Classical to Quantum
Classical to Quantum | Intro to Harmonic Analysis on a Circle (Rationally!) | Wild Egg Maths
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54 views Mar 19, 2024
The harmonic analysis of spheres, particularly the circle S1, the 2 dimensional sphere S2, and the 3 dimensional sphere S3, figure importantly in QM. The usual story for the circle is Fourier series, where functions of an angle theta parametrizing the circle are expanded into sums of cosine and sine functions of multiples of that angle. However this requires both the transcendental notion of an "angle" and the transcendental circular functions. In the spirit of Rational Trigonometry, we look for an alternative description which stays inside the elementary domain of algebra, polynomials and there ongoing extensions power series. But we will use quite different language: we prefer to work with polynumbers, which are arrays of coefficients, and incorporate some aspects of the very new "Box Arithmetic" developed in the Math Foundations series at Insights into Mathematics to deal with bi-polynumbers and Derivatives of them in a systematic way. The Laplacian and the concept of harmonic polynumbers are introduced used novel factorial notation which makes understanding them quite a lot simpler. By the end of the lecture we will have the essentials for a purely rational form of Fourier series, and will be well prepared for the exciting addition of complex number algebra into the story in the next video in the series! This is part of the Playlist Classical To Quantum, where we look at numerous issues in the borderlands of physics and mathematics, but from a mathematician's point of view. Many thanks to all Members of the Channel, and to Patreon supporters.
Transcript
0:01
hello everyone I'm noan wurger thanks for joining me on this adventure classical to Quantum we're looking at
0:07
physics from a mathematical point of view today uh introducing harmonic analysis on a circle moving towards the
0:14
quantum mechanics of uh the harmonic oscillator the hydrogen atom uh angular
0:22
momentum and other things um but we're going to be looking at this harmonic
0:28
analysis from a rational point of view so a very novel approach to things so
0:33
this is a topic that uh impinges a lot on rational trigonometry because people always ask me well what about you know
0:41
4A series how are you going to study um functions on a circle if you're going to
0:46
dispense with cosiness and signs so from my point of view trigonometry and
0:51
harmonic analysis or function theory on a circle are separate topics and they
0:57
don't have to be covered by the same technology but both of them can be developed purely rationally and
1:03
computably that's what I want to explain to you today so classically the harmonic analysis on a
1:10
circle going back to forier um involves the parameterization of the circle involving an angle typically from 0 to 2
1:17
pi and then there are these fora series like we see here where we express a function on the circle say F in terms of
1:25
a series of well these sort of basic functions cosine Theta cosine 2 Theta
1:31
cosine 3 Theta well along with sin Theta sin 2 Theta sin 3 Theta and there's also
1:36
a constant here a0 a half a z so the ingredients here are first of all the
1:42
angle which is essentially a transcendental concept that you really require calculus Notions to pin down and
1:49
then these corresponding transcendental functions cosine Theta and sin Theta which are given analytically by power
1:58
series like these ones here okay but in particular these uh infinite series
2:03
cannot actually be exactly evaluated so they're not honestly speaking really
2:09
functions in the in the sort of official sense of a function as being something that you input a number and you get out
2:15
a number there's also a sort of geometrical understanding of cosine and sign of course via right triangle where
2:23
we assume we have a meaning to the angle here okay and then the these opposite
2:28
and uh and a adjacent sides are sin Theta and Co Theta if we've normalized the hypotenuse to be equal to one so
2:34
here is where the traditional connection with trigonometry appears so this is the standard story and we are going to be
2:41
dispensing with it almost entirely all right so I want to present a new understanding of the circle which is
2:48
a fundamental object in mathematics but from a rational point of view where we
2:54
are always dealing with objects that can actually be concretely evaluated
3:00
exactly okay so how do we do that well let's start first of all with just some basic understanding uh that you'll have
3:08
seen me talk about a lot so we have the unit circle its equation is X2 + y2 = 1
3:14
here it is in the usual XY plane the standard parameterization given e of T =
3:19
2T over 1 + t2 and 1 - t2 over 1 + T squ in terms of this parameter T which
3:26
you can think about as being a position on the horizontal y- AIS we're connecting that position with the point
3:32
