technological inventions operate at a scale where quantum effects are significant. Examples include the <a href="https://en.wikipedia.org/wiki/Laser" title="Laser" >laser</a>, the <a href="https://en.wikipedia.org/wiki/Transistor" title="Transistor" >transistor</a>(and thus the <a href="https://en.wikipedia.org/wiki/Integrated_circuit" title="Integrated circuit" >microchip</a>), the <a href="https://en.wikipedia.org/wiki/Electron_microscope" title="Electron microscope" >electron microscope</a>, and <a href="https://en.wikipedia.org/wiki/Magnetic_Resonance_Imaging" title="Magnetic Resonance Imaging" class="mw-redirect" >magnetic resonance imaging</a> (MRI). The study of <a href="https://en.wikipedia.org/wiki/Semiconductor" title="Semiconductor" >semiconductors</a> led to the invention of the <a href="https://en.wikipedia.org/wiki/Diode" title="Diode" >diode</a> and the <a href="https://en.wikipedia.org/wiki/Transistor" title="Transistor" >transistor</a>, which are indispensable parts of modern <a href="https://en.wikipedia.org/wiki/Electronics" title="Electronics" >electronics</a> systems and devices.

Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to more fully develop <a href="https://en.wikipedia.org/wiki/Quantum_cryptography" title="Quantum cryptography" >quantum cryptography</a>, which will theoretically allow guaranteed secure transmission of <a href="https://en.wikipedia.org/wiki/Information" title="Information" >information</a>. A more distant goal is the development of <a href="https://en.wikipedia.org/wiki/Quantum_computer" title="Quantum computer" class="mw-redirect" >quantum computers</a>, which are expected to perform certain computational tasks exponentially faster than classical <a href="https://en.wikipedia.org/wiki/Computer" title="Computer" >computers</a>. Instead of using classical bits, quantum computers use <a href="https://en.wikipedia.org/wiki/Qubits" title="Qubits" class="mw-redirect" >qubits</a>, which can be in <a href="https://en.wikipedia.org/wiki/Quantum_superposition" title="Quantum superposition" >superpositions</a> of states. Another active research topic is <a href="https://en.wikipedia.org/wiki/Quantum_teleportation" title="Quantum teleportation" >quantum teleportation</a>, which deals with techniques to transmit quantum information over arbitrary distances.

<a href="https://en.wikipedia.org/wiki/Quantum_tunneling" title="Quantum tunneling" class="mw-redirect" >Quantum tunneling</a> is vital to the operation of many devices. Even in the simple <a href="https://en.wikipedia.org/wiki/Light_switch" title="Light switch" >light switch</a>, the electrons in the <a href="https://en.wikipedia.org/wiki/Electric_current" title="Electric current" >electric current</a> could not penetrate the potential barrier made up of a layer of oxide without quantum tunneling. <a href="https://en.wikipedia.org/wiki/Flash_memory" title="Flash memory" >Flash memory</a> chips found in <a href="https://en.wikipedia.org/wiki/USB_flash_drive" title="USB flash drive" >USB drives</a> use quantum tunneling to erase their memory cells.

While quantum mechanics primarily applies to the smaller atomic regimes of matter and energy, some systems exhibit <a href="https://en.wikipedia.org/wiki/Mechanics#Classical_versus_quantum" title="Mechanics" >quantum mechanical effects</a> on a large scale. <a href="https://en.wikipedia.org/wiki/Superfluidity" title="Superfluidity" >Superfluidity</a>, the frictionless flow of a liquid at temperatures near <a href="https://en.wikipedia.org/wiki/Absolute_zero" title="Absolute zero" >absolute zero</a>, is one well-known example. So is the closely related phenomenon of <a href="https://en.wikipedia.org/wiki/Superconductivity" title="Superconductivity" >superconductivity</a>, the frictionless flow of an electron gas in a conducting material (an <a href="https://en.wikipedia.org/wiki/Electric_current" title="Electric current" >electric current</a>) at sufficiently low temperatures.

Quantum theory also provides accurate descriptions for many previously unexplained phenomena, such as <a href="https://en.wikipedia.org/wiki/Black-body_radiation" title="Black-body radiation" >black-body radiation</a> and the stability of the <a href="https://en.wikipedia.org/wiki/Atomic_orbital" title="Atomic orbital" >orbitals</a> of electrons in atoms. It has also given insight into the workings of many different <a href="https://en.wikipedia.org/wiki/Biological_systems" title="Biological systems" class="mw-redirect" >biological systems</a>, including <a href="https://en.wikipedia.org/wiki/Smell_receptors" title="Smell receptors" class="mw-redirect" >smell receptors</a> and <a href="https://en.wikipedia.org/wiki/Protein_structure" title="Protein structure" >protein structures</a>.<sup id="cite_ref-71" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-71" >[71]</a></sup> Recent work on <a href="https://en.wikipedia.org/wiki/Photosynthesis" title="Photosynthesis" >photosynthesis</a> has provided evidence that quantum correlations play an essential role in this fundamental process of plants and many other organisms.<sup id="cite_ref-72" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-72" >[72]</a></sup> Even so, <a href="https://en.wikipedia.org/wiki/Classical_physics" title="Classical physics" >classical physics</a> can often provide good approximations to results otherwise obtained by quantum physics, typically in circumstances with large numbers of particles or large <a href="https://en.wikipedia.org/wiki/Quantum_number" title="Quantum number" >quantum numbers</a>. Since classical formulas are much simpler and easier to compute than quantum formulas, classical approximations are used and preferred when the system is large enough to render the effects of quantum mechanics insignificant.</p>

<h2 >Examples[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=11" title="Edit section: Examples" >edit</a>]</h2>

<h3 >Free particle[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=12" title="Edit section: Free particle" >edit</a>]</h3>

<p >For example, consider a <a href="https://en.wikipedia.org/wiki/Free_particle" title="Free particle" >free particle</a>. In quantum mechanics, there is <a href="https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality" >wave–particle duality</a>, so the properties of the particle can be described as the properties of a wave. Therefore, its <a href="https://en.wikipedia.org/wiki/Quantum_state" title="Quantum state" >quantum state</a> can be represented as a <a href="https://en.wikipedia.org/wiki/Wave" title="Wave" >wave</a> of arbitrary shape and extending over space as a <a href="https://en.wikipedia.org/wiki/Wave_function" title="Wave function" >wave function</a>. The position and momentum of the particle are <a href="https://en.wikipedia.org/wiki/Observables" title="Observables" class="mw-redirect" >observables</a>. The <a href="https://en.wikipedia.org/wiki/Uncertainty_Principle" title="Uncertainty Principle" class="mw-redirect" >Uncertainty Principle</a> states that both the position and the momentum cannot simultaneously be measured with complete precision. However, one <i>can</i> measure the position (alone) of a moving free particle, creating an eigenstate of position with a wavefunction that is very large (a <a href="https://en.wikipedia.org/wiki/Dirac_delta" title="Dirac delta" class="mw-redirect" >Dirac delta</a>) at a particular position <i>x</i>, and zero everywhere else. If one performs a position measurement on such a wavefunction, the resultant <i>x</i> will be obtained with 100% probability (i.e., with full certainty, or complete precision). This is called an eigenstate of position—or, stated in mathematical terms, a <i>generalized position eigenstate (<a href="https://en.wikipedia.org/wiki/Distribution_(mathematics)" title="Distribution (mathematics)" >eigendistribution</a>)</i>. If the particle is in an eigenstate of position, then its momentum is completely unknown. On the other hand, if the particle is in an eigenstate of momentum, then its position is completely unknown.<sup id="cite_ref-73" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-73" >[73]</a></sup> In an eigenstate of momentum having a <a href="https://en.wikipedia.org/wiki/Plane_wave" title="Plane wave" >plane wave</a> form, it can be shown that the <a href="https://en.wikipedia.org/wiki/Wavelength" title="Wavelength" >wavelength</a> is equal to <i>h/p</i>, where <i>h</i>is <a href="https://en.wikipedia.org/wiki/Planck%27s_constant" title="Planck's constant" class="mw-redirect" >Planck's constant</a> and <i>p</i> is the momentum of the <a href="https://en.wikipedia.org/wiki/Eigenstate" title="Eigenstate" class="mw-redirect" >eigenstate</a>.<sup id="cite_ref-74" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-74" >[74]</a></sup></p>

