Classical versus Quantum

954 views Apr 17, 2023

Quantum mechanics (QM) is a powerful yet perplexing development of 20th century physics that forces us to rethink many traditional beliefs about the world around us. Is physics ultimately reduced to probabilities behind which there is no deterministic unfolding, as the Copenhagen school would have us believe? Or is there a possibility of some kind of non-local hidden variables theory, as Einstein hope and de Broglie and Bohm tried to develop? How should we try to understand this unintuitive but crucial explanation of the microscopic world, which is now at the core of our understanding also of chemistry, and increasingly also of astrophysics? In this series, we put such questions a bit to the side, and inquire rather in how we can find aspects, or reflections, of the quantum world in the otherwise predictable and logical world of pure mathematics (at least potentially -- if we also put all those logical weaknesses and ambiguities of modern set theory and real numbers aside!). It turns out that the fascinating relations between classical physics and quantum physics have parallels in the pure mathematics world. It is our goal to be able to set out some of these fascinating parallels, and perhaps even to advance our understanding of some of them. Along the way, we will find many questions, challenges and even research projects that can inspire amateur researchers. In this video we give an introduction to the general direction of this series, describing briefly some of the early history of the subject, featuring the remarkable insights of Planck, Einstein, Bohr, Heisenberg, Schroedinger, Dirac and many others. And we introduce the famous two slit experiment which in some sense is the granddaddy of all speculation about QM, boiling down the essence of the "measurement / observer" problem.

Introduction

0:00

Hello everyone I'm Norman Wildberger thanks for joining me this is a new series on classical and

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quantum mathematics we might even say classical versus quantum maths because there is a bit of

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a tension or a bit of a dichotomy between the mathematics associated maybe more with

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classical physics and with 20th century quantum mechanics now quantum mechanics is really a

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remarkable development it really ranks up there with the 17th century introduction of Newtonian

0:33

mechanics as sort of a pivotal point in in the history of physics and it happened in the first

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half of the 20th century but its influence is still very much with us today and it it really

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explains so many aspects of modern technology and and chemistry and Modern Life but there's

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undeniably a mysterious strongly confusing aspect to it because it appears that what's

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going on at the quantum level is so completely different and unusual and almost incomprehensible

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to us and so there's still a lot of debate in physics circles as to the you know the meaning of quantum mechanics if indeed that even makes sense so I'm interested in that and

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I'm interested in that question but as a peer mathematician I'm also interested in how that

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that challenge of understanding quantum physics relates or connects with mathematics so it turns

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out that there's a lot of ways in which this dichotomy between classical and quantum physics

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is reflected in the mathematical world so I'm proposing to try to explore this range of of

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topics and uh and maybe try to get at the physics from a mathematical point of view so trying to

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shed some light on what's going on starting with purely the mathematics so we're going to think

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of mathematics as a potential laboratory in which we might think about some of these physical ideas

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so let's start out with a brief description of some of the differences between the classical world and the quantum world starting originally from physics but we can see that a lot of these

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distinctions are also somehow manifest in the mathematical world so classical

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somehow collected with continuous phenomena Quantum more associated with discrete phenomena

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commutative classical mathematics typically grounded in commutative mathematics quantum

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physics in non-commutative mathematics there's an exactness to the classical World which is

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lost in the quantum World becomes more approximate and correspondingly there's a predictable aspect to classical thinking which becomes more probabilistic in the quantum world

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In classical physics we concern ourselves with little particles and Newton's laws

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while in quantum physics waves start to play at least an equal role in the description of things

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there's some sense in which we're going from finite dimensional stories to infinite dimensional stories and going from local things to non-local things so

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especially having to say with entanglement of particles over vast distances and there is this

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um almost philosophical role of the you know the The Observer and the measurement problem and what

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um what effect does observation have on the story in classical physics classical mathematics The

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Observer is maybe part of the story but you can sort of identify or separate the role of the