minus one over here notice I'm using one bar for minus one and this point here that we get then is given by this so
3:38
this is a rational parameterization okay of the circle which is to be contrasted
3:44
with the more usual physics engineering kind of
3:50
parameterization in terms of cosine Theta and sin Theta so this this
3:55
parameterization is not exact right if you plug in a value for Theta and actually calculate cosine Theta sin thet
4:01
you're not actually getting a point on the circle you're just getting a point nearly on the circle because you're not
4:08
able to calculate these infinite decimals exactly so we want to stick
4:13
with this exact computable story now the circle is so important
4:18
first of all it's important from a geometrical historical point of view but it's also an important classical and
4:25
Quantum object so it sort of lives in both worlds and we can get um sort of a
4:31
preliminary idea of the transition from classical to Quantum by looking at just
4:37
the circle so classically the circle typically we'll say S1 consists of a
4:44
Continuum of points okay and the Transformations that we consider are
4:49
essentially geometric like Reflections rotations and maybe more
4:54
sophisticated things like projective Transformations like mobious transformations
5:00
in the quantum point of view um S1 consists rather of a linear space of
5:07
functions or the emphasis is on the space of functions on the circle and
5:13
that's a linear space because you could add functions and you can scale or multiply them in this case the
5:19
Transformations that previously were geometric end up manifesting themselves
5:24
as linear transformations of this underlying linear space of functions so
5:30
we end up getting Matrix versions of Reflections rotations projective Transformations whatever so the picture
5:36
that we might have is that a function this thing here in green sort of roughly suggesting you know a function which
5:43
which has variable values at different points on the circle okay we're thinking about that function as itself a single
5:49
point in some higher dimensional linear space of of functions okay so that's an
5:57
essential aspect of going from classical to Quantum uh going from a
6:04
geometrical object consisting of points to a Quantum corresponding object where
6:10
we're emphasizing a corresponding linear space of
6:16
functions okay so when we have linear spaces we have linear algebra and so
6:21
linear algebra becomes important okay so these two views of S1 are intimately connected of course and
6:28
they lie at the core of a lot of quantum mechanics especially when we extend the story to the two sphere S2 and the three
6:35
sphere S3 and each one of these steps going from the circle to the two sphere
6:41
and going from the two sphere to the three sphere is a big step but first
6:46
then we want to understand S1 so a natural question is how do we understand
6:52
functions on S1 well what we are not going to do is
6:58
introduce a parameter of the angle and then talk about functions of that angle
7:04
that's not the way we're going to proceed we're going to proceed instead by considering functions as just
7:12
restrictions to the circle S1 of poly
7:17
series on the ambient apine space A2 so the circle is sitting inside this Aline
7:25
space just consisting of Aline points of the form XY and and for us X and Y are
7:31
typically rational numbers this the ambient aine space in which S1 sits so
7:38
by far the simplest and most fundamental kinds of functions on the circle that we
7:43
can consider are just the restrictions of the simplest kinds of sort of
7:50
functions on the ambient apine two-dimensional space and those are the
7:56
bipo numbers so we're going to to be talking about some some things which are
8:02
quite familiar to many of my viewers okay if you've been watching my various series but I'm going to briefly review
8:08
these Concepts so what is a bipo well it's a basically in it's a way of
8:14
thinking about a polinomial in two variables without emphasizing the word variable okay so what we want to do is
8:21
we want to restrict our attention rather to the coefficients involved and think about
8:26
the array of coefficients as being the fun FAL object which encodes the poly
8:32
number so for example here we see a generic poly number as an array a
8:39
two-dimensional array of rational numbers so of starting up here in the
8:44
top left with the degree zero term so this a z0 here would correspond to a
8:50
constant term and then here in this next level degree one terms this would
8:55
correspond to A10 Alpha plus a01 beta so we have increasing powers of alpha as we
9:01
go down and increasing powers of beta as we go across this next diagonal here the degree 2 terms corresponds to a20 Alpha
9:09
squ plus a11 Alpha Beta plus a 02 beta squ and so
9:15
on so uh we always agree there's a finite number of nonzero terms for these
9:21
poly numbers although later on we also want to consider sequences so we can talk about poly
9:27
