Classical to Quantum
Classical to Quantum | Chebyshev polynomials and circular harmonic analysis | Wild Egg Maths
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38 views Mar 24, 2024
It is possible, and indeed preferable, to develop Fourier series, and more generally harmonic analysis on spheres, in a purely algebraic, rational fashion, without resort to "real numbers" and "transcendental functions" like "cos x" and "sin x", as these objects are actually fictions in the sense that they do not correspond to computational reality, except in an approximate, applied way. In my last video I showed how elementary considerations allowed us to develop harmonic analysis on the unit circle x2+y2=1; replacing the usual cos x, cos 2x, ... sinx, sin2x, ... functions with homogeneous harmonic polynomials coming from the affine structure of the ambient (rational ) plane. In this video, I explain how the Chebyshev polynomials of both the first and second kind naturally emerge when we restrict these harmonic polynomials (actually polynumbers in our formulation!) to the unit circle, and try to express them mostly in terms of just x. We also introduce a physical model of these functions that demonstrate their essential sinusoidal nature without reliance on the transcendental functions; for this a cardboard cylinder from a roll of toilet paper works quite well. This is Fourier series made simple!
Transcript
0:02
hello everyone thanks for joining me
0:03
we're looking at
0:05
physics classical to Quantum and I'm
0:09
going to give you a novel approach to
0:12
forier series and spherical harmonics on
0:16
Circle and then the Spheres which will
0:18
have a lot of implications for quantum
0:22
mechanics and um and I think also
0:25
reconfigures that subject in modern
0:28
analysis and my main Orient ation is to
0:30
try to remove any dependence on
0:32
so-called infinite procedures or real
0:36
numbers so I want to frame everything in
0:39
the context of just essentially high
0:40
school algebra polom arithmetic and uh
0:45
sort of Elementary
0:47
considerations okay so last time I
0:50
introduced uh various uh sort
0:53
foundational directions uh today we're
0:55
going to look at how the chish
0:57
polinomial arise naturally from the
1:00
point of view that I'm proposing and the
1:05
starting point is our somewhat novel
1:07
convention having to do with factorial
1:09
Pol numers which are defined as follows
1:12
so for an integer n not just a natural
1:14
number but an integer we Define Alpha to
1:17
the N with an underscore so I might just
1:19
say Alpha to the N underscore to be
1:22
Alpha to the N Over N factorial if N is
1:25
a positive that is greater than or equal
1:27
to zero integer and it's zero otherwise
1:30
so zero if N is a strictly negative
1:33
integer this way we get the pleasant
1:35
rule that the vile H derivative of alpha
1:39
to the ncore is Alpha to the N minus1
1:42
underscore and that's valid for an
1:45
arbitrary integer so we don't have to
1:47
distinguish what happens when we cross
1:49
the zero boundary it's just uniform okay
1:51
and we saw last time that U we had this
1:54
theorem that if K is greater than or
1:56
equal to one then the space of
1:58
homogeneous harmonic bipol numbers so A
2:01
bipo number being something in both
2:03
Alpha and beta harmonic meaning that
2:06
it's sent to Zero by the llas operator
2:10
and homogeneous means that all the terms
2:11
are of the same degree in this case
2:13
degree k then the space of such harmonic
2:16
homogeneous bipol numbers is
2:19
two-dimensional with basis consisting of
2:21
these two guys here which are U actually
2:25
quite Elementary in a way we just taking
2:26
sort of alternating sum starting with Al
2:30
to the K underscore and then further
2:32
terms where we decrease the power of
2:36
alpha by two and subsequently increase
2:39
the power of beta by two and this is not
2:42
an infinite or ongoing series because as
2:44
soon as one of these Powers say Alpha
2:47
power gets to be negative well then it's
2:49
zero and all subsequent terms are zero
2:52
and then there's a corresponding one
2:53
where we start with Alpha to the K
2:55
minus1 * beta and do the same kind of
2:58
descending um powers of alpha increasing
3:01
powers of beta alternating signs again a
3:05
finite sum so we call these things C
