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PROFESSOR: So today,
let me remind you,
00:00:25.880 --> 00:00:28.000
for the convenience
of also the people
00:00:28.000 --> 00:00:29.690
that weren't here
last time, we don't
00:00:29.690 --> 00:00:32.600
need too much of what
we did last time, except
00:00:32.600 --> 00:00:34.970
to know, more or
less, what's going on.
00:00:34.970 --> 00:00:38.130
We were solving central
potential problems
00:00:38.130 --> 00:00:42.600
in which you have a potential
that just depends on r.
00:00:42.600 --> 00:00:46.230
And at the end of the
day, the wave functions
00:00:46.230 --> 00:00:51.080
were shown to take the
form of a radial part
00:00:51.080 --> 00:00:56.600
and an angular part with
the spherical harmonic here.
00:00:56.600 --> 00:00:59.980
The radial part was
very conveniently
00:00:59.980 --> 00:01:03.890
presented as a U
function divided by r.
00:01:03.890 --> 00:01:07.940
That's another function, but
the differential equation for U
00:01:07.940 --> 00:01:08.980
is nice.
00:01:08.980 --> 00:01:13.050
It takes the form of a
1-dimensional Schrodinger
00:01:13.050 --> 00:01:16.980
equation for a particle
under the influence
00:01:16.980 --> 00:01:19.030
of an effective potential.
00:01:19.030 --> 00:01:22.280
This potential,
effective potential,
00:01:22.280 --> 00:01:25.890
has the potential that you
have in your Hamiltonian
00:01:25.890 --> 00:01:29.630
plus an extra term, a barrier.
00:01:29.630 --> 00:01:33.190
It's a potential that
grows as r goes to 0,
00:01:33.190 --> 00:01:38.830
so it's a barrier that
explodes at r equals 0.
00:01:38.830 --> 00:01:42.110
And this being the
effective potential that
00:01:42.110 --> 00:01:45.710
enters into this 1-dimensional
Schrodinger equation,
00:01:45.710 --> 00:01:51.070
we made some observations
about this function U.
00:01:51.070 --> 00:01:53.530
The normalization of
this wave function
00:01:53.530 --> 00:01:59.920
is guaranteed if the integral of
U squared over r is equal to 1.
00:01:59.920 --> 00:02:02.320
So that's a pretty nice thing.
00:02:02.320 --> 00:02:07.300
U squared meaning absolute
value squared of U.
00:02:07.300 --> 00:02:15.550
And we also noticed that U
must go like r to the l plus 1
00:02:15.550 --> 00:02:18.540
near r going to 0.
00:02:18.540 --> 00:02:23.300
So those were the
general properties of U.
00:02:23.300 --> 00:02:25.890
I'm trying to catch
up with notes.
00:02:25.890 --> 00:02:30.360
I hope to put some
notes out today.
00:02:30.360 --> 00:02:33.680
But this material, in fact,
you can find parts of it
00:02:33.680 --> 00:02:35.440
in almost any book.
00:02:35.440 --> 00:02:37.970
It will just be presented
a little differently.
00:02:37.970 --> 00:02:42.860
But this is not
very unusual stuff.
00:02:42.860 --> 00:02:47.580
Now, the diagram that I wanted
to emphasize to you last time
00:02:47.580 --> 00:02:50.410
was that if you're
trying to discuss
00:02:50.410 --> 00:02:54.160
the spectrum of a
central state potential,
00:02:54.160 --> 00:02:56.090
you do it with a
diagram in which you
00:02:56.090 --> 00:02:59.641
list the energies
as a function of l.
00:02:59.641 --> 00:03:04.560
And it's like a histogram
in which for l equals 0,
00:03:04.560 --> 00:03:08.350
you have to solve some
1-dimensional Schrodinger
00:03:08.350 --> 00:03:09.770
equation.
00:03:09.770 --> 00:03:12.070
This 1-dimensional
Schrodinger equation
00:03:12.070 --> 00:03:17.050
will have a bound state
spectrum that is non-degenerate.
00:03:17.050 --> 00:03:20.090
So for l equal 0, there
will be one solution,
00:03:20.090 --> 00:03:21.450
two solutions, three.
00:03:21.450 --> 00:03:25.640
I don't know how many before
the continuous spectrum sets in,
00:03:25.640 --> 00:03:27.700
or if there is a
continuous spectrum.
00:03:27.700 --> 00:03:30.150
But there are some solutions.
00:03:30.150 --> 00:03:33.100
For l equal 1, there will
be some other solutions.
00:03:33.100 --> 00:03:36.060
For l equal 2, there might
be some other solutions.
00:03:36.060 --> 00:03:41.000
And that depends on which
problem you are solving.
00:03:41.000 --> 00:03:45.830
In general, there's no rhyme
or reason in this diagram,
00:03:45.830 --> 00:03:50.470
except that the lowest energy
state for each level goes up.
00:03:50.470 --> 00:03:54.560
And that's because the
potential goes up and up
00:03:54.560 --> 00:03:55.830
as you increase l.
00:03:55.830 --> 00:03:58.340
Notice this is totally positive.
00:03:58.340 --> 00:04:00.810
So whatever potential
you have, it's
00:04:00.810 --> 00:04:03.530
just going up as you increase l.
00:04:03.530 --> 00:04:08.620
So the ground
state should go up.
00:04:08.620 --> 00:04:10.980
The ground state
energy should go up.
00:04:10.980 --> 00:04:13.320
So this diagram looks like this.
00:04:13.320 --> 00:04:17.170
We also emphasized
that for every l,
00:04:17.170 --> 00:04:22.260
there are 2l plus 1 solutions
obtained by varying M,
00:04:22.260 --> 00:04:25.990
because M goes
from l to minus l.
00:04:25.990 --> 00:04:29.530
Therefore, this
bar here represents
00:04:29.530 --> 00:04:35.470
a single multiplate of l equals
1, therefore three states.
00:04:35.470 --> 00:04:39.990
This is a single multiplate of
l equals 1, three more states.
00:04:39.990 --> 00:04:43.240
Here is five
states, five states,
00:04:43.240 --> 00:04:49.100
but only one l equal 1
multiplate, one l equal
00:04:49.100 --> 00:04:51.730
1 multiplate, 1, 1.
00:04:51.730 --> 00:04:53.640
There are no cases
in which you have
00:04:53.640 --> 00:04:58.440
two multiplates because that
would contradict our known
00:04:58.440 --> 00:05:05.770
statement that the spectrum of
the potential of bound states
00:05:05.770 --> 00:05:08.990
in one dimensions
is non-degenerate.
00:05:08.990 --> 00:05:13.220
So that was one thing we did.