series now for Simplicity and to connect with the more usual story we will often
9:32
identify these kinds of things with polinomial functions in X and Y but I
9:38
want to emphasize that the alpha and beta here are not variables in the usual sense they're just sort of placeholders
9:45
so for example the poly number 3 + Alpha + 5 beta - Alpha 2 + 2 Alpha Beta plus
9:51
beta 4th okay um that's not given in terms of a variables Alpha Beta the Pol
9:57
num's name is just p this is what it is but it corresponds to what more traditionally would be called a
10:04
polinomial function say F of X and Y where you just replace the alpha with an X and the beta with a Y so then what are
10:13
these alphas and betas officially well actually Alpha is just this particular poly number that has a zero in the top
10:20
and then a one below it and beta is the poly number that has a zero on the top and the one beside it now for various
10:27
reasons we want to ultimately extend the story to higher dimensions and for that it's useful to um have an alternate
10:35
notation and call this alpha alpha 0 and call this beta alpha 1 so this connects
10:41
with my theory of box arithmetic okay where I developing a whole new U sort of
10:47
convention for understanding basic arithmetic uh that's in the math Foundation series so anyway after we've
10:54
gotten to poly numbers then there are these higher dimensional things called multi- numbers and we get a whole
11:00
sequence of things starting with this Alpha zero and then this Alpha One and there's a whole sequence of them so
11:06
there's a natural ordering to the variables or the poly numbers that naturally pair in this
11:12
story but we're only going to stick with these two for for now so um we'll
11:17
sometimes use alpha beta for convenience but we can replace them at any time with alpha0 Alpha 1 so our notational
11:24
convention is that Alpha to the m beta to the N this term which is the same as Alp Sub 0 to the m alpha 1 to the N can
11:32
be now more succinctly written as just Alpha to the m comma n where now this
11:38
Alpha is just representing this um sort of possible index of variables right so
11:44
it's really a certain power of alpha 0er followed or multiplied by a certain
11:49
power of alpha 1 we don't have to put the zero and one here because these things are naturally ordered starting
11:55
from zero so for example uh this uh poly number that we talked
12:02
about before so we can now rewrite it as three times strictly speaking Alpha 0
12:08
comma 0 plus the alpha is really alpha 1 comma 0 the beta is Alpha 0 1 this Alpha
12:14
squar is Alpha 2 comma 0 this Alpha multiplied by Beta is just Alpha 1 comma
12:21
1 this beta 4th is Alpha 0 comma 4 and
12:26
then we could become a little bit more concise and remove this Alpha 0 comma 0
12:32
just replace it with one so this becomes three this thing here is just Alpha and
12:38
U and Alpha squar is is that so we so we could write it a little bit more succinctly but you see there's uh some
12:45
potential here for having a sort of consistent um and rather concise notation to deal with uh functions of
12:53
higher more variables which is going to be important uh potentially when we get to the two sphere the 3 because we have
13:00
to go up in Dimensions okay so now we've got some poly numers some basic algebra when we
13:05
want to do calculus the starting point is with the derivative in the algebraic calculus we learn that the derivative
13:11
was really introduced essentially by foul H before liit and Newton and so on
13:16
and um there's a purely algebraic way of thinking about it so our derivative is capital D and D of alpha to the N is n *
13:24
alpha n minus 1 so that's for Nal 1 2 and 3 and strictly speaking when n is
13:30
zero this doesn't really make sense because we have not defined Alpha to the minus one that's not in the world of
13:35
polers so we sort of have to separately declare that alpha or D of alpha to zero
13:41
which is D of one is is zero okay so this all appears prominently in the algebraic calculus and we can also talk
13:48
about Powers so if we look at d2 Alpha to the end that's same as D of D of
13:53
alpha the N so we perform this thing twice then we get n * n -1 * alpha n
13:59
minus 2 at least when n is equal to 2 and 3 then we have to make some sort of separate uh statement saying that uh
14:06
when we are applying D2 to Alpha 0 or to Alpha 1 we get zero and just extending
14:12
it in a natural way the Ki derivative D to the K of alpha n well we just apply
14:18
it K times then we get an n * an N minus one all the way up to n- k + 1 * Alpha
14:25
to the N minus K with the obvious meanings again when this uh this K that we're we're
14:32
operating on is a natural number which is bigger than n we're going to get zero okay so we' got derivatives but
14:41
that was just derivatives with respect to one variable the alpha the poly numbers
14:48
themselves but we actually have these two variables floating around Alpha and beta so we want the algebraic calculus
14:54