3D confined electron wave functions for each eigenstate in a Quantum Dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more ‘s-type’ and ‘p-type’. However, in a triangular dot, the wave functions are mixed due to confinement symmetry.</div>

</div>

<h3 >Step potential[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=13" title="Edit section: Step potential" >edit</a>]</h3>

<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/Solution_of_Schr%C3%B6dinger_equation_for_a_step_potential" title="Solution of Schrödinger equation for a step potential" >Solution of Schrödinger equation for a step potential</a></div>

Scattering at a finite potential step of height <i>V</i><sub >0</sub>, shown in green. The amplitudes and direction of left- and right-moving waves are indicated. Yellow is the incident wave, blue are reflected and transmitted waves, red does not occur. <i>E</i> > <i>V</i><sub >0</sub> for this figure.</div>

</div>

<p >The potential in this case is given by:</p>

<dl >

</dl>

<p >The solutions are superpositions of left- and right-moving waves:</p>

<dl >

</dl>

<p >where the <a href="https://en.wikipedia.org/wiki/Wave_vector" title="Wave vector" >wave vectors</a> are related to the energy via</p>

<dl >

</dl>

<p >with coefficients A and B determined from the <a href="https://en.wikipedia.org/wiki/Boundary_condition" title="Boundary condition" class="mw-redirect" >boundary conditions</a> and by imposing a continuous <a href="https://en.wikipedia.org/wiki/Derivative" title="Derivative" >derivative</a> on the solution.

Each term of the solution can be interpreted as an incident, reflected, or transmitted component of the wave, allowing the calculation of transmission and reflection coefficients. Notably, in contrast to classical mechanics, incident particles with energies greater than the potential step are partially reflected.</p>

<h3 >Rectangular potential barrier[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=14" title="Edit section: Rectangular potential barrier" >edit</a>]</h3>

<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/Rectangular_potential_barrier" title="Rectangular potential barrier" >Rectangular potential barrier</a></div>

<p >This is a model for the <a href="https://en.wikipedia.org/wiki/Quantum_tunneling" title="Quantum tunneling" class="mw-redirect" >quantum tunneling</a> effect which plays an important role in the performance of modern technologies such as<a href="https://en.wikipedia.org/wiki/Flash_memory" title="Flash memory" >flash memory</a> and <a href="https://en.wikipedia.org/wiki/Scanning_tunneling_microscope" title="Scanning tunneling microscope" >scanning tunneling microscopy</a>. Quantum tunneling is central to physical phenomena involved in <a href="https://en.wikipedia.org/wiki/Superlattice" title="Superlattice" >superlattices</a>.</p>

<h3 >Particle in a box[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=15" title="Edit section: Particle in a box" >edit</a>]</h3>

1-dimensional potential energy box (or infinite potential well)</div>

</div>

<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/Particle_in_a_box" title="Particle in a box" >Particle in a box</a></div>

<p >The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere <i>inside</i> a certain region, and infinite potential energy everywhere <i>outside</i> that region. For the one-dimensional case in the <img src="/c0f78e6e-d5ba-56f1-b767-5a5b6256dd6d.png"/> direction, the time-independent Schrödinger equation may be written<sup id="cite_ref-75" class="reference" ><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-75" >[75]</a></sup></p>

<dl >

</dl>

<p >With the differential operator defined by</p>

<dl >

</dl>

<p >the previous equation is evocative of the <a href="https://en.wikipedia.org/wiki/Kinetic_energy#Kinetic_energy_of_rigid_bodies" title="Kinetic energy" >classic kinetic energy analogue</a>,</p>

<dl >

</dl>

<p >with state <img src="/05244957-2a1c-5326-9f89-94c22a1fb634.png"/> in this case having energy <img src="/7a6c950a-d621-5170-85a9-06d9ceec27ec.png"/> coincident with the kinetic energy of the particle.

The general solutions of the Schrödinger equation for the particle in a box are</p>

<dl >

</dl>

<p >or, from <a href="https://en.wikipedia.org/wiki/Euler%27s_formula" title="Euler's formula" >Euler's formula</a>,</p>

<dl >

</dl>

<p >The infinite potential walls of the box determine the values of <i>C</i>, <i>D</i>, and <i>k</i> at <i>x</i> = 0 and <i>x</i> = <i>L</i> where <i>ψ</i> must be zero. Thus, at <i>x</i> = 0,</p>

<dl >

</dl>

<dl >

</dl>

<p >in which <i>C</i> cannot be zero as this would conflict with the Born interpretation. Therefore, since sin(<i>kL</i>) = 0, <i>kL</i> must be an integer multiple of π,</p>

<dl >

</dl>

<p >The quantization of energy levels follows from this constraint on <i>k</i>, since</p>

<dl >

</dl>

<h3 >Finite potential well[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=16" title="Edit section: Finite potential well" >edit</a>]</h3>

<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/Finite_potential_well" title="Finite potential well" >Finite potential well</a></div>

<p >A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth.

The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wavefunction is not pinned to zero at the walls of the well. Instead, the wavefunction must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well.</p>

<h3 >Harmonic oscillator[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=17" title="Edit section: Harmonic oscillator" >edit</a>]</h3>

<div class="hatnote relarticle mainarticle" >Main article: <a href="https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator" >Quantum harmonic oscillator</a></div>

Some trajectories of a <a href="https://en.wikipedia.org/wiki/Harmonic_oscillator" title="Harmonic oscillator" >harmonic oscillator</a> (i.e. a ball attached to a <a href="https://en.wikipedia.org/wiki/Hooke%27s_law" title="Hooke's law" >spring</a>) in <a href="https://en.wikipedia.org/wiki/Classical_mechanics" title="Classical mechanics" >classical mechanics</a> (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a<a href="https://en.wikipedia.org/wiki/Wave" title="Wave" >wave</a> (called the <a href="https://en.wikipedia.org/wiki/Wavefunction" title="Wavefunction" class="mw-redirect" >wavefunction</a>), with the <a href="https://en.wikipedia.org/wiki/Real_part" title="Real part" class="mw-redirect" >real part</a>shown in blue and the <a href="https://en.wikipedia.org/wiki/Imaginary_part" title="Imaginary part" class="mw-redirect" >imaginary part</a> shown in red. Some of the trajectories (such as C,D,E,and F) are<a href="https://en.wikipedia.org/wiki/Standing_wave" title="Standing wave" >standing waves</a> (or "<a href="https://en.wikipedia.org/wiki/Stationary_state" title="Stationary state" >stationary states</a>"). Each standing-wave frequency is proportional to a possible <a href="https://en.wikipedia.org/wiki/Energy_level" title="Energy level" >energy level</a> of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have <i>any</i> energy.</div>

</div>

<p >As in the classical case, the potential for the quantum harmonic oscillator is given by</p>

<dl >

</dl>

<p >This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The <a href="https://en.wikipedia.org/wiki/Eigenstate" title="Eigenstate" class="mw-redirect" >eigenstates</a> are given by</p>

<dl >

</dl>

<p >where <i>H<sub >n</sub></i> are the <a href="https://en.wikipedia.org/wiki/Hermite_polynomials" title="Hermite polynomials" >Hermite polynomials</a>,</p>

<dl >

</dl>

<p >and the corresponding energy levels are</p>

<dl >

</dl>

<p >This is another example illustrating the quantization of energy for bound states.</p>

<tr >

<td ><a href="https://en.wikipedia.org/wiki/Portal:Physics" title="Portal:Physics" >Physics portal</a></td>

</tr>

</tbody></table>

</div>

<ul >

<li ><a href="https://en.wikipedia.org/wiki/Angular_momentum_diagrams_(quantum_mechanics)" title="Angular momentum diagrams (quantum mechanics)" >Angular momentum diagrams (quantum mechanics)</a></li>

<li ><a href="https://en.wikipedia.org/wiki/EPR_paradox" title="EPR paradox" >EPR paradox</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Fractional_quantum_mechanics" title="Fractional quantum mechanics" >Fractional quantum mechanics</a></li>

<li ><a href="https://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions" title="List of quantum-mechanical systems with analytical solutions" >List of quantum-mechanical systems with analytical solutions</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Macroscopic_quantum_phenomena" title="Macroscopic quantum phenomena" >Macroscopic quantum phenomena</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Phase_space_formulation" title="Phase space formulation" >Phase space formulation</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Regularization_(physics)" title="Regularization (physics)" >Regularization (physics)</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Spherical_basis" title="Spherical basis" >Spherical basis</a></li>