00:05:13.220 --> 00:05:15.250
And the other thing
that we concluded
00:05:15.250 --> 00:05:18.400
that ties up with what
I want to talk now
00:05:18.400 --> 00:05:25.495
was a discussion of the free
particle, free particle.
00:05:28.850 --> 00:05:30.870
And in the case
of a free particle
00:05:30.870 --> 00:05:32.830
you say, well, so
what are you solving?
00:05:32.830 --> 00:05:34.980
Well, we're solving
for solutions
00:05:34.980 --> 00:05:37.530
that have radial symmetry.
00:05:37.530 --> 00:05:41.200
So they are functions
of r [INAUDIBLE]
00:05:41.200 --> 00:05:43.150
angular distribution.
00:05:43.150 --> 00:05:51.450
So what do you find is UEl
of r is equal to rJl of kr,
00:05:51.450 --> 00:05:55.190
as we explained, where these
were the spherical Bessel
00:05:55.190 --> 00:05:55.690
functions.
00:05:58.860 --> 00:06:05.570
And those are not as bad as
the usual Bessel functions,
00:06:05.570 --> 00:06:06.870
not that complicated.
00:06:06.870 --> 00:06:10.220
They're finite
series constructed
00:06:10.220 --> 00:06:16.590
with sines and cosines, so
these are quite tractable.
00:06:16.590 --> 00:06:18.380
And that was for
a free particle.
00:06:18.380 --> 00:06:20.620
So we decided that
we would solve
00:06:20.620 --> 00:06:33.310
the case of an infinite
spherical well, which
00:06:33.310 --> 00:06:44.492
is a potential V of r, which is
equal to 0 if r is less than a,
00:06:44.492 --> 00:06:48.846
and infinity if r is
greater or equal than a.
00:06:48.846 --> 00:06:53.400
It's a small-- well, a
is whatever size it is.
00:06:53.400 --> 00:06:58.810
It's a cavity, spherical
cavity where you can live.
00:06:58.810 --> 00:07:01.775
And outside you can't be there.
00:07:01.775 --> 00:07:04.980
This is the analog of
the infinite square
00:07:04.980 --> 00:07:08.090
well in one dimension.
00:07:08.090 --> 00:07:09.610
But this is in three dimensions.
00:07:09.610 --> 00:07:12.420
An infinite
spherical well should
00:07:12.420 --> 00:07:20.320
be imagined as some sort of hole
in the material and electrons
00:07:20.320 --> 00:07:26.380
or particles can move inside
and nothing can escape this.
00:07:26.380 --> 00:07:34.660
So this is a hollow thing.
00:07:34.660 --> 00:07:39.450
So this is a classic problem.
00:07:39.450 --> 00:07:42.480
You would say this must
be as simple to solve
00:07:42.480 --> 00:07:45.950
as the infinite square well.
00:07:45.950 --> 00:07:49.320
And no, it's more complicated.
00:07:49.320 --> 00:07:54.390
Not conceptually much more
complicated, but mathematically
00:07:54.390 --> 00:07:55.145
more work.
00:07:57.710 --> 00:08:03.200
You will consider some aspects
of the finite spherical
00:08:03.200 --> 00:08:05.790
well in the homework.
00:08:05.790 --> 00:08:09.270
The finite square
well, you remember,
00:08:09.270 --> 00:08:11.050
is a bit more complicated.
00:08:11.050 --> 00:08:13.020
You can't solve it exactly.
00:08:13.020 --> 00:08:16.710
The finite spherical
well, of course,
00:08:16.710 --> 00:08:19.850
you can't solve exactly either.
00:08:19.850 --> 00:08:23.260
But you will look at
some aspects of it,
00:08:23.260 --> 00:08:27.870
the most famous result
of which is the statement
00:08:27.870 --> 00:08:32.430
that while any attractive
potential in one dimension
00:08:32.430 --> 00:08:36.679
has a bound state
in three dimensions.
00:08:36.679 --> 00:08:42.530
An attractive potential,
so a finite spherical well,
00:08:42.530 --> 00:08:46.220
may not have a bound state,
even a single bound state.
00:08:46.220 --> 00:08:48.990
So that's a very
interesting thing
00:08:48.990 --> 00:08:54.360
that you will understand in
the homework in several ways.
00:08:54.360 --> 00:08:57.520
You will also understand some
things about delta functions,
00:08:57.520 --> 00:08:58.600
that they're important.
00:08:58.600 --> 00:09:04.320
So we'll touch base with that.
00:09:04.320 --> 00:09:08.840
So that's as far as I got
last time and just a review.
00:09:08.840 --> 00:09:12.350
If there are any
questions, don't be shy
00:09:12.350 --> 00:09:15.765
if you weren't here and
you have a question.
00:09:15.765 --> 00:09:16.265
Yes.
00:09:16.265 --> 00:09:21.601
AUDIENCE: Is there any reason to
expect [INAUDIBLE] intuitively
00:09:21.601 --> 00:09:25.740
should be like [INAUDIBLE]?
00:09:25.740 --> 00:09:29.720
PROFESSOR: Well, the reason,
intuitively the reason
00:09:29.720 --> 00:09:37.810
is basically the conspiracy
between this UEl,
00:09:37.810 --> 00:09:47.810
as I was saying,
UEl as r goes to 0
00:09:47.810 --> 00:09:51.080
goes like r to the l plus 1.
00:09:51.080 --> 00:09:58.650
So first of all, this
potential is very repulsive.
00:09:58.650 --> 00:09:59.700
Is that right?
00:09:59.700 --> 00:10:02.130
So that tends to ruin things.
00:10:02.130 --> 00:10:06.260
So you could say, oh, well,
this thing is probably not
00:10:06.260 --> 00:10:08.770
going to get anything
because near r equal 0,
00:10:08.770 --> 00:10:09.830
you're being repelled.
00:10:09.830 --> 00:10:13.480
But you cay say, no, let's
look at that l equal 0.
00:10:13.480 --> 00:10:16.950
So you don't have
that, so just V of r.
00:10:16.950 --> 00:10:25.630
But we take l equals 0--
I'm sorry, U here, U of El
00:10:25.630 --> 00:10:27.270
has to go like that.
00:10:27.270 --> 00:10:31.810
So actually, U will
vanish for r equals 0.
00:10:31.810 --> 00:10:37.500
So the effective potential
for the 1-dimensional problem
00:10:37.500 --> 00:10:41.850
may look like a finite square
well, that is like that.
00:10:41.850 --> 00:10:47.140
But the wave function has
to vanish on this side.