equivalent of partial derivatives okay and we're not going to use liit notation
15:00
we're just going to uh expand our use of the capital D by having some suitable
15:06
subscripts so we're going to introduce d sub z and d sub1 as essentially the
15:11
partial derivatives if you like with respect to Alpha and beta remember Alpha
15:17
is the same as Alpha Subzero beta is the same as Alpha sub one so it makes sense
15:22
that these operators are D Sub 0 D sub1 and um so we've already talked about D
15:29
so D sub1 is the the derivative that uh that takes beta to the n and gives us n
15:36
beta to the N minus one so it's a like a derivative with respect to Beta and again there's conditions um if the power
15:45
is zero then we have to separately specify that the derivative of a constant is is zero so now we can extend
15:54
both d0 and D1 to bipo numbers so combination of alphas and betas so we
16:01
declare that d sub Z okay which is like the derivative with respect to Alpha you can think about that of alpha to the m
16:09
beta to the N remember this is more concisely written as just Alpha to the m comma n that this is going to be we're
16:16
just going to treat the beta as a constant and just take the derivative with respect to Alpha so we get m alpha
16:22
the M minus one n so we get m alpha to the m minus1 comma n and similarly d
16:28
sub1 of the same thing we get n * Alpha the M beta to the N minus one so we have
16:35
you know a reasonable World in which we can apply these called partial
16:41
derivative operators but they're just derivatives with respect to our um our
16:46
basic poly numbers Alpha and beta okay so here's a little exercise
16:51
for you show that if you have two bipoles u and v that we have a familiar
16:57
kind of product rule for any one of these um partial derivatives D subi of UV
17:04
equals u d sub of V plus v d subi u okay and now that we've got these D
17:12
zos and d1s well we can apply powers of them so we can apply D Sub 0 to the k d
17:18
sub1 to the L apply to one of these generic terms the at and we can
17:25
introduce this notation sort of consistent with what we were talking about right we don't really need the the
17:30
zeros and ones because we know what the ordering is so we can just encode the information with a capital d superscript
17:37
k comma L acting on Alpha to the m comma n and sort of implicit here are the are
17:43
the underlying zeros and ones which are s of the building blocks of these things
17:49
so what happens when you when you do this well then you get M * m minus 1 all the way up to M minus k + 1 applying the
17:56
D operator K times to the essentially the alpha to the m component and that
18:02
gives us an alpha to the m minus K and then applying the D1 to the L to the beta to the N Get n * nus 1 all the way
18:10
to nus L + 1 and there's a beta to the N minus L okay so just a little bit of new
18:18
notation okay it's something to think about we're using the natural ordering
18:24
that we get with multi- numberers so the Box arithmetic is actually quite important for Oran izing all of this and
18:30
gives us a framework for doing this all sort of rather consistently okay so now let's look at
18:38
some examples so here is a nice little bipol number let's call it little Q let's call Alpha squ plus beta squ or in
18:46
our ordered notation Alpha sub 0^ 2 plus Alpha sub 1^ squ okay so if we apply d
18:54
sub Z to this thing well we just differentiating with respect to Alpha so we get 2 alpha or 2 Alpha 0 if we apply
19:03
D sub1 to this thing well we get 2 Alpha sub 1 or if you like two
19:08
beta if we apply d sub 0^ squared to it we differentiate once again we get two
19:14
if we apply d sub 1^ squared we apply D again here we get
19:20
two okay so that's uh fine and now we can introduce a gradient operator which
19:26
is what we get when we comb combine a d0 and a D1 as a little Vector so if p is a
19:33
bipol we can Define this maybe NAA symbol um Vector
19:41
d0 comma D1 by the way it acts on a bipol p is its uh first component is d0
19:50
of p and its second component is D1 of P so for
19:55
example applying this to this Q up here this uh gradient of Q okay so it's the
20:01
gradient of alpha 0 squ plus alpha 1^ s that's going to be the derivative with
20:06
respect to Alpha 0 is two alpha 0 and with respect to Alpha 1 it's 2 alpha 1
20:13
which also we can write as 2 Alpha 0 comma alpha
20:18
1 now next is the Lan which is going to play a very big role here which is
20:26
another kind of combination but it's the operator d sub 0^ 2+ d sub 1^
20:35
2 so this is the laian operator so um for example the leoan
20:43
acting on Q so we have to apply db0 squared to it and we add the result of
20:50
applying D sub1 Squared to it so D Sub 0 squ of Q was 2 d sub 1 squ of Q is also
20:56
two so the total is four we get another another bipol in general how about the lopo of q^
21:05
sared okay so we have to apply D Sub 0 twice to here's the square of Q and we
21:10
have to add result of applying D sub1 twice to again here's the square okay so