</ul>

<h2 >Notes[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=19" title="Edit section: Notes" >edit</a>]</h2>

<li id="cite_note-Born.26_Wolf-1" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Born.26_Wolf_1-0" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Max_Born" title="Max Born" >Max Born</a> & Emil Wolf, Principles of Optics, 1999, Cambridge University Press</li>

<li id="cite_note-2" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-2" >Jump up^{</a></b> Mehra}, J.; Rechenberg, H. (1982). <i>The historical development of quantum theory</i>. New York: Springer-Verlag. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0387906428" title="Special:BookSources/0387906428" >0387906428</a>.</li>

<li id="cite_note-3" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-3" >Jump up^{</a></b> Kragh}, Helge (2002). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=ELrFDIldlawC&pg=PA58" ><i>Quantum Generations: A History of Physics in the Twentieth Century</i></a>. Princeton University Press. p. 58. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-691-09552-3" title="Special:BookSources/0-691-09552-3" >0-691-09552-3</a>., <a rel="nofollow" class="external text" href="http://books.google.com/books?id=ELrFDIldlawC&pg=PA58" >Extract of page 58</a></li>

<li id="cite_note-4" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-4" >Jump up^{</a></b> Ben-Menahem}, Ari (2009). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=9tUrarQYhKMC" ><i>Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1</i></a>. Springer. p. 3678.<a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/3540688315" title="Special:BookSources/3540688315" >3540688315</a>., <a rel="nofollow" class="external text" href="http://books.google.com/books?id=9tUrarQYhKMC&pg=PA3678" >Extract of page 3678</a></li>

<li id="cite_note-5" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-5" >Jump up^{</a></b>&#160};E Arunan (2010). <a rel="nofollow" class="external text" href="http://www.ias.ac.in/resonance/Volumes/15/12/1056-1059.pdf" >"Peter Debye"</a> (PDF). <i>Resonance (journal)</i>(Indian Academy of Sciences) <b>15</b> (12).</li>

<li id="cite_note-6" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-6" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Thomas_Samuel_Kuhn" title="Thomas Samuel Kuhn" class="mw-redirect" >Kuhn, T. S.</a> (1978). <i>Black-body theory and the quantum discontinuity 1894-1912</i>. Oxford: Clarendon Press.<a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0195023838" title="Special:BookSources/0195023838" >0195023838</a>.</li>

<li id="cite_note-Kragh-7" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Kragh_7-0" >Jump up^{</a></b> Kragh}, Helge (1 December 2000), <a rel="nofollow" class="external text" href="http://physicsworld.com/cws/article/print/373" ><i>Max Planck: the reluctant revolutionary</i></a>, PhysicsWorld.com</li>

<li id="cite_note-8" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-8" >Jump up^{</a></b> Einstein}, A. (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" [On a heuristic point of view concerning the production and transformation of light]. <i><a href="https://en.wikipedia.org/wiki/Annalen_der_Physik" title="Annalen der Physik" >Annalen der Physik</a></i> <b>17</b> (6): 132–148. <a href="https://en.wikipedia.org/wiki/Bibcode" title="Bibcode" >Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1905AnP...322..132E" >1905AnP...322..132E</a>.<a href="https://en.wikipedia.org/wiki/Digital_object_identifier" title="Digital object identifier" >doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1002%2Fandp.19053220607" >10.1002/andp.19053220607</a>. Reprinted in <i>The collected papers of Albert Einstein</i>, John Stachel, editor, Princeton University Press, 1989, Vol. 2, pp. 149-166, in German; see also<i>Einstein's early work on the quantum hypothesis</i>, ibid. pp. 134-148.</li>

<li id="cite_note-9" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-9" >Jump up^{</a></b>&#160};van Hove, Leon (1958). <a rel="nofollow" class="external text" href="http://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10206-2/S0002-9904-1958-10206-2.pdf" >"Von Neumann's contributions to quantum mechanics"</a> (PDF). <i>Bulletin of the American Mathematical Society</i> <b>64</b>: Part2:95–99. <a href="https://en.wikipedia.org/wiki/Digital_object_identifier" title="Digital object identifier" >doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2Fs0002-9904-1958-10206-2" >10.1090/s0002-9904-1958-10206-2</a>.</li>

<li id="cite_note-10" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-10" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.nature.com/ncomms/journal/v2/n4/full/ncomms1263.html" >"Quantum interference of large organic molecules"</a>. Nature.com. Retrieved April 20, 2013.</li>

<li id="cite_note-11" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-11" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.merriam-webster.com/dictionary/quantum" >"Quantum - Definition and More from the Free Merriam-Webster Dictionary"</a>. Merriam-webster.com. Retrieved 2012-08-18.</li>

<li id="cite_note-12" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-12" >Jump up^{</a></b> }<a rel="nofollow" class="external free" href="http://mooni.fccj.org/_{ethall/quantum/quant.htm}" >http://mooni.fccj.org/_{ethall/quantum/quant.htm</a></li}>

<li id="cite_note-13" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-13" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://ysfine.com/" >"ysfine.com"</a>. <i>ysfine.com</i>. Retrieved 11 September 2015.</li>

<li id="cite_note-14" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-14" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="https://web.archive.org/web/20091026095410/http://geocities.com/mik_malm/quantmech.html" >Oocities.com</a> at the <a href="https://en.wikipedia.org/wiki/Wayback_Machine" title="Wayback Machine" >Wayback Machine</a> (archived October 26, 2009)<sup class="noprint Inline-Template" >[<i><a href="https://en.wikipedia.org/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot" >dead link</a></i>]</sup></li>

<li id="cite_note-15" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-15" >Jump up^{</a></b> P.A.M}. Dirac, <i>The Principles of Quantum Mechanics</i>, Clarendon Press, Oxford, 1930.</li>

<li id="cite_note-16" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-16" >Jump up^{</a></b> D}. Hilbert <i>Lectures on Quantum Theory</i>, 1915–1927</li>

<li id="cite_note-17" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-17" >Jump up^{</a></b> J}. von Neumann, <i>Mathematische Grundlagen der Quantenmechanik</i>, Springer, Berlin, 1932 (English translation:<i>Mathematical Foundations of Quantum Mechanics</i>, Princeton University Press, 1955).</li>

<li id="cite_note-18" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-18" >Jump up^{</a></b> H}.Weyl "The Theory of Groups and Quantum Mechanics", 1931 (original title: "Gruppentheorie und Quantenmechanik").</li>

<li id="cite_note-19" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-19" >Jump up^{</a></b> Greiner}, Walter; Müller, Berndt (1994). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=gCfvWx6vuzUC&pg=PA52" ><i>Quantum Mechanics Symmetries, Second edition</i></a>. Springer-Verlag. p. 52. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/3-540-58080-8" title="Special:BookSources/3-540-58080-8" >3-540-58080-8</a>., <a rel="nofollow" class="external text" href="http://books.google.com/books?id=gCfvWx6vuzUC&pg=PA52" >Chapter 1, p. 52</a></li>

<li id="cite_note-20" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-20" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.aip.org/history/heisenberg/p08a.htm" >"Heisenberg - Quantum Mechanics, 1925–1927: The Uncertainty Relations"</a>. Aip.org. Retrieved 2012-08-18.</li>

<li id="cite_note-google215-21" >^ <a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-google215_21-0" >Jump up to:<sup ><i><b>a</b></i></sup></a> <a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-google215_21-1" ><sup ><i><b>b</b></i></sup></a> Greenstein, George; Zajonc, Arthur (2006). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215" ><i>The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics, Second edition</i></a>. Jones and Bartlett Publishers, Inc. p. 215. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-7637-2470-X" title="Special:BookSources/0-7637-2470-X" >0-7637-2470-X</a>., <a rel="nofollow" class="external text" href="http://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215" >Chapter 8, p. 215</a></li>

<li id="cite_note-22" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-22" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.actapress.com/PaperInfo.aspx?PaperID=25988&reason=500" >"[Abstract] Visualization of Uncertain Particle Movement"</a>. Actapress.com. Retrieved 2012-08-18.</li>

<li id="cite_note-23" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-23" >Jump up^{</a></b> Hirshleifer}, Jack (2001). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=W2J2IXgiZVgC&pg=PA265" ><i>The Dark Side of the Force: Economic Foundations of Conflict Theory</i></a>. Campbridge University Press. p. 265. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-521-80412-4" title="Special:BookSources/0-521-80412-4" >0-521-80412-4</a>., <a rel="nofollow" class="external text" href="http://books.google.com/books?id=W2J2IXgiZVgC&pg=PA265" >Chapter , p.</a></li>