00:10:47.140 --> 00:10:53.970
Even though you would say,
it's a finite spherical well,
00:10:53.970 --> 00:10:56.280
why does it have to
vanish Here well,
00:10:56.280 --> 00:11:00.090
it's the unusual behavior
of this U function.
00:11:00.090 --> 00:11:03.790
So the wave function that
you can sort of imagine
00:11:03.790 --> 00:11:05.640
must vanish here.
00:11:05.640 --> 00:11:07.710
So in order to
get a bound state,
00:11:07.710 --> 00:11:10.080
it has to have enough
time to sort of curve
00:11:10.080 --> 00:11:14.090
so that it can fall, and it's
sometimes difficult to do it.
00:11:14.090 --> 00:11:16.680
So basically, it's the fact
that the wave function has
00:11:16.680 --> 00:11:21.370
to vanish at the origin, the
U wave function has to vanish.
00:11:21.370 --> 00:11:23.200
Now, the whole wave
function doesn't
00:11:23.200 --> 00:11:26.380
vanish because
it's divided by r.
00:11:26.380 --> 00:11:27.620
But the U does.
00:11:27.620 --> 00:11:34.980
So it's the reason why you don't
have bound states in general.
00:11:34.980 --> 00:11:38.470
And then there's also funny
things like a delta function.
00:11:38.470 --> 00:11:41.850
You would say, well, a
3-dimensional delta function,
00:11:41.850 --> 00:11:45.160
how many bound states do
you get, or what's going on?
00:11:45.160 --> 00:11:47.820
With a 1-dimensional
delta function,
00:11:47.820 --> 00:11:51.290
you have one bound
state, and that's it.
00:11:51.290 --> 00:11:54.930
With a 3-dimensional delta
function, as you will find,
00:11:54.930 --> 00:11:59.690
it's [INAUDIBLE]
is rather singular,
00:11:59.690 --> 00:12:03.610
and you tend to get
infinitely many bound states.
00:12:03.610 --> 00:12:08.110
And you cannot even calculate
them because they fall off all
00:12:08.110 --> 00:12:14.510
the way through r and go
to minus infinity energy.
00:12:14.510 --> 00:12:17.230
It's a rather strange situation.
00:12:17.230 --> 00:12:18.030
All right.
00:12:18.030 --> 00:12:19.045
Any other questions?
00:12:26.740 --> 00:12:31.105
So let's do this
infinite spherical well.
00:12:33.650 --> 00:12:38.590
Now, the reason we did
the free particle first
00:12:38.590 --> 00:12:41.740
was that inside here,
this is all free,
00:12:41.740 --> 00:12:45.830
so the solutions will
be sort of simple.
00:12:45.830 --> 00:12:52.820
Nevertheless, we can begin with
looking at the differential
00:12:52.820 --> 00:12:57.280
equation directly for inside.
00:12:57.280 --> 00:13:09.470
So r less than a, you would
have minus d second UEl
00:13:09.470 --> 00:13:15.240
over d rho squared,
actually, plus l times l
00:13:15.240 --> 00:13:21.310
plus 1 over rho
squared UEl equals
00:13:21.310 --> 00:13:26.690
UEl, where rho is equal to kr.
00:13:26.690 --> 00:13:30.740
And k-- I'm sorry, I
didn't write it there--
00:13:30.740 --> 00:13:35.330
is 2mE over h squared as usual.
00:13:38.110 --> 00:13:41.660
So here I didn't say what k was.
00:13:41.660 --> 00:13:46.430
That was 2mE over h squared.
00:13:49.460 --> 00:13:53.760
And this doesn't quite look
like the differential equation
00:13:53.760 --> 00:13:54.570
you have here.
00:13:58.930 --> 00:14:02.010
Well, V of r is 0
for r less than a,
00:14:02.010 --> 00:14:04.230
so you just have this term.
00:14:04.230 --> 00:14:07.570
The h squared's
over 2m and the E
00:14:07.570 --> 00:14:12.430
have been rescaled
by changing r to rho.
00:14:12.430 --> 00:14:14.320
So the differential
equation becomes
00:14:14.320 --> 00:14:18.460
simple and looking like this.
00:14:18.460 --> 00:14:22.540
So that was a manipulation that
was done in detail last time,
00:14:22.540 --> 00:14:24.530
but you can redo it.
00:14:24.530 --> 00:14:29.550
Now, this, as I mentioned,
is not a simple differential
00:14:29.550 --> 00:14:30.240
equation.
00:14:30.240 --> 00:14:33.540
If you didn't have this, it
would have a power solution.
00:14:33.540 --> 00:14:38.440
If you don't have this,
it's just a sine or cosines.
00:14:38.440 --> 00:14:41.040
But if you have
both, it's Bessel.
00:14:41.040 --> 00:14:43.040
So having a
differential with two
00:14:43.040 --> 00:14:48.170
derivatives, 1 over
rho squared and 1,
00:14:48.170 --> 00:14:50.880
brings you into
Bessel territory.
00:14:50.880 --> 00:14:53.450
Anyway, this is
the equation that,
00:14:53.450 --> 00:14:56.530
in fact, is solved
by these functions
00:14:56.530 --> 00:15:00.920
because it's a free
Schrodinger equation,
00:15:00.920 --> 00:15:05.260
and you can take
it for l equal 0.
00:15:05.260 --> 00:15:08.550
This is the only
case we can do easily
00:15:08.550 --> 00:15:11.100
without looking up
any Bessel functions
00:15:11.100 --> 00:15:12.710
or anything like that.
00:15:12.710 --> 00:15:20.256
You then have d second UE0
d rho squared is equal UE0.
00:15:22.910 --> 00:15:39.060
And therefore, UE0 goes like A
sine of rho plus B cosine rho.
00:15:39.060 --> 00:15:42.290
Rho is kr.
00:15:42.290 --> 00:15:46.570
UEl must behave like
r to the l plus 1,
00:15:46.570 --> 00:15:49.700
so UE0 must behave like r.
00:15:49.700 --> 00:15:54.270
So for this thing to
behave, must behave like r.
00:15:54.270 --> 00:15:59.980
So it must behave like
rho as rho goes to 0.
00:15:59.980 --> 00:16:03.510
Therefore, this term
cannot be there.
00:16:03.510 --> 00:16:10.920
The only solution is
UE0 is equal to sine
00:16:10.920 --> 00:16:14.260
of rho, which is kr.
00:16:14.260 --> 00:16:21.990
So UE0 of r must
be of this form.
00:16:21.990 --> 00:16:28.060
Then in order to have a
solution of the 1-dimensional
00:16:28.060 --> 00:16:30.610
Schrodinger equation, it's
true that the potential
00:16:30.610 --> 00:16:33.070
becomes infinite for r equal a.