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Observer from what factually being observed not so easily in Quantum mathematics or quantum physics

Can we understand QM

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anyway so can we understand quantum mechanics well that's a big challenge Feynmann I think

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famously said you know that nobody basically understands quantum mechanics and I think that situation hasn't changed all that much over the decades there are actually a quite a wide range

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of opinions that the physicists have about the meaning of quantum mechanics and there's also

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you know widely different views as to what's really going on there but I'm not necessarily

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wanting to come to some different conclusion about that I think that might be too ambitious although we might aim towards that but I'm sort of interested in the mathematical side of this entire

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story so I believe that there is a mathematical side to the physical world of quantum mechanics

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and that it manifests itself in all kinds of interesting ways and that there's really a

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very powerful potential for putting some of these things together and trying to get an overview of

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quantum mechanics from a mathematical point of view which is not to say that we're going to go start doing physics and looking at the mathematics involved in the physics rather I want to look at

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those aspects of mathematics pure mathematics which ultimately have a strong sort of quantum

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aspect to them and I want to investigate those things and try to understand them more deeply and

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more carefully and maybe or maybe not but maybe that will shed some light on the physics as well

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so I claim that they're in fact there is a Quantum side to mathematics and

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that it's actually logically independent of modern physics you see that it's rather an interesting kind of coincidence that in the 1920s and

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1930s when quantum mechanics in its somewhat modern form was was really being developed

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in this remarkable burst of Brilliance due to quite a lot of people correspondingly pretty

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well at the same time there were some really important investigations being made in in pure mathematics in the direction of Lie groups and the algebras and and representations Associated

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to names like E Cartan and Hermann Weyl and sort of spectacularly culminating in vials character

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formula for compact simple lead groups and it turns out after the fact like decades later

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we started to realize that Weyl's theory in some sense could be reframed to make it very close or

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you know having a lot of parallels with this sort of transition from classical mechanics to quantum

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mechanics that was sort of being developed at the same time as vile was was working on this pure

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mathematics side while it's a very interesting character in this story because he actually plays a an important role in the development and the understanding the mathematical side

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of quantum mechanics so he has this his feet in both camps we'll have to talk about Hermann bile

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um but anyway I'm I'm interested in in this sort of coincidence and for me it's

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um it's at least theoretically possible that it could have been if history had been slightly

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different that we could have sort of discovered quantum mechanics from a mathematical point

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of view quite earlier and independently of the physicists so that that could have happened and

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so it's interesting to think about you know the logical structure mathematics is not necessarily

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just a tool that the the physicists are using it can also in fact be a separate laboratory

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for exploring issues which are directly related to this physics but have a completely independent

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existence in the world of pure mathematics that's what we're interested in in this series

Mathematical areas close to QM

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so what are some of these mathematical areas that are pretty close to Quantum Theory well here is

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a rough list of them and probably I could come up with some more and maybe we will come up with some more during the course of these investigations but uh first one is the theory of symplectic manifolds

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this is geometry and associated Lagrangian and Hamiltonian formulations of Newtonian mechanics

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in what's also sort of a remarkable coincidence so Newton developed his Laws of Motion using his

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his newborn calculus in the 17th century and then eventually like a hundred years later Lagrange came around Euler also contributed and sort of reformulated Newton's Laws now called

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Lagrangian point of view based on principle of least action which turns out also to be really crucial for modern physics and then the Hamilton later on came around and gave another

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kind of reformulation which probably LaGrange had already sort of put his finger on in some sense

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but it turns out that when quantum mechanics then arose in the 20th century in the first

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part of the 20th century now these reformulations of Newton's mechanics were just exactly somehow

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what the physicists needed they were the sort of mathematical tools that they could then use to try

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to formulate what's going on in quantum mechanics so there's a sense in which there's already a sort

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of coincidence here at this level quite a bit earlier before real quantum mechanics developed