<li id="cite_note-24" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-24" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.dict.cc/german-english/eigen.html" >"dict.cc dictionary :: eigen :: German-English translation"</a>.<i>dict.cc</i>. Retrieved 11 September 2015.</li>

<li id="cite_note-25" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-25" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.phy.olemiss.edu/_{luca/Topics/qm/collapse.html}" >"Topics: Wave-Function Collapse"</a>. Phy.olemiss.edu. 2012-07-27. Retrieved 2012-08-18.</li>

<li id="cite_note-26" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-26" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/qmech/lectures/node28.html" >"Collapse of the wave-function"</a>. Farside.ph.utexas.edu. Retrieved 2012-08-18.</li>

<li id="cite_note-27" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-27" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.reddit.com/r/philosophy/comments/8p2qv/determinism_and_naive_realism/" >"Determinism and Naive Realism : philosophy"</a>. Reddit.com. 2009-06-01. Retrieved 2012-08-18.</li>

<li id="cite_note-28" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-28" >Jump up^{</a></b>&#160};Michael Trott. <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/TimeEvolutionOfAWavepacketInASquareWell/" >"Time-Evolution of a Wavepacket in a Square Well — Wolfram Demonstrations Project"</a>. Demonstrations.wolfram.com. Retrieved 2010-10-15.</li>

<li id="cite_note-29" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-29" >Jump up^{</a></b>&#160};Michael Trott. <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/TimeEvolutionOfAWavepacketInASquareWell/" >"Time Evolution of a Wavepacket In a Square Well"</a>. Demonstrations.wolfram.com. Retrieved 2010-10-15.</li>

<li id="cite_note-30" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-30" >Jump up^{</a></b> Mathews}, Piravonu Mathews; Venkatesan, K. (1976). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=qzs1DD3TcsC&pg=PA36" ><i>A Textbook of Quantum Mechanics</i></a>. Tata McGraw-Hill. p. 36.<a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-07-096510-2" title="Special:BookSources/0-07-096510-2" >0-07-096510-2</a>., <a rel="nofollow" class="external text" href="http://books.google.com/books?id=qzs1DD3TcsC&pg=PA36" >Chapter 2, p. 36</a></li>

<li id="cite_note-31" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-31" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://physics.ukzn.ac.za/_{petruccione/Phys120/Wave%20Functions%20and%20the%20Schr%F6dinger%20Equation.pdf}" >"Wave Functions and the Schrödinger Equation"</a> (PDF). Retrieved 2010-10-15.<sup class="noprint Inline-Template" >[<i><a href="https://en.wikipedia.org/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot" >dead link</a></i>]</sup></li>

<li id="cite_note-33" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-33" >Jump up^{</a></b>&#160};Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (Basic Books, 2005), pp. 124-8 and 285-6.</li>

<li id="cite_note-34" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-34" >Jump up^{</a></b> }<a rel="nofollow" class="external free" href="http://ocw.usu.edu/physics/classical-mechanics/pdf_lectures/06.pdf" >http://ocw.usu.edu/physics/classical-mechanics/pdf_lectures/06.pdf</a></li>

<li id="cite_note-35" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-35" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://nobelprize.org/nobel_prizes/physics/laureates/1979/index.html" >"The Nobel Prize in Physics 1979"</a>. Nobel Foundation. Retrieved 2010-02-16.</li>

<li id="cite_note-36" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-36" >Jump up^{</a></b>&#160};Carl M. Bender, Daniel W. Hook, Karta Kooner (2009-12-31). "Complex Elliptic Pendulum". <a href="https://en.wikipedia.org/wiki/ArXiv" title="ArXiv" >arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1001.0131" >1001.0131</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/hep-th" >hep-th</a>].</li>

<li id="cite_note-37" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-37" >Jump up^{</a></b> See}, for example, <a href="https://en.wikipedia.org/wiki/Precision_tests_of_QED" title="Precision tests of QED" >Precision tests of QED</a>. The relativistic refinement of quantum mechanics known as <a href="https://en.wikipedia.org/wiki/Quantum_electrodynamics" title="Quantum electrodynamics" >quantum electrodynamics</a> (QED) has been shown to agree with experiment to within 1 part in 10<sup >8</sup> for some atomic properties.</li>

<li id="cite_note-Tipler-38" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Tipler_38-0" >Jump up^{</a></b> Tipler}, Paul; Llewellyn, Ralph (2008). <i>Modern Physics</i> (5 ed.).<a href="https://en.wikipedia.org/wiki/W.H.Freeman_and_Company" title="W. H. Freeman and Company" >W. H. Freeman and Company</a>. pp. 160–161. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/978-0-7167-7550-8" title="Special:BookSources/978-0-7167-7550-8" >978-0-7167-7550-8</a>.</li>

<li id="cite_note-39" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-39" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.scribd.com/doc/5998949/Quantum-mechanics-course-iwhatisquantummechanics" >"Quantum mechanics course iwhatisquantummechanics"</a>. Scribd.com. 2008-09-14. Retrieved 2012-08-18.</li>

<li id="cite_note-40" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-40" >Jump up^{</a></b> A}. Einstein, B. Podolsky, and N. Rosen, <a rel="nofollow" class="external text" href="http://www.drchinese.com/David/EPR.pdf" ><i>Can quantum-mechanical description of physical reality be considered complete?</i></a> Phys. Rev. <b>47</b> 777 (1935). <a rel="nofollow" class="external autonumber" href="http://prola.aps.org/abstract/PR/v47/i10/p777_1" >[2]</a></li>

<li id="cite_note-41" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-41" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/2328/1/handbook.pdf" >"Between classical and quantum�"</a> (PDF). Retrieved2012-08-19.</li>

<li id="cite_note-42" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-42" >Jump up^{</a></b> (see }<a href="https://en.wikipedia.org/wiki/Macroscopic_quantum_phenomena" title="Macroscopic quantum phenomena" >macroscopic quantum phenomena</a>, <a href="https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate" >Bose–Einstein condensate</a>, and <a href="https://en.wikipedia.org/wiki/Quantum_machine" title="Quantum machine" >Quantum machine</a>)</li>

<li id="cite_note-43" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-43" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://academic.brooklyn.cuny.edu/physics/sobel/Nucphys/atomprop.html" >"Atomic Properties"</a>. Academic.brooklyn.cuny.edu. Retrieved2012-08-18.</li>

<li id="cite_note-44" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-44" >Jump up^</a></b><a rel="nofollow" class="external free" href="http://assets.cambridge.org/97805218/29526/excerpt/9780521829526_excerpt.pdf" >http://assets.cambridge.org/97805218/29526/excerpt/9780521829526_excerpt.pdf</a></li>

<li id="cite_note-45" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-45" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Max_Born" title="Max Born" >Born, M.</a>, <a href="https://en.wikipedia.org/wiki/Werner_Heisenberg" title="Werner Heisenberg" >Heisenberg, W.</a>, <a href="https://en.wikipedia.org/wiki/Pascual_Jordan" title="Pascual Jordan" >Jordan, P.</a> (1926). <i>Z. Phys.</i> <b>35</b>: 557–615. Translated as 'On quantum mechanics II', pp. 321–385 in<a href="https://en.wikipedia.org/wiki/Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden" >Van der Waerden, B.L.</a> (1967), <i>Sources of Quantum Mechanics</i>, North-Holland, Amsterdam, "The basic difference between the theory proposed here and that used hitherto ... lies in the characteristic kinematics ...", p. 385.</li>

<li id="cite_note-46" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-46" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Paul_Dirac" title="Paul Dirac" >Dirac, P.A.M.</a> (1930/1958). <i>The Principles of Quantum Mechanics</i>, fourth edition, Oxford University Press, Oxford UK, p. 5: "A question about what will happen to a particular photon under certain conditions is not really very precise. To make it precise one must imagine some experiment performed having a bearing on the question, and enquire what will be the result of the experiment. Only questions about the results of experiments have a real significance and it is only such questions that theoretical physics has to consider."</li>