00:16:33.070 --> 00:16:34.880
So that is familiar.
00:16:34.880 --> 00:16:38.380
It's not the point r
equal 0 that is unusual.
00:16:38.380 --> 00:16:41.520
r equal a, this must vanish.
00:16:41.520 --> 00:16:50.280
So we need that UE0
of a will equal to 0.
00:16:50.280 --> 00:17:04.040
So this requires k equal some
kn so that kna is equal to n pi.
00:17:04.040 --> 00:17:10.599
So for k is equal to kn,
where kn,a is equal to n pi,
00:17:10.599 --> 00:17:17.010
a multiple of pi, then the
wave function will vanish at r
00:17:17.010 --> 00:17:17.990
equals a.
00:17:20.829 --> 00:17:23.050
So easy enough.
00:17:23.050 --> 00:17:25.220
We've found the values of k.
00:17:25.220 --> 00:17:31.150
This is quite analogous to
the infinite square well.
00:17:31.150 --> 00:17:37.690
And now the energies
from this formula En
00:17:37.690 --> 00:17:44.060
will be equal to h squared
kn squared over 2m.
00:17:44.060 --> 00:17:46.950
And it's convenient,
of course, to divide
00:17:46.950 --> 00:17:55.240
by ma squared so that
you have kna squared.
00:17:55.240 --> 00:17:59.636
So the energies are h
squared over 2ma squared.
00:18:03.790 --> 00:18:08.870
Here we have n pi squared.
00:18:08.870 --> 00:18:10.370
I'll put them like this.
00:18:10.370 --> 00:18:19.360
En,0 for l equal
0, En,l's energies.
00:18:19.360 --> 00:18:23.020
Now, if you want to
remember something
00:18:23.020 --> 00:18:25.470
about this, of course,
all these constants
00:18:25.470 --> 00:18:28.340
are kind of irrelevant.
00:18:28.340 --> 00:18:33.070
But the good thing
is that this carries
00:18:33.070 --> 00:18:35.230
the full units of energy.
00:18:35.230 --> 00:18:38.950
And you know in a system
with length scale a,
00:18:38.950 --> 00:18:41.140
this is the typical energy.
00:18:41.140 --> 00:18:45.560
So the energies are essentially
that typical energy times
00:18:45.560 --> 00:18:47.620
n squared pi squared.
00:18:47.620 --> 00:18:56.530
So it's convenient to define,
in general, En,l to be En,l,
00:18:56.530 --> 00:18:59.490
for any l that you
may be solving,
00:18:59.490 --> 00:19:03.693
divided by h squared
over 2ma squared.
00:19:07.030 --> 00:19:09.970
So that this thing has no units.
00:19:09.970 --> 00:19:14.260
And it tells you for any
level, the calligraphic E,
00:19:14.260 --> 00:19:19.300
roughly how much bigger it
is than the natural energy
00:19:19.300 --> 00:19:22.020
scale of your problem.
00:19:22.020 --> 00:19:24.230
So it's a nice definition.
00:19:24.230 --> 00:19:32.660
And in this way, we've learned
that En,0 is equal to n pi
00:19:32.660 --> 00:19:35.510
squared.
00:19:35.510 --> 00:19:54.640
And a few values of this are
E1,0 about 9,869 [INAUDIBLE],
00:19:54.640 --> 00:20:12.940
E2,0 equal 39,478, and
E3,0 is equal 88,826.
00:20:12.940 --> 00:20:17.940
Not very dramatic
numbers, but they're
00:20:17.940 --> 00:20:21.700
still kind of interesting.
00:20:21.700 --> 00:20:25.930
So what else about this problem?
00:20:25.930 --> 00:20:29.740
Well, we can do
the general case.
00:20:29.740 --> 00:20:36.520
Let me erase a little here
so that we can proceed.
00:20:36.520 --> 00:20:40.470
The general case
is based on knowing
00:20:40.470 --> 00:20:44.710
the zeroes of this
spherical Bessel function.
00:20:44.710 --> 00:20:48.140
So this is something that the
first one you can do easily.
00:20:51.238 --> 00:20:57.200
The zeroes of J1
of rho are points
00:20:57.200 --> 00:21:01.370
at which tan rho
is equal to rho.
00:21:01.370 --> 00:21:06.520
That is a short calculation
if you ever want to do it.
00:21:06.520 --> 00:21:08.615
That's not that
difficult, of course,
00:21:08.615 --> 00:21:11.210
but you have to
do it numerically.
00:21:11.210 --> 00:21:14.840
So the zeroes of
the Bessel functions
00:21:14.840 --> 00:21:17.100
are known and are tabulated.
00:21:17.100 --> 00:21:22.440
You can find them on the web,
little programs that do it
00:21:22.440 --> 00:21:24.760
on the web and give you
[? directly those ?] zeroes.
00:21:24.760 --> 00:21:27.000
So how are they defined?
00:21:27.000 --> 00:21:39.850
Basically, people define Zn,l to
be the n-th zero with n equals
00:21:39.850 --> 00:21:45.770
1 like that of Jl.
00:21:45.770 --> 00:21:54.050
So more precisely, Jl
of Zn,l is equal to 0.
00:21:54.050 --> 00:21:57.590
And all the Z and l's
are different from 0.
00:21:57.590 --> 00:22:02.000
There's a trivial zero at 0.
00:22:02.000 --> 00:22:05.160
And nevertheless,
that is not counted.
00:22:05.160 --> 00:22:09.170
It's just too trivial
for it to be interesting.
00:22:09.170 --> 00:22:16.070
So these numbers, Z and
l, are basically it.
00:22:16.070 --> 00:22:16.620
Why?
00:22:16.620 --> 00:22:19.600
Because what you
need is, if you're
00:22:19.600 --> 00:22:28.000
looking for the l-th solution,
you need UEl of a equal 0.
00:22:30.880 --> 00:22:44.260
And UEa of that equal 0 means
that you need kn,l times a be
00:22:44.260 --> 00:22:45.300
equal to Zn,l.
00:22:48.110 --> 00:22:54.960
So kn,l is the value of k.
00:22:54.960 --> 00:23:00.310
And just like we quantized
here, we had kn, well,
00:23:00.310 --> 00:23:03.980
if you have various
l's, put the kn,l.
00:23:03.980 --> 00:23:11.410
So for every value of l, you
have kn,l's that are given
00:23:11.410 --> 00:23:12.360
by this.
00:23:12.360 --> 00:23:17.905
And the energy's like this.
00:23:21.620 --> 00:23:24.310
Let me copy what this would be.
00:23:27.070 --> 00:23:40.290
En,l would be En,l
over this ratio.