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those entire areas of modern analysis having to do with operators operator algebra's Hilbert

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spaces spectral Theory okay that are very close to Quantum Mechanics indeed are important tools

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for certain approaches to Quantum Mechanics but also I think quantum mechanics sort of naturally has something to say about them there's a Q Series which is a branch of algebra or slash

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number Theory or slash combinatorics which often also has a sort of a Quantum aspect

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there's some sense in which Information Theory as developed by quote Shannon also is an approach

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which these days is some people considered a very important way of thinking about a possible way of

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thinking about quantum mechanics for me this one here is is very prominent important I've

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mentioned it already the representation theory of Lie groups and Lee algebras which is a huge

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subject which just you know exploded in the 20th century and partially at least because of

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these Newfound strong connections with physics and indeed quantum mechanics but also with relativity

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so this is a very important topic even for the theoretical physicists these days they have to

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become quite conversant with this topic and very interestingly this is going back again

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to a while very interestingly there's aspects of quantum mechanics that are appearing in this in

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this Theory completely independently of any physics and I think that's not well enough

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appreciated and it's something I would like to talk about and try to explain there's a theory

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of quantum groups okay which obviously has some connection with issues of quantization

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um of course probability Theory measure Theory because quantum mechanics is some sense inherently

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probabilistic there's Fourier analysis there's a sort of corresponding uncertainty principle

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in in the Fourier Theory World which sort of reflects or is a version a mathematical

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version of the Heisenberg uncertainty principle there's something called geometric quantization

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associated with names of Kirillov and Kostant and other people and then there's another sort of

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geometrical Theory having to do with moment maps and then sort of a reverse arrow from quantization

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called dequantization that I've had some occasion to to think about so in in fact in some sense

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I've been thinking about these kinds of things for a long time because my PhD thesis which I did at Yale University under the supervision of Roger Howe that was many years ago 40 years ago

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probably or so the title of that was quantization and harmonic analysis on no potent Lie groups

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so already at that uh early time I was quite interested in quantization so it's been a kind

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of a recurring thing every so often I I think you know I'd like to think more about quantum mechanics and try to understand what's going on there especially the connections with mathematics

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so these are some of the areas that we will try to explore and I will try to to give you some

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kind of you know introduction to these things even though some of them are quite sophisticated but

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I'm going to try to do that in a way which is also consistent with my general philosophy of avoiding

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infinite things and avoiding axiomatics and set theory and indeed a lot of the limiting processes

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of analysis so I have to find alternate ways of you know stating what's going on with these things

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and I hope to do that also to promote the idea that it is possible to look at important parts of

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mathematics you know that are important in physics not necessarily with the usual status quo belief

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system that's currently wedded onto the education of most mathematicians these days now my thinking

Other series

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is that this series is also going to overlap and complement some of the other series that I

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currently have going such as the exceptional structures and Dynamics on graphs which is

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available for members on the wild egg Channel which is also the main place where this series

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is going to be so it's primarily for members of the wild egg mathematics courses YouTube channel

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there's also a series called diffusion symmetry we're talking about hyper groups okay which is

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at the insights into mathematics YouTube channel there's another members series called Explorations

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with Q series so that's also in the wild egg and there's are obviously going to be connections here

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and there's also uh none other intended series that I'm hoping to eventually start where I want

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to sort squarely deal with the representation theory of Lee algebra and Lee groups sort of as a separate topic which is a huge thing but I think it's well worth trying to give

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an exposition of that in you know what I consider sort of relatively simple-minded accessible terms

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so I'm hoping that there will be resonances here and that's some of the things that you could be

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exposed to in this series will will overlap maybe or at least resonate with some of the things that

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you're learning about in these other Series so there's a lot to do here lots of interesting

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things to to learn so in the rest of the video I want to attempt a very brief and cursory sort

History of QM

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of overview of the historical aspects of the development of quantum mechanics in the physical world so this is a really interesting story with you know that requires a huge explanation but I'm