3:08
subk and S
3:10
subk okay so each of these may be viewed
3:14
as a polinomial function on the apine
3:16
plane in other words we can just take
3:18
the Pol C subk for example and replace
3:22
the Alpha and the beta with numbers X
3:25
and Y representing a point in the Aline
3:27
plane and similarly for S subk so we
3:29
have these two polom functions if we
3:32
like on the avine plane and then we can
3:35
consider the restrictions of these
3:37
functions to the unit circle X2 + y2 =
3:40
1 and we're going to refer to those
3:42
restrictions with a Tilda so CK
3:46
Tilda and SK til are the restrictions of
3:50
those functions initially defined on the
3:52
plane to the unit circle so they are
3:54
functions on the unit circle and it's by
3:57
looking at these restrictions that we
3:59
see that we get another way of thinking
4:01
about what these special harmonic pols
4:05
look like and connect with the cherev
4:07
pooms of both the first and second kind
4:10
so let's look at the polinomial of the
4:13
first kind what we're going to do is
4:15
we're going to assume that the point x y
4:19
is on the unit circle and then this
4:21
function of X and Y okay which is just
4:25
the same one just talked about C subk
4:27
now Tilda can be Rewritten because on
4:30
the unit circle we have the relation X2
4:32
y2 = 1 so the y2 can be replaced
4:36
with 1 - x² so we can replace this thing
4:39
with 1 - x2 but there's a a bar
4:42
underneath here an underscore right so
4:43
that means there's really a two
4:44
factorial in the denominator so let's
4:47
put that there explicitly now and
4:49
similarly with this y 4th that becomes 1
4:52
x2 squared because this is just y^ 2^
4:55
squar and the underscore here has a four
4:59
factorial underneath it so we put it
5:00
there explicitly if we just had an
5:02
underscore here then we would probably
5:04
think well this is maybe it means uh you
5:06
know with a two factorial here so I want
5:07
to be explicit there's a four factorial
5:09
here and the powers of X are not
5:11
changing so it's a very obvious pattern
5:13
we just keep going like this we get
5:15
increasing powers of 1 -
5:17
x2 and these factorials in the bottom
5:20
now this way we see that we actually
5:23
getting a function or a polinomial in X
5:26
and it actually is a polinomial okay
5:29
because again we have the in terms of
5:31
the X's here with these underscores
5:33
those powers are decreasing and
5:35
eventually the exponents become negative
5:37
then all subsequent terms are are zero
5:40
so these are indeed real polinomial in X
5:44
and let's have a look at what they look
5:46
like so if K is equal to 1 well there's
5:49
an x to the 1 underscore and then
5:52
there's a minus x to the minus1
5:54
underscore so that's zero all subsequent
5:56
terms are zero we just get X and I'm
5:58
going to write X as 1 / 1 * X okay when
6:02
K is 2 then this term is non zero and
6:05
this term is also non zero but these
6:08
further terms are zero and we just get
6:10
x2 uncore minus 1 - x2 over 2
6:14
factorial and we just simplify that to
6:16
get x2 minus 12 and I want to rewrite
6:19
it as 1/ 12 I factored out the 1/2 *
6:22
2x2 minus one I do that because this is
6:26
going to be the chubby Chef polinomial
6:28
that ends up being called T sub 2 it has
6:31
integer coefficients whereas the
6:34
original C2 Tilda only has a rational
6:37
coefficients okay what about K is three
6:41
in this case there's a x to the 3
6:43
underscore there's also an x to the
6:45
1core but this term here is still
6:47
contributing zero so we get this thing
6:49
here which becomes this which becomes 1
6:51
6 * 4X Cub -
6:54
3x that turns out to be T
6:58
sub3 and C sub4 well in that case we get
7:02
three terms we get the x 4ore - x^ 2core
7:06
1 - X2 2 factorial plus 1 - x^ 2^ 2 O
7:11
under over 4 factorial that turns out to
7:14
be this which simplifies to this and
7:16
that becomes 1 24 * 8X 4 - 8 x2 + 1
7:21
that turns out to be the 4th Chev
7:24
polinomial of the first
7:26
kind okay and so it's pretty clear that
7:30
we have a kind of a pattern here and if
7:32
we connect the formula for what we're
7:36
getting here with the classical chbf
7:38
polinomial we see that we are actually
7:40
getting the chbf polinomial that have
7:42
been classically defined going back to
7:44