<li id="cite_note-47" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-47" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Niels_Bohr" title="Niels Bohr" >Bohr, N.</a> (1939). The Causality Problem in Atomic Physics, in<i>New Theories in Physics, Conference organized in collaboration with the International Union of Physics and the Polish Intellectual Co-operation Committee, Warsaw, May 30th – June 3rd 1938</i>, International Institute of Intellectual Co-operation, Paris, 1939, pp. 11–30, reprinted in <i>Niels Bohr, Collected Works</i>, volume 7 (1933 – 1958) edited by J. Kalckar, Elsevier, Amsterdam, <a href="https://en.wikipedia.org/wiki/Special:BookSources/0444898921" class="internal mw-magiclink-isbn" >ISBN 0-444-89892-1</a>, pp. 303–322. "The essential lesson of the analysis of measurements in quantum theory is thus the emphasis on the necessity, in the account of the phenomena, of taking the whole experimental arrangement into consideration, in complete conformity with the fact that all unambiguous interpretation of the quantum mechanical formalism involves the fixation of the external conditions, defining the initial state of the atomic system and the character of the possible predictions as regards subsequent observable properties of that system. Any measurement in quantum theory can in fact only refer either to a fixation of the initial state or to the test of such predictions, and it is first the combination of both kinds which constitutes a well-defined phenomenon."</li>

<li id="cite_note-48" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-48" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Niels_Bohr" title="Niels Bohr" >Bohr, N.</a> (1948). On the notions of complementarity and causality, <i>Dialectica</i> <b>2</b>: 312–319. "As a more appropriate way of expression, one may advocate limitation of the use of the word<i>phenomenon</i> to refer to observations obtained under specified circumstances, including an account of the whole experiment."</li>

<li id="cite_note-49" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-49" >Jump up^{</a></b> Ludwig}, G. (1987). <i>An Axiomatic Basis for Quantum Mechanics</i>, volume 2, <i>Quantum Mechanics and Macrosystems</i>, translated by K. Just, Springer, Berlin, <a href="https://en.wikipedia.org/wiki/Special:BookSources/9783642718991" class="internal mw-magiclink-isbn" >ISBN 978-3-642-71899-1</a>, Chapter XIII, Special Structures in Preparation and Registration Devices, §1, Measurement chains, p. 132.</li>

<li id="cite_note-Heisenberg_1927-50" >^ <a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Heisenberg_1927_50-0" >Jump up to:<sup ><i><b>a</b></i></sup></a> <a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Heisenberg_1927_50-1" ><sup ><i><b>b</b></i></sup></a> <a href="https://en.wikipedia.org/wiki/Werner_Heisenberg" title="Werner Heisenberg" >Heisenberg, W.</a> (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, <i>Z. Phys.</i> <b>43</b>: 172–198. Translation as 'The actual content of quantum theoretical kinematics and mechanics' <a rel="nofollow" class="external text" href="http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19840008978.pdf" >here</a>, "But in the rigorous formulation of the law of causality, — "If we know the present precisely, we can calculate the future" — it is not the conclusion that is faulty, but the premise."</li>

<li id="cite_note-51" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-51" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Herbert_S._Green" title="Herbert S. Green" >Green, H.S.</a> (1965). <i>Matrix Mechanics</i>, with a foreword by <a href="https://en.wikipedia.org/wiki/Max_Born" title="Max Born" >Max Born</a>, P. Noordhoff Ltd, Groningen. "It is not possible, therefore, to provide 'initial conditions' for the prediction of the behaviour of atomic systems, in the way contemplated by classical physics. This is accepted by quantum theory, not merely as an experimental difficulty, but as a fundamental law of nature", p. 32.</li>

<li id="cite_note-52" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-52" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/L%C3%A9on_Rosenfeld" title="Léon Rosenfeld" >Rosenfeld, L.</a> (1957). Misunderstandings about the foundations of quantum theory, pp. 41–45 in <i>Observation and Interpretation</i>, edited by S. Körner, Butterworths, London. "A phenomenon is therefore a process (endowed with the characteristic quantal wholeness) involving a definite type of interaction between the system and the apparatus."</li>

<li id="cite_note-53" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-53" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Paul_Dirac" title="Paul Dirac" >Dirac, P.A.M.</a> (1973). Development of the physicist's conception of nature, pp. 1–55 in <i>The Physicist's Conception of Nature</i>, edited by J. Mehra, D. Reidel, Dordrecht, <a href="https://en.wikipedia.org/wiki/Special:BookSources/9027703450" class="internal mw-magiclink-isbn" >ISBN 90-277-0345-0</a>, p. 5: "That led Heisenberg to his really masterful step forward, resulting in the new quantum mechanics. His idea was to build up a theory entirely in terms of quantities referring to two states."</li>

<li id="cite_note-54" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-54" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Max_Born" title="Max Born" >Born, M.</a> (1927). Physical aspects of quantum mechanics,<i>Nature</i> <b>119</b>: 354–357, "These probabilities are thus dynamically determined. But what the system actually does is not determined ..."</li>

<li id="cite_note-55" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-55" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Albert_Messiah" title="Albert Messiah" >Messiah, A.</a> (1961). <i>Quantum Mechanics</i>, volume 1, translated by G.M. Temmer from the French <i>Mécanique Quantique</i>, North-Holland, Amsterdam, p. 157.</li>

<li id="cite_note-56" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-56" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Niels_Bohr" title="Niels Bohr" >Bohr, N.</a> (1928). The Quantum postulate and the recent development of atomic theory, <i>Nature</i> <b>121</b>: 580–590.</li>

<li id="cite_note-57" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-57" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Werner_Heisenberg" title="Werner Heisenberg" >Heisenberg, W.</a> (1930). <i>The Physical Principles of the Quantum Theory</i>, translated by C. Eckart and F.C. Hoyt, University of Chicago Press.</li>

<li id="cite_note-58" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-58" >Jump up^{</a></b> }<a href="https://en.wikipedia.org/wiki/Herbert_Goldstein" title="Herbert Goldstein" >Goldstein, H.</a> (1950). <i>Classical Mechanics</i>, Addison-Wesley,<a href="https://en.wikipedia.org/wiki/Special:BookSources/0201025108" class="internal mw-magiclink-isbn" >ISBN 0-201-02510-8</a>.</li>

<li id="cite_note-59" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-59" >Jump up^{</a></b> }"There is as yet no logically consistent and complete relativistic quantum field theory.", p. 4.  — V. B. Berestetskii, <a href="https://en.wikipedia.org/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz" >E. M. Lifshitz</a>, L P Pitaevskii (1971). J. B. Sykes, J. S. Bell (translators).<i>Relativistic Quantum Theory</i> <b>4, part I</b>. <i>Course of Theoretical Physics (Landau and Lifshitz)</i> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0080160255" class="internal mw-magiclink-isbn" >ISBN 0-08-016025-5</a></li>

<li id="cite_note-60" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-60" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.damtp.cam.ac.uk/strings02/dirac/hawking/" >"Stephen Hawking; Gödel and the end of physics"</a>. <i>cam.ac.uk</i>. Retrieved 11 September 2015.</li>

<li id="cite_note-61" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-61" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://books.google.com/books?id=6a-agBFWuyQC&pg=PA61&lpg=PA61&dq=most+accurately+tested+physical+theory+pulsar&source=bl&ots=rsrPvowECD&sig=S9MaY509U0pOgF86h2uvcpQQ_wU&hl=en&sa=X&ei=TwkTT6j9M8LXiAKSwNjTDQ&sqi=2&ved=0CE0Q6AEwBw#v=onepage&q=most%20accurately%20tested%20physical%20theory%20pulsar&f=false" >"The Nature of Space and Time"</a>. <i>google.com</i>. Retrieved11 September 2015.</li>

<li id="cite_note-62" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-62" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://latticeqcd.blogspot.com/2005/06/most-accurate-theory-we-have.html" >"Life on the lattice: The most accurate theory we have"</a>. Latticeqcd.blogspot.com. 2005-06-03. Retrieved 2010-10-15.</li>

<li id="cite_note-63" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-63" >Jump up^{</a></b> Parker}, B. (1993). <i>Overcoming some of the problems</i>. pp. 259–279.</li>

<li id="cite_note-64" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-64" >Jump up^{</a></b>&#160};The Character of Physical Law (1965) Ch. 6; also quoted in The New Quantum Universe (2003), by Tony Hey and Patrick Walters</li>

<li id="cite_note-65" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-65" >Jump up^{</a></b> Weinberg}, S. <a rel="nofollow" class="external text" href="http://arxiv.org/abs/1109.6462" >"Collapse of the State Vector"</a>, Phys. Rev. A 85, 062116 (2012).</li>