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just going to give you an overview just sort of to position us the physics is not really my main

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aim at least you know in terms of exposition I want to explain the corresponding mathematics

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which is sort of adjacent to the physics but having at least some cursory understanding of the physics and the issues involved there is is going to be necessary to orient us

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so the story of quantum mechanics really begins pretty well exactly or in the year 1900 with the

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work of Max Planck but before that we have to understand that 19th century physics had these

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very big successes and physicists at that time were very confident that they more or less had a

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grip on you know the nature of reality that there were some things still to be figured out but more

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or less the the basic edifice was there and it was really based on perhaps three things the

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the Newtonian laws of mechanics that we've already talked about along with the the reformulations of

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LaGrange and Hamilton and then there was Maxwell's laws of electromagnetism having to do with light

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electric Fields magnetic fields Etc and there was also a theory of thermodynamics that had

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been developed Clausius, Boltzmann and others so there was a now a second law of Thermodynamics a

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very important law so these these things sort of provided a theoretical structure but it

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turned out that the 20th century you would see all these Revolutions in thinking many of them

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Associated ultimately with Albert Einstein special Theory relativity general theory of relativity but he was also a key contributor to Quantum Mechanics but you know I mean so basically

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all bets were off in the 20th century things were just became hugely more complicated and

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and weird and strange and the certainty that the 19th century physicists had had to be let go of

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okay so the story probably starts with Max Planck okay in his study of black body radiation and this

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is around the year 1900 so the story is we have some some box which is sort of at some uniform temperature so it's a black box and inside there's some gas which is uh you know

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at the same temperature as this thing and we have a little hole so this thing is emitting radiation so we can see how hot it is so see you know what what color it's it's hot

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in there maybe there's a red coming out so if we actually look at the frequencies of light coming out well there's a range of frequencies okay and the range of frequencies the intensity

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of the light for different frequencies turns out to have some graph that looks like this

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and um that's not really something that was predicted by the current theory so it was a bit

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of a struggle to figure out you know what's the shape of this thing here and why is it this way

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and um and that's what Planck did he found an equation for this thing okay in terms of

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the frequency f okay frequency of the of the the light being emitted or the radiation being emitted

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um the intensity I okay and then there were two constants that end up playing a huge role in

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this entire story um Planck's constant H which is some really small quantity in in traditional units

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and Boltzmann's constant having uh coming from thermodynamics and it's uh that this this curve

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is given by two h f cubed over X of HF over KT minus one where temperature is uh T so the whole

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curve depends on t now in his analysis of this and obtaining this equation uh plank had to make

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some some rather um unusual kind of postulates and he had to rethink things and eventually he

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came to this really fundamental realization that the energy of of light okay was in some sense sort

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of quantized so this is the original meaning of quanta meaning sort of uh some small quantity

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um of which integer multiples end up figuring so there's not an infinite divisibility like we might

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think so he found that that the energy was not in a continuous Spectrum it rather had discrete

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values in that those discrete values could be in terms of sort of some fundamental energy which was given by h f is this constant f is the frequency okay so this was sort of the original

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quanta of of of that started quantum mechanics and uh you know that's that was really quite a

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remarkable formula and a remarkable thinking that light had this different property than

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people had thought previously the next important contribution was by Albert Einstein in 1905 paper

Photoelectric effect

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on the photoelectric effect which ultimately gave him the Nobel Prize although 1905 was also

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the year of special relativity and in fact another paper uh this was like a super year in the history

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of physics and Einstein was looking at trying to explain a certain Effect called the photoelectric

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effect we have some some body with some electrons here and you shine light on this thing okay and

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that sort of dislodges electrons and then you have some voltage potential here and you you detect these electrons after they've emerged from this voltage separation so the um the voltage that

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you have here kind of allows you to determine the uh the energy of the electrons coming out so you