the uh probably mid 19th century uh by
7:47
Cherry Chef so the theorem is that for k
7:51
a natural number greater than or equal
7:52
to one C subk Tilda of XY is 1 K
7:55
factorial time T sub KX where T subk is
7:59
the CH polinomial defined by so a common
8:04
definition of this thing is in terms of
8:06
recurrence t0 of x = 1 T1 of x = x and t
8:10
the n + 1 of x = 2 T the N of x minus t
8:14
the nus 1 of X and if you are interested
8:17
in these things well I have a whole
8:18
series on orthogonal polinomial of
8:21
various kinds in the advice for amateur
8:23
mathematicians Series where I talk about
8:26
the Chev polinomial and interesting
8:29
perhaps Noble things about
8:31
them now as an aside and I want to make
8:35
you know sure that you realize this is
8:37
an aside this is not the essential
8:38
definition but these things are often
8:41
sometimes introduced via the cosign
8:43
functions and these cosine multiple
8:46
functions cosine KX so it turns out that
8:49
cosine
8:50
KX as a function is a polinomial in
8:54
cosine X and that's T subk so for
8:58
example cosine 2x which is um you know
9:03
like cos2
9:04
xus sin2 X but then you rewrite that as
9:07
2 cos2 X - 1 when you do that you see a
9:10
we have a polom in cosine X that's the
9:14
CHF polinomial T sub 2 of X which is 2
9:19
x2 - 1 and cos 3x is T sub3 of cosine X
9:23
that's 4 cos cubed x - 3 cos x and I put
9:28
these in quotes because this is not the
9:30
way I want you to think about these
9:31
things these formulas here require a
9:35
prior belief that the cosine actually is
9:39
a legitimate function but it's not a
9:42
legitimate function it's only an
9:43
approximate function it's it's an
9:46
important ingredient in the world of
9:48
Applied Mathematics but it's not so
9:50
easily an object in pure mathematics
9:53
especially when you want to try to
9:54
consider it as a function for the simple
9:56
reason that if you put an in an input
9:59
like x = 7 over3 and ask for the output
10:03
you cannot find the output you can get
10:07
your calculator to give some
10:09
approximation but the actual true output
10:12
cannot be accessed so strictly speaking
10:16
this is not a function according to the
10:18
standard definition of a function so
10:20
although this is you know quite useful
10:22
in sort of an engineering applied sense
10:24
it really is not part of pure
10:26
mathematics strictly speaking that's why
10:29
we we prefer this purely algebraic point
10:32
of view that I'm explaining to you here
10:34
today so these particular polinomial T
10:37
suban of X are often called the chbf
10:40
polinomial of the first
10:42
kind and here is a graph of some of them
10:46
okay so this is from minus one to one
10:48
this is from Wikipedia and
10:52
here um we see T sub one okay T Sub 0 is
10:58
is just the constant one t sub 1 is
11:01
function X and then you have t sub 2
11:04
that's that like
11:06
2x2 minus one and then here is T sub3
11:10
in red and then T sub4 in purple okay so
11:14
you see that they all bounce nicely
11:17
between minus one and one in the range X
11:21
going from minus1 to one so um they have
11:24
many beautiful um
11:26
properties okay so now we're going to
11:28
look at the the other kinds of harmonic
11:32
polinomial the S subk tildas and we're
11:35
going to see that the chbf polinomial of
11:37
the second kind naturally emerg in a
11:39
very parallel way to what we've just
11:41
done so if XY is a point on the unit
11:43
circle then we take the formula for S
11:45
subk we replace y2 again with 1 - x2
11:49
but this y here that's in the first term
11:52
is not replaced it's not a y^ squ it's
11:54
only a y and what we're going to do in
11:56
fact is we're going to take that y out
11:58
front so that these other ones here just
12:00
become even powers of Y so that we can
12:02
replace them fully with 1 - x2 and
12:06
powers of that so when we do that
12:08
however we have to sort of adjust for
12:09
the fact that these things here have a
12:12
factorial poly number nature they have
12:14
factorials in the bottom so we have to
12:16
be explicit about that okay so when we
12:18
pull out the Y the Y underscore is the
12:21
same as y we have an X to K 1core here
12:24
we have an - x^ the K - 3core and this
12:27
y^ 2 now is becoming 1 - x2 but there
12:31
is a 3 factorial in the bottom here