21:16
when we Square this thing here we're going to get an alpha 0 to the 4th we're going to get two times the product and we getting get alpha 1 to the 4th if we
21:23
apply d0 squared to that uh here's the thing
21:29
that we're applying it to the the term involving alpha 1 to the 4th is not
21:34
going to figure because that's going to get killed these are the ones that are not going to get killed and similarly
21:39
here when we Square these are the ones that are not going to be killed so the d0 squar acting on this well we multiply
21:46
by four and then by three so we get 12 Alpha 0^ squared the result applying it to this
21:52
and over here we just derivative with respect to Alpha zero so we get a four Alpha um Alpha 1^
22:00
2 and here 4 Alpha 0 2 + 12 alpha 1 s so a total I guess of uh 16 Alpha 0 S + 16
22:09
alpha 1 square so 16 * Q okay now a very important definition a
22:16
bipol p is harmonic precisely
22:21
when the Lao of p is zero so we're picking out a very important family of
22:29
bipol numbers the harmonic ones ones which are killed by the llas operator so for
22:36
example here these guys are all harmonic 1 Alpha Beta Alpha squ minus beta squ
22:43
it's is harmonic and so is Alpha Beta but Alpha squ plus beta squ which is
22:50
calling Q is not we've seen that already because the Lao of Q is four that's not equal to
22:56
zero now in degree 3 there happen to be two more there's Alpha cubus 3 Alpha
23:03
Beta s you can check that that's harmonic and also 3 Alpha 2 Beta minus
23:08
beta cubed you can check that's harmonic also so the ones that we've listed so
23:14
far are basically rational analoges of 1 cos Theta sin Theta cos 2 Theta sin 2
23:23
Theta cos 3 Theta and sin 3 Theta in that order
23:29
so what we're getting here is this potential this possibility of using
23:34
these harmonic polinomial to sort of get at these more traditional forier series
23:42
sort of ingredients but in a purely rational form okay so this is uh this is
23:48
a very good direction to go in so now to make further progress and
23:56
to really get at the heart of what's going on on I'm going to introduce a rather novel notation which I don't
24:03
think is part of the standard literature I'm going to introduce a variant of poly
24:08
numbers called say factorial poly numbers La better so now for an N an
24:16
integer not a natural number but actually an integer we're going to Define Alpha to the N
24:23
underscore okay that expression is going to be Alpha to the N Over N factorial
24:32
for n greater than or equal to Z and 0 otherwise in other words zero when N is
24:38
a strictly negative integer recall the convention that Zer
24:44
factorial is one so when n is equal to zero here okay then we're talking about
24:51
one underscore being just one because the denominator in that case is also one along with the
24:57
numerator okay so we now have
25:03
these variant poly numbers defined for General integers and that turns out
25:09
meaning that we can have a a consistency with respect to
25:14
differentiation which is very pleasant okay so given by essentially this rule
25:19
here that if you have an integer n then
25:25
the capital D derivative of Alpha to the N underscore is equal to Alpha to the
25:34
nus1 underscore you see because when n is
25:41
strictly negative then both of these things are zero and the derivative of 0 is zero when n is exactly equal to zero
25:49
this is one but this is Alpha to theus1 underscore which according to this is
25:55
zero so that is in fact correct and when n is strictly positive natural number then
26:03
this just coincides with the usual formula um except that the N factorial
26:09
here and the N minus one factorial uh sort of combine to to take care of that extra factor of n that you get when you
26:16
when you differentiate right there's really uh sort of an N factorial on the bottom here if you like and there's
26:22
really an N minus1 factorial we're just not not writing them and so if we just multiply through by those n factorials
26:29
that we are not writing down they' uh we just get an N over here because the N
26:34
factorial on the top and the N minus one factori on the bottom would just give us an N so this recovers the usual formula
26:41
but that means we have a a consistency of notation we have one of these guys for every integer and the capital
26:47
derivative operator now just just takes one of them and and sends it to the the the previous one in this ladder okay
26:54
with no exceptions right across the board so for example if you want to apply d2 of alpha to the ncore you get
27:03
Alpha to the N minus 2core and again there's no need to separate into well in
27:08
this case uh if n is positive we do this and if n is uh you know zero or one we
27:14
we do something else and with n is negative we do something else it's all just taken care
27:21
of okay so now it's not a big step to just extend our remember s of mixed
27:28
derivatives um so if M and N now are integers and K and L are natural numbers
27:35