00:23:40.290 --> 00:23:47.880
And En,l h squared, well,
let me do it this way.
00:23:47.880 --> 00:23:49.530
I'm sorry.
00:23:49.530 --> 00:24:02.190
En,l would be h squared
kn,l over 2ma squared,
00:24:02.190 --> 00:24:04.200
over 2m like that.
00:24:04.200 --> 00:24:07.110
Then you multiply
by a squared again.
00:24:07.110 --> 00:24:16.900
So you get kn,l a
squared over 2ma squared.
00:24:16.900 --> 00:24:25.020
So what you learn from
this is that En,l,
00:24:25.020 --> 00:24:32.206
you divide this by that, is
just kn,l times a, which is Zm,l
00:24:32.206 --> 00:24:32.706
squared.
00:24:36.160 --> 00:24:38.440
So that's the simple result.
00:24:38.440 --> 00:24:44.740
The En,l's are just the squares
of the zeroes of the Bessel
00:24:44.740 --> 00:24:45.240
function.
00:24:49.080 --> 00:24:52.200
So you divide it again
by h squared over 2ma,
00:24:52.200 --> 00:24:54.540
and that's all that was left.
00:24:54.540 --> 00:24:57.450
So you need to know the
zeroes of the Bessel function.
00:24:57.450 --> 00:25:01.050
And there's one,
you might say, well,
00:25:01.050 --> 00:25:02.960
what for do I care about this?
00:25:02.960 --> 00:25:07.830
But it's kind of
nice to see them.
00:25:07.830 --> 00:25:20.360
So Z1,1 is equal to 4.49
Z2,1 is equal to 7.72,
00:25:20.360 --> 00:25:32.420
and Z3,1 is 10.90, numbers that
may have no rhyme or reason.
00:25:32.420 --> 00:25:35.330
Now, you've done
here l equals 1.
00:25:35.330 --> 00:25:38.890
Of course, it continues
down, down, down.
00:25:38.890 --> 00:25:43.140
You can continue with the first
zero, first nontrivial zero,
00:25:43.140 --> 00:25:45.990
second nontrivial zero,
third nontrivial zero,
00:25:45.990 --> 00:25:48.070
and it goes on.
00:25:48.070 --> 00:25:50.290
The energies the squares.
00:25:50.290 --> 00:26:01.790
So the squared goes like 20.19.
00:26:01.790 --> 00:26:08.580
This goes like 59.7.
00:26:08.580 --> 00:26:12.320
And this goes like 119 roughly.
00:26:16.500 --> 00:26:19.810
Then you have the other zeroes.
00:26:19.810 --> 00:26:28.680
First zero for l equals
2, that is 5.76 roughly.
00:26:28.680 --> 00:26:37.670
Second zero for l equals
to 2 is 9.1 roughly.
00:26:40.800 --> 00:26:45.780
And if you square those to
see those other energies,
00:26:45.780 --> 00:26:59.750
you would get, by
squaring, 33.21 and 82.72.
00:26:59.750 --> 00:27:04.320
And finally, let me do one more.
00:27:04.320 --> 00:27:11.640
Z1,3, the first zero of the
l equal 3, and the Z2,3,
00:27:11.640 --> 00:27:21.450
the second zero,
are 6.99 and 10.4,
00:27:21.450 --> 00:27:33.890
which when squared give
you 48.83 and 108.5.
00:27:33.890 --> 00:27:34.390
OK.
00:27:37.180 --> 00:27:40.240
Why do you want to
see those numbers?
00:27:40.240 --> 00:27:42.120
I think the reason
you want to see
00:27:42.120 --> 00:27:45.710
them is to just look at
the diagram of energies,
00:27:45.710 --> 00:27:48.660
which is kind of interesting.
00:27:48.660 --> 00:27:51.910
So let's do that.
00:27:51.910 --> 00:27:55.960
So here I'll plot
energies, and here I put l.
00:27:58.750 --> 00:28:02.740
And now I need a big diagram.
00:28:02.740 --> 00:28:05.710
Here I'll put the
curly energies.
00:28:05.710 --> 00:28:18.350
And here is 10, 20, 30,
40, 50, 60-- and now
00:28:18.350 --> 00:28:31.382
I need the next blackboard,
let's see, we're 60,
00:28:31.382 --> 00:28:35.420
let's see, more or less,
here is about right--
00:28:35.420 --> 00:28:48.110
70, 80, 90, 100, 110.
00:28:48.110 --> 00:28:49.255
How far do I need?
00:28:51.810 --> 00:28:57.840
120, ooh, OK, 120.
00:28:57.840 --> 00:28:59.590
There we go.
00:28:59.590 --> 00:29:10.330
So just for the fun of it.
00:29:10.330 --> 00:29:13.110
Look at them to
see how they look,
00:29:13.110 --> 00:29:14.820
if you can see any pattern.
00:29:14.820 --> 00:29:24.070
So the first energy was
986, so that's roughly here.
00:29:24.070 --> 00:29:27.360
That's l equals 0
is the first state.
00:29:27.360 --> 00:29:32.395
Second is 39.47, so it's
a little below here.
00:29:34.990 --> 00:29:44.680
Next is 88.82, so we
are here, roughly.
00:29:47.550 --> 00:29:50.670
Then we go l equals 1.
00:29:50.670 --> 00:29:53.010
What are the values?
00:29:53.010 --> 00:30:00.580
This one's 20.19.
00:30:00.580 --> 00:30:06.950
L equals 1, 20.19,
so we're around here.
00:30:06.950 --> 00:30:13.410
Then 59.7 is almost 60.
00:30:13.410 --> 00:30:19.043
And then 119, so that's why
we needed to go that high.
00:30:24.610 --> 00:30:25.720
So here we are.
00:30:33.360 --> 00:30:45.760
And then l equals 3, you
have 48.83, so that's 50.
00:30:45.760 --> 00:30:49.888
I'm sorry, l equals 2.
00:30:49.888 --> 00:30:52.590
48.83.
00:30:52.590 --> 00:30:54.140
A little lower than that.
00:31:00.190 --> 00:31:00.900
No, I'm sorry.
00:31:00.900 --> 00:31:03.340
It's 33.21.
00:31:03.340 --> 00:31:06.490
I'm misreading that.
00:31:06.490 --> 00:31:11.310
33 over here.
00:31:11.310 --> 00:31:23.910
And then 82.72, so we are here.
00:31:23.910 --> 00:31:30.470
And then l equals
3, we have 48.83,
00:31:30.470 --> 00:31:51.640
so that was the one
I wanted, and 108.5.