12:33
because there was one over here and
12:35
similarly we replace the Y to the 4th
12:37
that's left over here with 1 - x2 s but
12:40
there was a 5 factorial here and now
12:42
there's still a 5 factorial there and
12:44
otherwise we get alternating uh signs
12:47
okay so we can similarly as before we
12:49
can evaluate some small cases of this
12:52
when K is equal to 1 well we get y *
12:56
just 1 and when k = 2 well we get y * X
13:03
to
13:05
1core which is the same as y / 2 *
13:09
2x when K = 3 then we get an additional
13:13
term here we get an x^ 2core minus
13:19
x to the 0 underscore that's just 1 * 1
13:23
x2 3 factorial that becomes y / 6 *
13:28
4x2 - one the things that are appearing
13:31
in Brackets here are the Trib polinomial
13:34
of the second kind they are denoted by
13:36
Capital U's so U Sub 0 is 1 U sub 1 is
13:39
2x U sub 2 is 4x2 - 1 and if we go a
13:44
little bit further we see here s sub4 is
13:48
y over 24 88x Cub - 4x you can check the
13:52
computation there and S sub5 is y over
13:55
120 * 16 x 4 - 12x2 + 1 and in general
14:02
the rule is that for K greater than or
14:03
equal to 1 this S Sub k + 1 Tilda of XY
14:07
is y over k + 1 factorial so the same
14:12
index that we have here times U sub K so
14:16
this s sub5 Tilda is equal to Y over 5
14:20
factorial * U sub4 ofx where U sub K of
14:24
X is the K Chev Chev polinomial of the
14:28
second kind
14:30
and here is a graph of them again from
14:35
minus1 to one and now they're not just
14:40
bouncing around between minus one to one
14:41
they they go further and here you see um
14:46
some various ones u0 U1 is linear U2 is
14:50
quadratic Etc these are the CH
14:52
polinomial of the second kind again from
14:55
Wikipedia okay so great so um what we
14:58
have now is this strong connection
15:02
between what's usually considered to be
15:03
4A series and usually in 4A series
15:06
you're thinking about cosine x cosine 2X
15:09
actually usually in terms of theta
15:10
cosine Theta cosine 2 Theta cosine 3
15:13
Theta cosine 4 Theta Etc and
15:15
correspondingly sin Theta sin 2 Theta
15:17
sin 3 Theta Etc but now we see that when
15:20
we actually look at the Circle as
15:22
embedded in the Aline plane where the
15:24
natural coordinates are really not
15:27
angles okay it's the Theta is a
15:29
transcendental notion that you cannot
15:31
actually pin down the natural algebraic
15:35
Notions there are just the cartisian
15:36
cordinates the X and Y and it's those
15:39
ones that you want to frame everything
15:41
in terms of when you do that the Smoke
15:44
Clears and everything becomes
15:46
dramatically simpler okay so if you want
15:48
to go from Sydney to Melbourne you don't
15:51
go via hell sinky okay that doesn't make
15:54
any sense that's what you're doing when
15:56
you're trying to understand the circle
15:58
by first introducing this transcendental
16:00
idea of an angle and saying okay we're
16:02
going to we're going to insist on doing
16:04
everything in terms of angles well then
16:06
naturally you end up having to use these
16:08
transcendental functions because you've
16:10
introduced some transcendental aspect
16:12
rate to begin with with the angles so
16:14
you have to introduce another
16:16
transcendental sort of function or
16:18
object to undo what you've just done
16:21
okay to get back to something but you
16:23
could have gotten from from where you
16:25
wanted to to where you you end up in a
16:27
purely algebraic way just with high
16:29
school algebra just by dispensing with
16:31
the Angles and just thinking about x's
16:33
and y's and the pols that I've shown you
16:36
today okay just pure basic Elementary
16:40
algebra now there's an important um sort
16:43
of geometrical physical object that goes
16:47
with this and um okay you could actually
16:49
use a toilet roll for this right it's
16:51
really instructive but I've got this
16:53
little object here so I want to show you
16:55
this this is like a cylinder okay and
16:58
you should think about this as being
17:00
like uh you know the the circle X2 + y =
17:03
1 in the base here right we're thinking
17:05
about a function above it and what I've
17:08
done here
17:09
is I've basically shown you what happens
17:12
when you just look at the first CHF
17:14
polinomial of the first kind which is