<li id="cite_note-66" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-66" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/qm-action-distance/" >"Action at a Distance in Quantum Mechanics (Stanford Encyclopedia of Philosophy)"</a>. Plato.stanford.edu. 2007-01-26. Retrieved 2012-08-18.</li>

<li id="cite_note-67" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-67" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/qm-everett/" >"Everett's Relative-State Formulation of Quantum Mechanics (Stanford Encyclopedia of Philosophy)"</a>. Plato.stanford.edu. Retrieved 2012-08-18.</li>

<li id="cite_note-Cramer-68" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Cramer_68-0" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://www.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html" >The Transactional Interpretation of Quantum Mechanics</a> by John Cramer. <i>Reviews of Modern Physics</i> 58, 647-688, July (1986)</li>

<li id="cite_note-69" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-69" >Jump up^{</a></b> See}, for example, <a href="https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics" title="The Feynman Lectures on Physics" >the Feynman Lectures on Physics</a> for some of the technological applications which use quantum mechanics, e.g., <a href="https://en.wikipedia.org/wiki/Transistor" title="Transistor" >transistors</a> (vol <b>III</b>, pp. 14-11 ff), <a href="https://en.wikipedia.org/wiki/Integrated_circuit" title="Integrated circuit" >integrated circuits</a>, which are follow-on technology in solid-state physics (vol <b>II</b>, pp. 8-6), and <a href="https://en.wikipedia.org/wiki/Laser" title="Laser" >lasers</a> (vol <b>III</b>, pp. 9-13).</li>

<li id="cite_note-70" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-70" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://books.google.com/books?id=vdXU6SD4_UYC" ><i>Introduction to Quantum Mechanics with Applications to Chemistry - Linus Pauling, E. Bright Wilson</i></a>. 1985-03-01.<a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/9780486648712" title="Special:BookSources/9780486648712" >9780486648712</a>. Retrieved 2012-08-18.</li>

<li id="cite_note-71" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-71" >Jump up^{</a></b> Anderson}, Mark (2009-01-13). <a rel="nofollow" class="external text" href="http://discovermagazine.com/2009/feb/13-is-quantum-mechanics-controlling-your-thoughts/article_view?b_start:int=1&-C" >"Is Quantum Mechanics Controlling Your Thoughts? | Subatomic Particles"</a>. DISCOVER Magazine. Retrieved 2012-08-18.</li>

<li id="cite_note-72" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-72" >Jump up^{</a></b> }<a rel="nofollow" class="external text" href="http://physicsworld.com/cws/article/news/41632" >"Quantum mechanics boosts photosynthesis"</a>. physicsworld.com. Retrieved 2010-10-23.</li>

<li id="cite_note-73" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-73" >Jump up^{</a></b> Davies}, P. C. W.; Betts, David S. (1984). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=XRyHCrGNstoC&pg=PA79" ><i>Quantum Mechanics, Second edition</i></a>. Chapman and Hall. p. 79. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-7487-4446-0" title="Special:BookSources/0-7487-4446-0" >0-7487-4446-0</a>., <a rel="nofollow" class="external text" href="http://books.google.com/books?id=XRyHCrGNstoC&pg=PA79" >Chapter 6, p. 79</a></li>

<li id="cite_note-74" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-74" >Jump up^{</a></b> Baofu}, Peter (2007-12-31). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=tKm-Ekwke_UC" ><i>The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos</i></a>.<a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/9789812708991" title="Special:BookSources/9789812708991" >9789812708991</a>. Retrieved 2012-08-18.</li>

<li id="cite_note-75" ><b><a href="https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-75" >Jump up^{</a></b>&#160};Derivation of particle in a box, <a rel="nofollow" class="external text" href="http://chemistry.tidalswan.com/index.php?title=Quantum_Mechanics" >chemistry.tidalswan.com</a></li>

</ol>

</div>

<h2 >References[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=20" title="Edit section: References" >edit</a>]</h2>

<p >The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus.</p>

<ul >

<li >Chester, Marvin (1987) <i>Primer of Quantum Mechanics</i>. John Wiley. <a href="https://en.wikipedia.org/wiki/Special:BookSources/0486428788" class="internal mw-magiclink-isbn" >ISBN 0-486-42878-8</a></li>

<li >Cox, Brian; Forshaw, Jeff (2011). <i><a href="https://en.wikipedia.org/wiki/The_Quantum_Universe" title="The Quantum Universe" >The Quantum Universe: Everything That Can Happen Does Happen</a>:</i>. Allen Lane. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/1-84614-432-9" title="Special:BookSources/1-84614-432-9" >1-84614-432-9</a>.</li>

<li ><a href="https://en.wikipedia.org/wiki/Richard_Feynman" title="Richard Feynman" >Richard Feynman</a>, 1985. <i><a href="https://en.wikipedia.org/wiki/QED:_The_Strange_Theory_of_Light_and_Matter" title="QED: The Strange Theory of Light and Matter" >QED: The Strange Theory of Light and Matter</a></i>, <a href="https://en.wikipedia.org/wiki/Princeton_University_Press" title="Princeton University Press" >Princeton University Press</a>. <a href="https://en.wikipedia.org/wiki/Special:BookSources/0691083886" class="internal mw-magiclink-isbn" >ISBN 0-691-08388-6</a>. Four elementary lectures on <a href="https://en.wikipedia.org/wiki/Quantum_electrodynamics" title="Quantum electrodynamics" >quantum electrodynamics</a> and <a href="https://en.wikipedia.org/wiki/Quantum_field_theory" title="Quantum field theory" >quantum field theory</a>, yet containing many insights for the expert.</li>

<li ><a href="https://en.wikipedia.org/wiki/Giancarlo_Ghirardi" title="Giancarlo Ghirardi" >Ghirardi, GianCarlo</a>, 2004. <i>Sneaking a Look at God's Cards</i>, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using <a href="https://en.wikipedia.org/wiki/Algebra" title="Algebra" >algebra</a>, <a href="https://en.wikipedia.org/wiki/Trigonometry" title="Trigonometry" >trigonometry</a>, and <a href="https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation" >bra–ket notation</a> can be passed over on a first reading.</li>

<li ><a href="https://en.wikipedia.org/wiki/N._David_Mermin" title="N. David Mermin" class="mw-redirect" >N. David Mermin</a>, 1990, "Spooky actions at a distance: mysteries of the QT" in his <i>Boojums all the way through</i>. <a href="https://en.wikipedia.org/wiki/Cambridge_University_Press" title="Cambridge University Press" >Cambridge University Press</a>: 110-76.</li>

<li ><a href="https://en.wikipedia.org/wiki/Victor_Stenger" title="Victor Stenger" class="mw-redirect" >Victor Stenger</a>, 2000. <i>Timeless Reality: Symmetry, Simplicity, and Multiple Universes</i>. Buffalo NY: Prometheus Books. Chpts. 5-8. Includes<a href="https://en.wikipedia.org/wiki/Cosmological" title="Cosmological" class="mw-redirect" >cosmological</a> and <a href="https://en.wikipedia.org/wiki/Philosophical" title="Philosophical" class="mw-redirect" >philosophical</a> considerations.</li>

</ul>

<p >More technical:</p>

<ul >

<li ><a href="https://en.wikipedia.org/wiki/Bryce_DeWitt" title="Bryce DeWitt" >Bryce DeWitt</a>, R. Neill Graham, eds., 1973. <i>The Many-Worlds Interpretation of Quantum Mechanics</i>, Princeton Series in Physics, <a href="https://en.wikipedia.org/wiki/Princeton_University_Press" title="Princeton University Press" >Princeton University Press</a>. <a href="https://en.wikipedia.org/wiki/Special:BookSources/069108131X" class="internal mw-magiclink-isbn" >ISBN 0-691-08131-X</a></li>

<li ><a href="https://en.wikipedia.org/wiki/P.A.M._Dirac" title="P. A. M. Dirac" class="mw-redirect" >Dirac, P. A. M.</a> (1930). <i><a href="https://en.wikipedia.org/wiki/Principles_of_Quantum_Mechanics" title="Principles of Quantum Mechanics" class="mw-redirect" >The Principles of Quantum Mechanics</a></i>. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-19-852011-5" title="Special:BookSources/0-19-852011-5" >0-19-852011-5</a>. The beginning chapters make up a very clear and comprehensible introduction.</li>