00:31:51.640 --> 00:31:57.980
That's it, and there's
no pattern whatsoever.
00:31:57.980 --> 00:32:00.120
The zeroes never match.
00:32:00.120 --> 00:32:07.140
The only thing that is
true is that 0, 1, 2, 3,
00:32:07.140 --> 00:32:09.900
they were ascending
as we predicted.
00:32:09.900 --> 00:32:15.300
But no level matches
with any other level.
00:32:15.300 --> 00:32:18.700
If you were trying to say, OK,
this potential is interesting,
00:32:18.700 --> 00:32:24.110
is special, it has magic to
it, a spherical square well,
00:32:24.110 --> 00:32:27.030
it doesn't seem to have
anything to it, in fact.
00:32:27.030 --> 00:32:30.480
It's totally random.
00:32:30.480 --> 00:32:35.550
I cannot prove for you,
but it's probably true,
00:32:35.550 --> 00:32:38.860
and probably not
impossible to prove,
00:32:38.860 --> 00:32:44.010
that these zeroes
are never the same.
00:32:44.010 --> 00:32:47.990
No l and l prime will
have the same zero.
00:32:52.920 --> 00:32:59.190
No degeneracy ever occurs
that needs an explanation.
00:32:59.190 --> 00:33:05.010
For example, this state could
have ended up equal to this one
00:33:05.010 --> 00:33:08.020
or equal to this one,
and it doesn't happen.
00:33:08.020 --> 00:33:10.760
And that's OK,
because at this level,
00:33:10.760 --> 00:33:13.430
we would not be able to
predict why it happened.
00:33:13.430 --> 00:33:20.980
We actually, apart from the fact
that this a round, nice box,
00:33:20.980 --> 00:33:23.620
what symmetries does
it have, that box,
00:33:23.620 --> 00:33:25.340
except rotational symmetry?
00:33:25.340 --> 00:33:28.590
Nothing all that dramatic.
00:33:28.590 --> 00:33:33.700
So you would say, OK, let's
look for a problem, which we'll
00:33:33.700 --> 00:33:40.290
deal now, that does have a
more surprising structure,
00:33:40.290 --> 00:33:41.965
and let's try to figure it out.
00:33:45.570 --> 00:33:52.500
Let's try the three dimensional
harmonic oscillator.
00:33:52.500 --> 00:33:56.980
So 3D SHO.
00:33:59.804 --> 00:34:00.303
Isotropic.
00:34:04.510 --> 00:34:06.180
What is the potential?
00:34:06.180 --> 00:34:10.689
It's 1/2 m omega
squared x squared
00:34:10.689 --> 00:34:15.750
for x plus y squared plus z
squared, all the same constant.
00:34:15.750 --> 00:34:21.532
So it's 1/2 m omega
squared r squared.
00:34:21.532 --> 00:34:24.690
You would say, this
potential may or may not
00:34:24.690 --> 00:34:26.880
be nicer than the
spherical well,
00:34:26.880 --> 00:34:30.820
but actually, it
is extraordinarily
00:34:30.820 --> 00:34:34.949
symmetric in a way that
the spherical well is not.
00:34:34.949 --> 00:34:36.640
So we'll see why is that.
00:34:39.980 --> 00:34:41.880
Let's look at the
states of this.
00:34:41.880 --> 00:34:45.120
Now, we're going to
do it with numerology.
00:34:45.120 --> 00:34:47.239
Everything will be
kind of numerology
00:34:47.239 --> 00:34:50.050
here because I don't
want to calculate
00:34:50.050 --> 00:34:52.639
things for this problem.
00:34:52.639 --> 00:34:57.010
So first thing, how
you build the spectrum?
00:34:57.010 --> 00:35:11.810
H is equal to h bar omega
N1 plus N2 plus N3 plus 3/2,
00:35:11.810 --> 00:35:18.850
where these are the three
number operators, and 0.
00:35:18.850 --> 00:35:23.920
Now, just for you to realize,
in the language of things
00:35:23.920 --> 00:35:27.850
that we've been doing,
what is the state space?
00:35:27.850 --> 00:35:42.370
If we call H1 the state space
of a one dimensional SHO,
00:35:42.370 --> 00:35:47.550
what is the state space of
the three dimensional SHO?
00:35:51.630 --> 00:36:00.410
Well, conceptually, how do you
build a three dimensional SHO?
00:36:00.410 --> 00:36:05.185
Well, you have the creation
annihilation operators
00:36:05.185 --> 00:36:09.040
that you had for
the x, y, and z.
00:36:09.040 --> 00:36:17.450
So you have the ax dagger, the
ay dagger, and the az dagger,
00:36:17.450 --> 00:36:21.640
and you could act on the vacuum.
00:36:21.640 --> 00:36:26.460
So the way you can think of
the state space of the one
00:36:26.460 --> 00:36:30.250
dimensional oscillator is this
is one dimensional oscillator
00:36:30.250 --> 00:36:32.200
and I have all these things.
00:36:32.200 --> 00:36:33.850
Here is the other
one dimensional,
00:36:33.850 --> 00:36:35.900
here is the last
one dimensional.
00:36:35.900 --> 00:36:40.570
But if I want to build a
state of the three dimensional
00:36:40.570 --> 00:36:45.272
oscillator, I have to say how
many ax's, how many ay's, how
00:36:45.272 --> 00:36:46.890
many az's.
00:36:46.890 --> 00:36:52.350
So you're really
multiplying the states
00:36:52.350 --> 00:36:55.110
in the sense of tensor products.
00:36:55.110 --> 00:37:03.616
So the H, for a 3D SHO,
is the tensor product
00:37:03.616 --> 00:37:11.900
of three H1's, the H1
x, the 1y, and the z.
00:37:11.900 --> 00:37:14.190
You're really multiplying
all the states together.
00:37:14.190 --> 00:37:16.358
Yes?
00:37:16.358 --> 00:37:18.024
AUDIENCE: So this is
generalized to when
00:37:18.024 --> 00:37:19.360
you have a wave function
that's separable into products
00:37:19.360 --> 00:37:20.420
of different coordinates.
00:37:20.420 --> 00:37:22.087
Can you express those
as tensor products
00:37:22.087 --> 00:37:23.544
of the different
states, basically?
00:37:23.544 --> 00:37:25.410
BARTON ZWIEBACH: You
see, the separable
00:37:25.410 --> 00:37:29.170
into different coordinates,
it's yet another thing
00:37:29.170 --> 00:37:33.990
because it would be the
question of whether the state is
00:37:33.990 --> 00:37:37.880
separable or is entangled.