then D to the k l remember that's short form for D Sub 0 to the
27:41
K followed by d sub 1 to the L those in fact commutes it doesn't matter which order we do it in so here now is one of
27:48
these sort of factorial term but there's actually a product of this factorial polum times this factorial Pol so
27:56
there's actually an M factorial that's missing you know under here and there's an N factorial missing under
28:02
here so this is what another way of saying what this thing is and the result is just that well we're
28:09
differentiating the Alpha's K times so the m is going to go to m minus K we're
28:15
going to get Alpha to the N minus K underscore and the beta is being differentiated L times we get beta to
28:21
the N minus L underscore and there's no need for any uh particular state about special cases
28:30
and in particular with our laass operator which is d sub 0^ S Plus D1 s
28:37
when we apply that to one of these sort of mixed factorial poly numbers or bipl
28:44
number llin acting on Alpha to the mcore Time beta to the ncore well the d0 squ
28:53
is going to take this Alpha to the mcore and send it to a Al the N minus 2core
28:59
leave the beta to the ncore where it is and then this D1 2qu is going to do the
29:05
same kind of thing it's going to leave the alpha to the mcore where it is and send the beta to the ncore to Beta to
29:11
the N minus 2core so the action of the laian on the bip pooms in this basis is
29:21
particularly uh sort of pleasant okay because somehow the factorials that are floating around um take care of the the
29:29
various um sort of you know exponents that are dropping down when you take uh
29:35
repeated differentiations okay so now let's have
29:40
a look at an example so here is a bip polum written in this factorial form and
29:46
we're taking the Lin of it okay so Lin of alpha 3 Alpha cubed underscore beta
29:54
2core plus 2 Alpha 1core Beta to the fourth underscore okay well the Lin is
30:01
linear so it's going to be the Lin of this first term plus two times the lassin of this second term and what is
30:07
leosin on this thing well you lower the alpha exponent by two so the alpha to
30:15
the 3 becomes Alpha to the one but still an underscore leave the beta where it is
30:20
and then you have another term where you lower the beta by two giving us beta to
30:26
the 0o uncore which is just one and the alpha to the 3core remains so there it is there when we apply the Lelo to Alpha
30:34
to the 1 to the beta 4 when we drop the uh one here by two we get Alpha to the
30:40
minus one underscore and that's zero so that first term is zero and then when we drop the four by two we get Alpha 1core
30:48
Beta 2core so all together here we have an alpha to the 3core plus 3 * Alpha 1core
30:58
beta to the 2core okay so now we are in a very happy
31:04
position to more systematically understand how we get harmonic bip pooms
31:10
and so let's have a look at what happens when we look at degree four things but we're just going to look at
31:16
homogeneous terms so terms whose total degree is four and there are five of
31:22
them so there's Alpha the fourth Alpha cubed beta the one alpha squ beta squ Alpha Beta cubed and B to 4 but we're
31:29
considering the factorial versions of all of those okay so there's like there's a three factorial one factorial
31:36
really under in the denominator here but we're just not writing it okay so we want to think what happens to these five
31:43
individual terms when we apply Lan the Lan operator to each one
31:50
of them well the alpha to the fourth underscore that's going to what's going to happen well it's just going to drop
31:55
by two we're going to get an alpha Square uncore you can't see it I'll show you next one here probably go
32:02
right to this one here okay there we see what's happening so the alpha to the 4th we apply Theos we just get Alpha squar
32:08
underscore how about this one here well the three drops by two to give us
32:14
Alpha to the 1 beta the one underscore and the One Drops by two to give us a minus one so that's sort of going in
32:20
that direction but there's nothing there because that's zero this Alpha squ Beta squ with
32:25
underscores the two here if we drop that then we get a beta squared if we drop
32:33
this two by two we just get an alpha squar with underscores so the Lin acting on this
32:41
thing is actually the sum of this term and this term so each of these arrows indicates um a constituent in the lassan
32:50
so with positive plus sign so this thing plus this thing when we take the laass
32:56
in on this well we get Alpha to the 1 beta the one underscore and the plus of
33:03
this is this one here okay so now we want to ask under what conditions can we
33:11
cook up coefficients of these five terms so that the end result is
33:19
zero okay end result is zero so suppose that there's a an alpha to the 4th
33:24
underscore term so when we apply the Lum we're going to get this thing here but we want to get zero so we want to cancel
33:31