<li ><a href="https://en.wikipedia.org/wiki/Hugh_Everett" title="Hugh Everett" class="mw-redirect" >Hugh Everett</a>, 1957, "Relative State Formulation of Quantum Mechanics", <i>Reviews of Modern Physics</i> 29: 454-62.</li>

<li ><a href="https://en.wikipedia.org/wiki/Richard_Feynman" title="Richard Feynman" >Feynman, Richard P.</a>; <a href="https://en.wikipedia.org/wiki/Robert_B._Leighton" title="Robert B. Leighton" >Leighton, Robert B.</a>; Sands, Matthew (1965). <i><a href="https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics" title="The Feynman Lectures on Physics" >The Feynman Lectures on Physics</a></i> <b>1–3</b>. <a href="https://en.wikipedia.org/wiki/Addison-Wesley" title="Addison-Wesley" >Addison-Wesley</a>. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-7382-0008-5" title="Special:BookSources/0-7382-0008-5" >0-7382-0008-5</a>.</li>

<li >Griffiths, David J. (2004). <i>Introduction to Quantum Mechanics (2nd ed.)</i>. Prentice Hall. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-13-111892-7" title="Special:BookSources/0-13-111892-7" >0-13-111892-7</a>. <a href="https://en.wikipedia.org/wiki/OCLC" title="OCLC" >OCLC</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/40251748" >40251748</a>. A standard undergraduate text.</li>

<li ><a href="https://en.wikipedia.org/wiki/Max_Jammer" title="Max Jammer" >Max Jammer</a>, 1966. <i>The Conceptual Development of Quantum Mechanics</i>. McGraw Hill.</li>

<li ><a href="https://en.wikipedia.org/wiki/Hagen_Kleinert" title="Hagen Kleinert" >Hagen Kleinert</a>, 2004. <i>Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets</i>, 3rd ed. Singapore: World Scientific. <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/_kleinert/b5" >Draft of 4th edition.</a></li>

<li >Gunther Ludwig, 1968. <i>Wave Mechanics</i>. London: Pergamon Press. <a href="https://en.wikipedia.org/wiki/Special:BookSources/0082032041" class="internal mw-magiclink-isbn" >ISBN 0-08-203204-1</a></li>

<li ><a href="https://en.wikipedia.org/wiki/George_Mackey" title="George Mackey" >George Mackey</a> (2004). <i>The mathematical foundations of quantum mechanics</i>. Dover Publications. <a href="https://en.wikipedia.org/wiki/Special:BookSources/0486435172" class="internal mw-magiclink-isbn" >ISBN 0-486-43517-2</a>.</li>

<li ><a href="https://en.wikipedia.org/wiki/Albert_Messiah" title="Albert Messiah" >Albert Messiah</a>, 1966. <i>Quantum Mechanics</i> (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III.</li>

<li ><a href="https://en.wikipedia.org/wiki/Roland_Omn%C3%A8s" title="Roland Omnès" >Omnès, Roland</a> (1999). <i>Understanding Quantum Mechanics</i>. Princeton University Press. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-691-00435-8" title="Special:BookSources/0-691-00435-8" >0-691-00435-8</a>. <a href="https://en.wikipedia.org/wiki/OCLC" title="OCLC" >OCLC</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/39849482" >39849482</a>.</li>

<li >Scerri, Eric R., 2006. <i>The <a href="https://en.wikipedia.org/wiki/Periodic_Table" title="Periodic Table" class="mw-redirect" >Periodic Table</a>: Its Story and Its Significance</i>. Oxford University Press. Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. <a href="https://en.wikipedia.org/wiki/Special:BookSources/0195305736" class="internal mw-magiclink-isbn" >ISBN 0-19-530573-6</a></li>

<li ><a href="https://en.wikipedia.org/wiki/Transnational_College_of_Lex" title="Transnational College of Lex" class="mw-redirect" >Transnational College of Lex</a> (1996). <i>What is Quantum Mechanics? A Physics Adventure</i>. Language Research Foundation, Boston. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-9643504-1-6" title="Special:BookSources/0-9643504-1-6" >0-9643504-1-6</a>. <a href="https://en.wikipedia.org/wiki/OCLC" title="OCLC" >OCLC</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/34661512" >34661512</a>.</li>

<li ><a href="https://en.wikipedia.org/wiki/John_von_Neumann" title="John von Neumann" >von Neumann, John</a> (1955). <i>Mathematical Foundations of Quantum Mechanics</i>. Princeton University Press. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-691-02893-1" title="Special:BookSources/0-691-02893-1" >0-691-02893-1</a>.</li>

<li ><a href="https://en.wikipedia.org/wiki/Hermann_Weyl" title="Hermann Weyl" >Hermann Weyl</a>, 1950. <i>The Theory of Groups and Quantum Mechanics</i>, Dover Publications.</li>

<li >D. Greenberger, K. Hentschel, F. Weinert, eds., 2009. <i>Compendium of quantum physics, Concepts, experiments, history and philosophy</i>, Springer-Verlag, Berlin, Heidelberg.</li>

</ul>

</div>

<h2 >Further reading[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=21" title="Edit section: Further reading" >edit</a>]</h2>

<ul >

<li >Bernstein, Jeremy (2009). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=j0Me3brYOL0C&printsec=frontcover" ><i>Quantum Leaps</i></a>. Cambridge, Massachusetts: Belknap Press of Harvard University Press. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/978-0-674-03541-6" title="Special:BookSources/978-0-674-03541-6" >978-0-674-03541-6</a>.</li>

<li ><a href="https://en.wikipedia.org/wiki/David_Bohm" title="David Bohm" >Bohm, David</a> (1989). <i>Quantum Theory</i>. <a href="https://en.wikipedia.org/wiki/Dover_Publications" title="Dover Publications" >Dover Publications</a>. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-486-65969-0" title="Special:BookSources/0-486-65969-0" >0-486-65969-0</a>.</li>

<li >Eisberg, Robert; <a href="https://en.wikipedia.org/wiki/Robert_Resnick" title="Robert Resnick" >Resnick, Robert</a> (1985). <i>Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.)</i>. Wiley.<a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-471-87373-X" title="Special:BookSources/0-471-87373-X" >0-471-87373-X</a>.</li>

<li ><a href="https://en.wikipedia.org/wiki/Liboff,_Richard_L." title="Liboff, Richard L." class="mw-redirect" >Liboff, Richard L.</a> (2002). <i>Introductory Quantum Mechanics</i>. Addison-Wesley. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-8053-8714-5" title="Special:BookSources/0-8053-8714-5" >0-8053-8714-5</a>.</li>

<li >Merzbacher, Eugen (1998). <i>Quantum Mechanics</i>. Wiley, John & Sons, Inc. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-471-88702-1" title="Special:BookSources/0-471-88702-1" >0-471-88702-1</a>.</li>

<li ><a href="https://en.wikipedia.org/wiki/J.J.Sakurai" title="J. J. Sakurai" >Sakurai, J. J.</a> (1994). <i>Modern Quantum Mechanics</i>. Addison Wesley. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-201-53929-2" title="Special:BookSources/0-201-53929-2" >0-201-53929-2</a>.</li>

<li >Shankar, R. (1994). <i>Principles of Quantum Mechanics</i>. Springer. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/0-306-44790-8" title="Special:BookSources/0-306-44790-8" >0-306-44790-8</a>.</li>

<li >Stone, A. Douglas (2013). <i>Einstein and the Quantum</i>. Princeton University Press. <a href="https://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number" >ISBN</a> <a href="https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-13968-5" title="Special:BookSources/978-0-691-13968-5" >978-0-691-13968-5</a>.</li>

<li ><a href="https://en.wikipedia.org/wiki/Martinus_J.G.Veltman" title="Martinus J. G. Veltman" >Martinus J. G. Veltman</a>, 2003 <i>Facts and Mysteries in Elementary Particle Physics</i>.</li>

</ul>

<h2 >External links[<a href="https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=22" title="Edit section: External links" >edit</a>]</h2>

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<div >Find more about

<b>Quantum mechanics</b>

at Wikipedia's <a href="https://en.wikipedia.org/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects" >sister projects</a></div>

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<td ><a href="https://en.wiktionary.org/wiki/Special:Search/Quantum_mechanics" class="extiw" title="wikt:Special:Search/Quantum mechanics" >Definitions</a> from Wiktionary</td>