17:16
just X okay so if you imagine that
17:19
you're looking at um the um the the
17:22
plane is horizontal okay with an x
17:24
coordinate in this direction and a
17:25
y-coordinate in this direction okay and
17:27
you're looking at it this is the X
17:28
coordinate you know Min -1 to + one okay
17:32
and and let's say the actual coordinate
17:35
plane is here okay so the the graph of
17:40
you f ofx yals x that's a linear
17:43
function that's like this it's a plane
17:45
in in this sort of three-dimensional
17:48
space where you introduce a zed
17:49
coordinate and the intersection of that
17:52
plane with this cylinder given by X2 + y
17:54
s = 1 is is this ellipse here okay it's
17:57
like a conic section really okay and
18:00
this is what you're getting when you
18:03
look at the function X on the unit
18:07
circle okay so if this is if uh you know
18:11
this circle here was sort of the actual
18:13
unit circle then the height here this
18:15
sort of the Zed chord in this direction
18:17
is just telling you what this function X
18:18
is when you restrict it to the unit
18:21
circle right that's what it looks
18:23
like okay so it's purely we're just
18:25
taking the cylinder and this plane and
18:28
the result of intersecting those is this
18:31
ellipse here and it's this ellipse here
18:33
which is really responsible for what's
18:35
usually called cosine X you usually
18:37
there's a picture you know the usual
18:38
picture the sinos soidal picture going
18:40
up and down right that's just
18:42
corresponding if I had a an object here
18:45
and I just rotated this thing and
18:46
watched you know the the ascent of this
18:48
thing it goes up and then it goes down
18:52
and then it goes up and then it goes
18:55
down okay so if you look at it sort of
18:57
from an angular point of view in terms
18:58
of the angular coordinate which you
19:01
can't actually get a hold of but you
19:03
pretend that you can then it looks like
19:05
you've got um a vertical displacement
19:07
which is exactly sort
19:09
sinusoidal right so but there's really
19:12
not intrinsically a transcendental
19:14
aspect to this unless you introduce it
19:17
via insisting that we're going to
19:19
parameterize this thing with an angle
19:21
that's where the transcendental aspect
19:23
of the 4A series comes from but
19:25
intrinsically geometrically we're just
19:27
talking about
19:29
one object which is the cylinder given
19:31
by equation x2 + y2 = 1 and another is
19:35
the the the plane where Z equals
19:38
x okay now over here is sort of my uh
19:43
rather crude uh cutting out with
19:44
scissors okay this is my attempt at
19:46
doing a a parabolic version so this is
19:49
like the second chbf polinomial of of
19:53
the second of the first kind so this is
19:55
like the CHF polinomial T sub 2 that was
19:58
like 2x2 - 1 so imagine again that this
20:01
is the x coordinates and if you look
20:04
straight on like this you're seeing the
20:05
graph 2x2 - one I've just cut this
20:07
thing out here right so this
20:09
crosssection here is just like this
20:11
parabolic thing and the result of
20:14
that when you intersect it with the
20:17
cylinder is something that goes up if I
20:19
rotate here it goes up and then it goes
20:21
down and then it goes up and then it
20:23
goes down as I go through one turn we're
20:26
going up and down and up and down we're
20:30
getting sort of you know it's like
20:31
cosine of
20:32
2x okay that's the reason why we're
20:35
getting that sinusoidal thing because
20:37
the intersection of this cylinder with
20:40
these various chbby shev um polinomial
20:43
of X are giving us these sort of
20:46
oscillatory wav
20:48
like uh configurations once we cut out
20:53
the uh the corresponding functions on
20:55
the
20:56
cylinder and uh you know my view is that
20:59
this is a a pretty powerful educational
21:02
tool okay so this particular one is not
21:05
very good but you could easily make sort
21:07
of more accurate ones if you had the
21:08
technology and then you could really
21:10
illustrate how these various sinusoidal
21:13
things are appearing in a very pleasant
21:15
geometrical fashion just the interaction
21:18
of these polinomial functions of X with
21:21
the cylinder determined by x2 + y2
21:24
equal
21:26
1 it's for a series Made
21:31
Simple