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would think that if you turned up the intensity of the light okay it's brighter okay that's more you

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know it's hotter there's more energy coming out uh of the of this light on this light beam you would

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think that that would then increase the energy of the electrons that are being emitted from this uh

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in this photoelectric effect but that's not what turns out to happen it turns out that the energies

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are all sort of dependent really more on the the frequency of light rather than the intensity of it

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okay in fact here is uh Planck's relation again E equals HF it's this e the energy associated

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with the frequency F that's contributing to the uh the energy that the electrons are emitted so

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Einstein's explanation of this was very novel okay so people had understood light as a you

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know a wave in terms of Maxwell's electromagnetic Theory and they were very comfortable with the

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wave mechanics of light in that context but Einstein realized that there was some aspect

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here in this situation which suggested that light actually had a particle nature as well as a wave

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nature and so we see what he's he's saying is that you know the the light can be viewed as actually

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having a particle nature so these little particles of light called These Days photons and the photon

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is coming down here and an individual Photon which has a certain frequency just like depending on the

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color of the light will have a certain energy that's that's the whole point of the the plant

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law that the frequency determines the energy of this Photon and it's that single Photon hitting a

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single electron and dislodging the electron sort of bumping it out that generates this electron

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movement and so that the energy transfer from the photon to the electron is exactly of this amount

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that means if you shine more light on the story what you're doing is you're shining more photons

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there's now a lot more photons but they all still have the same frequency and that means

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that you're going to have more electrons being emitted but the actual energy is not

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going to be increased it's still going to be just dependent on the frequency of the incoming photon

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so this led uh Einstein to this remarkable postulate that light even though it obviously

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has a wave-like aspect that it also has a particle aspect and that these photons you

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know directly carry planks Planck's relation in terms of their energy

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so this work of Einstein was not taken on board for quite some time it took people many years

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before they started to realize that Einstein actually was right here and it also provided

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important confirmation for Planck's law another really important development and sort of the pre

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the the earlier quantum theory so the quantum theory sort of has two parts of the earlier

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quantum theory up to about 1920s 19 mid 1920s and then the later development and the other

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sort of really important development in the earlier part is uh the atomic model of the

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atom that's largely due to Niels Bohr you know working on modifications of previous theories

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due to Thompson and Rutherford so his model of an atom say the hydrogen atom which is the simplest

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one you just have a single small nucleus and one electron zipping around it was very much like a

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planetary kind of story where you have a nucleus like playing the role of the Sun and you have the

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electron playing role of a planet and it's going around and he uh you know sort of adopted these

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points of view of Planck and Einstein to to assert that these um the shells or the orbits

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on which these electrons are going around are are discrete and they are constrained in energy

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by that their their energies have to be just integral multiples of this this basic thing and

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so that ended up uh having the consequence that that electrons actually end up in certain specific

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shells so if you have larger atoms with more electrons and bigger nuclei then the electrons

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end up organizing themselves in in little shells okay and the the inner shell has room for two

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electrons and then the next Shell has room for eight electrons uh the next Shell room for 18

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electrons and and so on and the pattern is that if you divide all these by two you get one four

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nine sixteen you get the square numbers so there's a nice sort of relationship between sort of this

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quantum number and and the number of electrons in one of these orbits and also he had a nice

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geometrical interpretation of the various radii of these shells so um this was I mean ultimately

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this model had to be modified or abandoned to a certain extent with the the newer quantum theory

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but this was still a very compelling model because it allowed one to explain all kinds of aspects

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of the the chemical periodic table in terms of the properties of various elements in terms of

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the configurations of the electrons and then you know whether the shells are full or not you know

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if a shell is completely full of electrons and there aren't any sort of extra electrons around in the next Shell then that's a kind of an inert right relatively stable element like an inert gas

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Etc so this was a very important kind of model even though ultimately it was superseded mechanics