this Alpha squore that we've just got so we need to cancel it by having one of these guys in the original with a minus
33:40
sign there has to be a minus Alpha squore Beta squore that will then contribute a minus
33:48
one here but it will also contribute a minus uh one of of these so to cancel
33:53
that one we have to have a plus one of these to cancel the minus beta 2 that's
33:59
coming from here great so the combination that we require is an alpha 4ore minus Alpha sore beta squore plus
34:09
beta 4core but there's another possibility when we look at the other terms that we
34:15
haven't uh looked at these two terms here the lassin on this one is this
34:21
thing here and the L PL on this one is also this thing here so if we want to have a result of zero we have to take a
34:27
difference this one minus this one so there's a second solution and of course
34:32
any combination or any linear combination of solutions is also solution so there's not just two
34:37
solutions there's actually a two-dimensional family of solutions given by multiples of this solution
34:45
added to a multiple of this solution but in particular there's a two-dimensional family of homogeneous
34:53
degree for bipol numbers which are harmonic actually the it's the same thing that we found for the degree 3
35:00
case as well in fact also the same thing that we found in the degree 2 case and also actually same thing we found in the
35:06
degree 1 case how about the degree 5
35:11
case well in degree 5 here are the
35:17
six basis homogeneous bipo numbers of degree five and the action of the line
35:26
is pretty predictable right this Alpha the 5core goes to Alpha the 3core uh
35:32
this one here goes to this one this one here goes to a sum of this Alpha cubed and Alpha Beta squar with underscores
35:39
this one here goes to sum of this one and this one this one goes to this one this one goes this one exactly the same
35:45
story in order we to have cancellation if we're going to have an alpha to the 5ifth we have to have a minus one of
35:51
these and then we have to have a corresponding plus one of these so that everything cancels and the result is zero so that's one solution
35:58
and so of the the odd ones if these are the even ones the odd ones a combination
36:04
of this Alpha the fourth beta and this Alpha squ Beta cubed and this beta fth we need plus one of these a minus one of
36:11
these and a plus one of those and we get our
36:17
second harmonic homogeneous bip poly number of degree five and once we've gone through these
36:24
examples it's completely obvious that this is pretty well the general story right for any degree it's going to be
36:30
exactly the same story exactly the same picture of these back and forth cancellations we have to choose every
36:36
second uh one of these factorial Pol numbers with a plus and minus plus and
36:41
minus coefficients so in general for a degree equals K so
36:48
homogeneous polinomial of degree K there will be a two-dimensional space of
36:53
bipoles P of that degree satisfying the harmonic condition unless k equals z in
37:01
case k equals z well in that case the space is Spann only by the constant
37:06
function one and so now a great exercise is to
37:12
just convince yourself that you know if we do temporarily adopt the uh the
37:17
cosine Theta and S story and we sort of identify cosine Theta with Alpha and sin Theta with beta that what we're really
37:24
recovering here are exactly the usual fora constituents one cos Theta sin Theta
37:32
those are the linear ones that we found like Alpha and beta and here's cos 2 Theta and sin 2 Theta that's like the
37:39
alpha squar minus beta squ and the alpha beta 1es up to multiple and then you
37:45
know then there's the three ones cos 3 Theta sin 3 Theta and all way up to the
37:51
degree k1s which are going to be cos K Theta sin K Theta so convince yourself that those those expression of the
37:58
alternating factorial bip poly numbers that we got is really just another way of of sort of writing these things here
38:05
but in a finite algebraic fashion without any Reliance on
38:12
transcendental functions okay so that's great uh we're in a happy position basically we've now
38:18
got the the basic ingredients of traditional fora series but in a in a rational form but there's actually
38:24
something even more exciting ahead of us where we meet for the first time a a
38:32
really remarkable aspect of quantum mechanics which is that complex numbers
38:37
for some crazy reason seem to be playing an integral crucial aspect in our
38:45
understanding of the quantum mechanical World why is that well one kind of
38:51
direction of explanation is exactly here when we're studying the simple and most
38:58
arguably most fundamental kind of non-trivial kind of curve the circle it
39:04
will turn out that there is a variant to what I did today okay which is
39:09
dramatically simpler in in in in many ways but it requires this contact with complex
39:17
numbers and it's really a motivation for why complex numbers naturally appear in
39:25
in the quantum mechanical world