00:37:37.880 --> 00:37:40.610
If you choose, for example,
one term like that,
00:37:40.610 --> 00:37:52.710
a1, x, ax dagger, ay dagger, az
dagger, with two of those here,
00:37:52.710 --> 00:37:57.667
the wave function is the product
of an x wave function, a y wave
00:37:57.667 --> 00:37:59.000
function, and a z wave function.
00:37:59.000 --> 00:38:05.610
But if you add to this ax
dagger squared plus ay plus az,
00:38:05.610 --> 00:38:10.130
it will also be factorable,
but the sum is not factorable.
00:38:10.130 --> 00:38:13.700
So you get the
entanglement kind of thing.
00:38:13.700 --> 00:38:18.800
So this is the general
thing, and the basis vectors
00:38:18.800 --> 00:38:23.940
of this tensor product are
basis vectors of one times basis
00:38:23.940 --> 00:38:25.680
vectors of the
other basis vector.
00:38:25.680 --> 00:38:28.600
So basically, one
basis vector here,
00:38:28.600 --> 00:38:31.880
you pick some number of ax's,
some number of ay's, some
00:38:31.880 --> 00:38:33.840
number of az's.
00:38:33.840 --> 00:38:35.910
So this shows,
and it's a point I
00:38:35.910 --> 00:38:38.250
want to emphasize
at this moment,
00:38:38.250 --> 00:38:42.610
it's very important, that even
though we started thinking
00:38:42.610 --> 00:38:45.890
of tensor products of
two particles, here,
00:38:45.890 --> 00:38:49.585
there are no two particles in
the three dimensional harmonic
00:38:49.585 --> 00:38:52.130
oscillator, no three particles.
00:38:52.130 --> 00:38:56.610
There's just one particle where
there's one kind of attribute
00:38:56.610 --> 00:38:58.800
that that's doing in
the x direction, one
00:38:58.800 --> 00:39:00.670
kind of attribute
that it's doing
00:39:00.670 --> 00:39:03.920
in the y, one kind of attribute
that it's doing in the z.
00:39:03.920 --> 00:39:07.260
And therefore, you
need data for each one,
00:39:07.260 --> 00:39:10.680
and the right data is
the tensor product.
00:39:10.680 --> 00:39:13.150
You're just combining
them together.
00:39:13.150 --> 00:39:17.260
We mentioned that the basis
vectors of a tensor product
00:39:17.260 --> 00:39:21.010
are the products of
those basis vectors,
00:39:21.010 --> 00:39:25.440
of each one, so that's exactly
how you build states here.
00:39:25.440 --> 00:39:28.700
So I think, actually, you
probably have this intuition.
00:39:28.700 --> 00:39:36.140
I just wanted to make it a
little more explicit for you.
00:39:36.140 --> 00:39:39.320
So you don't need to
have two particles
00:39:39.320 --> 00:39:41.730
to get a tensor product.
00:39:41.730 --> 00:39:44.920
It can happen in simpler cases.
00:39:44.920 --> 00:39:49.240
So here's the thing that
I want to do with you.
00:39:49.240 --> 00:39:52.260
I would like to find that
diagram for the three
00:39:52.260 --> 00:39:54.370
dimensional SHO.
00:39:54.370 --> 00:39:57.740
That's our goal that we're
going to spend the next 15
00:39:57.740 --> 00:39:59.770
minutes probably doing.
00:39:59.770 --> 00:40:01.200
How does that diagram look?
00:40:03.730 --> 00:40:08.340
So I'll put it somewhere
here maybe, or maybe here.
00:40:08.340 --> 00:40:11.345
I won't need such a big diagram.
00:40:17.700 --> 00:40:19.842
So I'll have it here.
00:40:19.842 --> 00:40:24.260
Here is l, and here
are the energies.
00:40:24.260 --> 00:40:27.050
So ground states.
00:40:27.050 --> 00:40:32.565
The ground state, you can think
of it as a state like that.
00:40:35.440 --> 00:40:37.380
How should I write it?
00:40:37.380 --> 00:40:40.220
A state like that.
00:40:40.220 --> 00:40:46.300
No oscillators acting on it
whatsoever, so the N's are N1
00:40:46.300 --> 00:40:50.990
equals N2 equals N3
equals 0, and you
00:40:50.990 --> 00:40:57.660
get E equals h bar
omega times 3/2.
00:40:57.660 --> 00:40:59.620
So 3/2 h bar omega.
00:40:59.620 --> 00:41:05.275
So actually, we got one state,
and it's the lowest energy
00:41:05.275 --> 00:41:05.775
state.
00:41:08.320 --> 00:41:10.480
Energy lowest possible.
00:41:10.480 --> 00:41:18.010
So let me write here, energy
equals 3/2 h bar omega.
00:41:18.010 --> 00:41:24.445
We got one state over here.
00:41:27.520 --> 00:41:32.980
Now, can it be an l
equals 1 state or an l
00:41:32.980 --> 00:41:35.900
equals 2 state or
an l equals 3 state?
00:41:40.290 --> 00:41:43.530
How much is l for that state?
00:41:43.530 --> 00:41:46.770
You see, if it's a
spherically symmetric problem,
00:41:46.770 --> 00:41:48.810
it has to give you
a table like that.
00:41:48.810 --> 00:41:53.480
It's guaranteed by angular
momentum, so we must find.
00:41:53.480 --> 00:42:01.480
My question is whether it's l
equals 0, 1, 2, 3, or whatever.
00:42:01.480 --> 00:42:04.566
Anybody would like to say
what do they think it is?
00:42:04.566 --> 00:42:05.065
Kevin?
00:42:05.065 --> 00:42:06.204
AUDIENCE: It's 0, right?
00:42:06.204 --> 00:42:06.995
BARTON ZWIEBACH: 0.
00:42:06.995 --> 00:42:08.180
And why?
00:42:08.180 --> 00:42:13.270
AUDIENCE: Because we
wrote the operator for l
00:42:13.270 --> 00:42:18.035
in terms of ax, ay, and az, and
you need one to be non-zero.
00:42:18.035 --> 00:42:22.490
You need a difference between
them to generate a rotation.
00:42:22.490 --> 00:42:24.140
BARTON ZWIEBACH:
OK, that's true.
00:42:24.140 --> 00:42:26.110
It's a good answer.
00:42:26.110 --> 00:42:28.680
It's very explicit.
00:42:28.680 --> 00:42:32.400
Let me say it some other
way, why it couldn't possibly
00:42:32.400 --> 00:42:36.270
be l equals 1.
00:42:36.270 --> 00:42:37.234
Yes?
00:42:37.234 --> 00:42:39.162
AUDIENCE: Because
the ground state
00:42:39.162 --> 00:42:42.466
decreases for l decreasing.