New developments

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and then in the 1920s all kinds of new developments took place and the whole subject was elevated to a much more serious level and a lot of really very smart people made remarkable

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contributions here so it was like a gold in time for physics and so very briefly I'm just going to

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mention some things and you know in the course of the this series maybe I'll be able to talk

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a little bit more about some of these things so Wolfgang Pauley discovered something called

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an exclusion principle also sort of related to the spin of electron so it turns out an electron

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has some some internal kind of spinning also and this allowed him to somehow sort of partly

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explain what was going on with boars model of the uh of the atom the you know the reason why

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these shells couldn't support more than those number of electrons that we talked about because

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there was some principle that that different um particles had to of this kind had to uh sort of

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be in different states okay um Louis de brogli who was a French actually I think Prince okay

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some royalty in in France he also became physics royalty uh because he really took this particle

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wave Duality that was sort of you know initiated perhaps by Einstein and really made it much

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broader and you basically claimed that you know almost all particles maybe all particles have both

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a particle and a wave-like aspect okay the bigger the the particle the smaller the wavelengths

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or the harder to to see the associated wave but nevertheless this was a basic principle

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then Werner Heisenberg spectacularly came up with Matrix mechanics to try to understand you

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know what's going on with with these systems actually there were a couple of sort of almost equivalent formulations of the Dynamics of of a Quantum system uh also Erwin Schroedinger came

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up with the Schroedinger equation which is sort of the fundamental driver of how things evolve

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in the quantum world and Heisenberg's matrix mechanics turned out to be sort of something

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roughly equivalent to that and Paul Dirac British physicist who probably is is the 20th century uh

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second greatest physicist after uh Albert Albert Einstein um he uh also developed a sort of an

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equivalent algebraic version and sort of explained the the equivalence of these various stories and

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also come up with the first Quantum field Theory and many other things and um and I'll just also

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mention briefly Heisenberg's uncertainty principle which is also a sort of a key key aspect to this

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so there's a lot to this story obviously but I think that it's not so easy to to explain it

29:51

until we look at an example and there's a sort of a canonical example that you find in every text and every explanation and so we'll also have a look briefly at that too okay so you don't really

The two slit experiment

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appreciate the weirdness of quantum mechanics until you've spent a few weeks at least thinking

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about the two slit experiment which is sort of the canonical weird thing that people observed

30:13

and that is is still a conundrum I mean it's still absolutely remarkable and just baffles the mind

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you know um okay what is the two slit experiment it's a relatively simple thing that you can do

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in fact you know it's probably accessible at some level even in a very rudimentary way it involves a

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little a beam of of particles maybe photons or even electrons doesn't really matter what the

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particles are and you have a little piece of some surface here with two small slits in them okay two

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little slits and they're very close to each other I've pictured them as reasonably far apart but you should think of them as being really close to each other and there's a screen behind this

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this object the two slits okay and we're shining our particles having a beam of particles directed

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towards this thing and some of them are going through these slits and then they're showing

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up on the screen behind Okay that's that's what happens so the first thing that we might do is say

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well what happens if there's just one slit okay so maybe we cover the right most slits now this

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is only this one slit and we shine our our Beam on this slit well what happens is that um you're

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getting uh some some images on on the background screen and somewhat obviously sort of the the

31:39

the area directly behind the slit ends up getting the most hits okay but there is some kind of

31:45

diffraction that occurs okay so as you go a little bit further from the image right directly behind

31:51

the slit there's still some some more images so you you shine this light for a while and you count how many uh you know how many blips or where the light ends up emerging on the screen you find some

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pattern like this so some some distribution which is reasonably sort of uniform going down okay and

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the same thing happens if you do this thing with the other slit open so there's the right slit open

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you get a similar kind of pattern it's maybe a little bit shifted over but not much it'll still be the same and now the interesting thing is what happens if you have both slits open and you shine

32:25

this beam of light or electrons well then what happens is that you get an interference pattern