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<td><a href="https://commons.wikimedia.org/wiki/Special:Search/Quantum_mechanics" class="extiw" title="c:Special:Search/Quantum mechanics" >Media</a> from Commons</td>

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<td><a href="https://en.wikinews.org/wiki/Special:Search/Quantum_mechanics" class="extiw" title="n:Special:Search/Quantum mechanics" >News stories</a> from Wikinews</td>

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<td><a href="https://en.wikiquote.org/wiki/Special:Search/Quantum_mechanics" class="extiw" title="q:Special:Search/Quantum mechanics" >Quotations</a> from Wikiquote</td>

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<td><a href="https://en.wikisource.org/wiki/Quantum_mechanics" class="extiw" title="s:Quantum mechanics" >Source texts</a> from Wikisource</td>

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<td><a href="https://en.wikibooks.org/wiki/Special:Search/Quantum_mechanics" class="extiw" title="b:Special:Search/Quantum mechanics" >Textbooks</a> from Wikibooks</td>

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<td><a href="https://en.wikiversity.org/wiki/Special:Search/Quantum_mechanics" class="extiw" title="v:Special:Search/Quantum mechanics" >Learning resources</a> from Wikiversity</td>

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<ul >

<li ><a rel="nofollow" class="external text" href="http://www.toutestquantique.fr/" >3D animations, applications and research for basic quantum effects</a> (animations also available in <a class="external text" href="https://commons.wikimedia.org/wiki/Special:Contributions/Jubobroff" >commons.wikimedia.org</a> (Université paris Sud))</li>

<li ><a rel="nofollow" class="external text" href="http://oyc.yale.edu/sites/default/files/notes_quantum_cookbook.pdf" >Quantum Cook Book</a> by R. Shankar, Open Yale PHYS 201 material (4pp)</li>

<li ><a rel="nofollow" class="external text" href="http://www.lightandmatter.com/lm/" >The Modern Revolution in Physics</a> - an online textbook.</li>

<li >J. O'Connor and E. F. Robertson: <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/The_Quantum_age_begins.html" >A history of quantum mechanics.</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.quantiki.org/wiki/index.php/Introduction_to_Quantum_Theory" >Introduction to Quantum Theory at Quantiki.</a></li>

<li ><a rel="nofollow" class="external text" href="http://bethe.cornell.edu/" >Quantum Physics Made Relatively Simple</a>: three video lectures by <a href="https://en.wikipedia.org/wiki/Hans_Bethe" title="Hans Bethe" >Hans Bethe</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.freebookcentre.net/Physics/Quantum-Mechanics-Books.html" >Quantum Mechanics Books Collection</a>: Collection of free books</li>

</ul>

<dl >

<dt >Course material</dt>

</dl>

<ul >

<li ><a rel="nofollow" class="external text" href="http://qpdb.eu/" >Quantum Physics Database - Fundamentals and Historical Background of Quantum Theory.</a></li>

<li ><a rel="nofollow" class="external text" href="http://arxiv.org/abs/quant-ph/0605180" >Doron Cohen: Lecture notes in Quantum Mechanics (comprehensive, with advanced topics).</a></li>

<li ><a href="https://en.wikipedia.org/wiki/MIT_OpenCourseWare" title="MIT OpenCourseWare" >MIT OpenCourseWare</a>: <a rel="nofollow" class="external text" href="http://ocw.mit.edu/OcwWeb/Chemistry/index.htm" >Chemistry</a>.</li>

<li >MIT OpenCourseWare: <a rel="nofollow" class="external text" href="http://ocw.mit.edu/OcwWeb/Physics/index.htm" >Physics</a>. See <a rel="nofollow" class="external text" href="http://ocw.mit.edu/OcwWeb/Physics/8-04Spring-2006/CourseHome/index.htm" >8.04</a></li>

<li ><a rel="nofollow" class="external text" href="https://www.youtube.com/stanford#g/c/84C10A9CB1D13841" >Stanford Continuing Education PHY 25: Quantum Mechanics</a> by <a href="https://en.wikipedia.org/wiki/Leonard_Susskind" title="Leonard Susskind" >Leonard Susskind</a>, see <a rel="nofollow" class="external text" href="http://continuingstudies.stanford.edu/courses/course.php?cid=20072_PHY%2025" >course description</a> Fall 2007</li>

<li ><a rel="nofollow" class="external text" href="http://www.physics.csbsju.edu/QM/" >5½ Examples in Quantum Mechanics</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.imperial.ac.uk/quantuminformation/qi/tutorials" >Imperial College Quantum Mechanics Course.</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.sparknotes.com/testprep/books/sat2/physics/chapter19section3.rhtml" >Spark Notes - Quantum Physics.</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.quantum-physics.polytechnique.fr/" >Quantum Physics Online : interactive introduction to quantum mechanics (RS applets).</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.didaktik.physik.uni-erlangen.de/quantumlab/english/index.html" >Experiments to the foundations of quantum physics with single photons.</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.nanohub.org/topics/AQME" >AQME</a> : Advancing Quantum Mechanics for Engineers — by T.Barzso, D.Vasileska and G.Klimeck online learning resource with simulation tools on <a href="https://en.wikipedia.org/wiki/Nanohub" title="Nanohub" class="mw-redirect" >nanohub</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.lsr.ph.ic.ac.uk/_{plenio/lecture.pdf}" >Quantum Mechanics</a> by Martin Plenio</li>

<li ><a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/qm/389.pdf" >Quantum Mechanics</a> by Richard Fitzpatrick</li>

<li ><a rel="nofollow" class="external text" href="http://nanohub.org/resources/2039" >Online course on <i>Quantum Transport</i></a></li>

</ul>

<dl >

</dl>

<ul >

<li ><a rel="nofollow" class="external text" href="http://www.hedweb.com/manworld.htm" >Many-worlds or relative-state interpretation.</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.mtnmath.com/faq/meas-qm.html" >Measurement in Quantum mechanics.</a></li>

</ul>

<dl >

<dt >Media</dt>

</dl>

<ul >

<li ><a rel="nofollow" class="external text" href="http://oyc.yale.edu/physics/phys-201#sessions" >PHYS 201: Fundamentals of Physics II</a> by Ramamurti Shankar, Open Yale Course</li>

<li ><a rel="nofollow" class="external text" href="https://www.youtube.com/view_play_list?p=84C10A9CB1D13841" >Lectures on Quantum Mechanics</a> by <a href="https://en.wikipedia.org/wiki/Leonard_Susskind" title="Leonard Susskind" >Leonard Susskind</a></li>

<li ><a rel="nofollow" class="external text" href="http://www.newscientist.com/channel/fundamentals/quantum-world" >Everything you wanted to know about the quantum world</a> — archive of articles from <i><a href="https://en.wikipedia.org/wiki/New_Scientist" title="New Scientist" >New Scientist</a></i>.</li>

<li ><a rel="nofollow" class="external text" href="http://www.sciencedaily.com/news/matter_energy/quantum_physics/" >Quantum Physics Research</a> from <i><a href="https://en.wikipedia.org/wiki/Science_Daily" title="Science Daily" >Science Daily</a></i></li>

<li >Overbye, Dennis (December 27, 2005). <a rel="nofollow" class="external text" href="http://www.nytimes.com/2005/12/27/science/27eins.html?scp=1&sq=quantum%20trickery&st=cse" >"Quantum Trickery: Testing Einstein's Strangest Theory"</a>. <i>The New York Times</i>. Retrieved April 12, 2010.</li>

<li ><a rel="nofollow" class="external text" href="http://www.astronomycast.com/physics/ep-138-quantum-mechanics/" >Audio: Astronomy Cast</a> Quantum Mechanics — June 2009. <a href="https://en.wikipedia.org/wiki/Fraser_Cain" title="Fraser Cain" class="mw-redirect" >Fraser Cain</a> interviews <a href="https://en.wikipedia.org/wiki/Pamela_L._Gay" title="Pamela L. Gay" >Pamela L. Gay</a>.</li>

</ul>

<dl >

<dt >Philosophy</dt>

</dl>

<ul >

<li ><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/qm" >"Quantum Mechanics"</a> entry by Jenann Ismael in the <i><a href="https://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy" >Stanford Encyclopedia of Philosophy</a></i></li>

<li ><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/qt-measurement" >"Measurement in Quantum Theory"</a> entry by Henry Krips in the <i><a href="https://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy" >Stanford Encyclopedia of Philosophy</a></i></li>

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