32:31

just as if these things were waves so maybe it's simpler to think of light you just have a beam of

32:37

light okay so we think of light as being waves and it's a kind of a classical experiment that

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if you have some situation like this the waves are going through and then they end up having some interference so on the screen behind you will have some bands of dark bands and light

32:52

bands light bands dark bands Etc and reflecting the interference and the superposition of these

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things okay so that's kind of almost a classical physics kind of experiment but now the interesting

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thing is that you can slow down the intensity of this light so you can you can dim the light by

33:14

having fewer and fewer photons being emitted in fact what you can end up doing is you can

33:23

end up slowing this thing so far down that you're basically just emitting single photons at a time

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now you might think well single Photon that's not going to be you know I can be able to see that well in fact it's human eye if you're a young person and you're in a really dark room

33:39

it's possible to actually see or register a single Photon so our eyes are pretty sensitive so that's

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not impossible to see so you just shine a single Photon on this now you're showing the photon in

33:51

this direction towards these slits you don't know which of the two slits it's going through but you can still record where it ends up you see a little blip okay and then you record

34:01

um where that lip was and then you shine another photons and record where it was and then you shine another Photon to record where it was and what you find is that this same interference pattern

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is preserved even though you're just shining one Photon at a time

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so there will be bands in here okay like the white here that represent no hits

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even though if we just had one of them closed and there was a single slit

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and we just shot one Photon at a time we still get a pattern like this we did many of them okay

34:40

and just recorded where all the images were we would find some some gradual uh decrease as we

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go from behind the uh the screen but in this case with two slits open we seem to have some

34:54

interference kind of behavior even though we're only sending in one Photon at a time

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so that's telling us that this single photon even though we're thinking of it as a particle

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has a wave like nature it's like the wave is going somehow through both slits now it turns

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out that's not just happening though with photons it's also happening with electrons okay electrons

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are different in fact it's happening pretty well with any any particle it's just going to be a

35:30

phenomena that you that you're going to see that even if you have a particle that doesn't seem to have anything to do with light and you shoot it at a two-slit thing like this and you look

35:37

to see what are the various possibilities behind you're going to see interference

35:43

okay so that's pretty weird but then to make it even weirder is what happens when you decide to

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try to determine which of the two slits this electron say is going through maybe you have

35:58

some detectors just at the slits that register whether the electron is going through the right

36:04

slit or the left slit so it lights up so it's just been green that means we've known that the electron is just that's gone through there has just gone through the right slit if we have these

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detectors in place and we can see left right right left then this interference pattern disappears

36:25

and we end up with just the same kind of pattern that we had in these cases here even though the two slits are still open

36:33

so if there's a wave-like aspect they can it can still do the wave-like thing but somehow the the

36:39

presence of this detector has the physicists say collapsed the wave function okay so these days

36:47

physicists mostly think that this particle even if it's an electron has an Associated wave the wave

36:54

function of the electron which is like a wave and it's somehow going through two slits But If

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You observe it if you make a measurement then that wave function collapses and it becomes just like a

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a particle and you end up with a sort of an ordinary distribution like this

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so explaining what's going on here has been a huge challenge for physicists and all kinds of

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very different explanations have emerged and maybe Feynman said something like well you

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know that almost all of quantum mechanics have somehow encoded in in this experiment

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you can see all kinds of things in it so that's that's really interesting um and so that's sort

37:41

of like a starting point for trying to figure out quantum mechanics but we'll see that on the mathematical side there's other things that we can do that other things that we can look at

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purely mathematical things that also shed some light on the quantum story eventually

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so this is Introduction to this series we'll we'll see where it goes and we'll uh we'll see

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if we can understand the mathematical side of this classical versus quantum dichotomy a little bit

38:11

uh more thoroughly I hope you'll you'll join me for this adventure join up to become a member of

38:17

the wild egg mathematics courses Channel and we'll do some Quantum mathematics thank you