00:42:42.466 --> 00:42:44.590
BARTON ZWIEBACH: The ground
state energy does what?
00:42:44.590 --> 00:42:46.482
AUDIENCE: It's
smaller for smaller l,
00:42:46.482 --> 00:42:49.740
and so for l equals 0, you
have to have a smaller ground
00:42:49.740 --> 00:42:51.850
state than for l equals 1.
00:42:51.850 --> 00:42:54.080
BARTON ZWIEBACH: That's true.
00:42:54.080 --> 00:42:55.330
Absolutely true.
00:42:55.330 --> 00:42:58.480
The energy increases so
it cannot be l equals 1,
00:42:58.480 --> 00:43:00.500
because then there will
be something below which
00:43:00.500 --> 00:43:01.580
doesn't exist.
00:43:01.580 --> 00:43:04.220
But there may be a
more plain answer.
00:43:04.220 --> 00:43:06.226
AUDIENCE: The state
is non-degenerative.
00:43:06.226 --> 00:43:07.100
BARTON ZWIEBACH: Yes.
00:43:07.100 --> 00:43:09.440
There's just one state here.
00:43:09.440 --> 00:43:12.510
We built one state.
00:43:12.510 --> 00:43:17.180
If it would be l equals 1,
there should be three states
00:43:17.180 --> 00:43:22.450
because l equals 1 comes with
m equals 1, 0, and minus 1.
00:43:22.450 --> 00:43:26.610
So unless there are three
states, you cannot have that.
00:43:26.610 --> 00:43:27.370
All right.
00:43:27.370 --> 00:43:31.280
So then we go to the next level.
00:43:31.280 --> 00:43:35.420
So I can build a state with
ax dagger on the vacuum,
00:43:35.420 --> 00:43:38.380
a state with ay
dagger on the vacuum,
00:43:38.380 --> 00:43:41.180
and a state with az
dagger on the vacuum
00:43:41.180 --> 00:43:43.210
using one oscillator.
00:43:43.210 --> 00:43:46.870
Here, the N's are
1, different ones,
00:43:46.870 --> 00:43:55.320
and the energy is h bar
omega 1 plus 3/2, so 5/2.
00:43:55.320 --> 00:43:58.880
And I got three states.
00:43:58.880 --> 00:44:01.920
What can that be?
00:44:01.920 --> 00:44:11.360
Well, could it be three
states of l equals 0?
00:44:11.360 --> 00:44:11.860
No.
00:44:11.860 --> 00:44:14.800
We said there's never
a degeneracy here.
00:44:14.800 --> 00:44:18.740
There's always one thing, so
there would be one state here,
00:44:18.740 --> 00:44:21.160
one state here, one
state here maybe.
00:44:21.160 --> 00:44:24.370
We don't know, but they would
not have the same energy,
00:44:24.370 --> 00:44:27.140
so it cannot be l equals 0.
00:44:27.140 --> 00:44:33.000
Now, you probably remember that
l equals 1 has three states.
00:44:33.000 --> 00:44:37.220
So without doing
any computation,
00:44:37.220 --> 00:44:40.980
I think I can argue that
this must be l equals 1.
00:44:40.980 --> 00:44:43.590
That cannot be any other thing.
00:44:43.590 --> 00:44:47.180
It cannot be l equals 2
because you need five states.
00:44:47.180 --> 00:44:49.860
Cannot be anything
with l equals 0.
00:44:49.860 --> 00:44:52.520
So it must be l equals 1.
00:44:52.520 --> 00:44:59.150
So here is l equals 0,
and here is l equals 1,
00:44:59.150 --> 00:45:04.770
and there's no state here, but
there's one at 5/2 h bar omega.
00:45:04.770 --> 00:45:07.160
So we obtain one state here.
00:45:09.950 --> 00:45:13.650
And this corresponds
to a degeneracy.
00:45:13.650 --> 00:45:18.580
This must correspond to l equals
1 because it's three states.
00:45:18.580 --> 00:45:21.740
And that degeneracy
is totally explained
00:45:21.740 --> 00:45:25.630
by angular momentum's
central potential.
00:45:25.630 --> 00:45:29.870
It has to group in that way.
00:45:29.870 --> 00:45:33.860
Of course, if my oscillator
had not been isotopic,
00:45:33.860 --> 00:45:35.860
it would not group that way.
00:45:35.860 --> 00:45:40.640
So we've got that one and we're,
I think, reasonably happy.
00:45:40.640 --> 00:45:45.580
Now, let's list the various l's.
00:45:45.580 --> 00:45:50.860
l equals 0, l equals 1,
l equals 2, l equals 3,
00:45:50.860 --> 00:45:53.720
l equals 4, l equals 5.
00:45:53.720 --> 00:45:54.750
How many states?
00:45:54.750 --> 00:46:01.660
1, 3, 5, 7, 9, 11.
00:46:01.660 --> 00:46:04.150
OK, good enough.
00:46:04.150 --> 00:46:08.360
So we succeeded, so let's
proceed to one more level.
00:46:08.360 --> 00:46:10.180
Let's see how we do.
00:46:10.180 --> 00:46:14.450
Here, I would have
ax dagger squared
00:46:14.450 --> 00:46:20.640
on the vacuum, ay dagger
squared on the vacuum, az
00:46:20.640 --> 00:46:23.890
dagger squared on the vacuum.
00:46:23.890 --> 00:46:26.800
Three states, but then
I have three more,
00:46:26.800 --> 00:46:39.130
ax ay, both dagger on the
vacuum, ax az, and ay az,
00:46:39.130 --> 00:46:41.385
for a total of six states.
00:46:48.040 --> 00:46:57.710
So at N equals 2,
the next level,
00:46:57.710 --> 00:47:05.350
let's call N equals
N1 plus N2 plus N3.
00:47:05.350 --> 00:47:09.690
So this is N equals 2.
00:47:09.690 --> 00:47:11.740
This is N equals 1.
00:47:11.740 --> 00:47:13.255
You've got six states.
00:47:16.710 --> 00:47:20.510
They must organize themselves
into representations
00:47:20.510 --> 00:47:27.120
of angular momentum, so they
must be billed by these things.
00:47:27.120 --> 00:47:29.240
So I cannot have l equals 3.
00:47:29.240 --> 00:47:32.790
I don't have that many states.
00:47:32.790 --> 00:47:38.340
I could have two l equals
1 states, three and three.
00:47:38.340 --> 00:47:40.925
That would give six states,
or a five and a one.
00:47:45.020 --> 00:47:48.410
So what are we looking at?
00:47:48.410 --> 00:47:52.990
Let's see what we could have.
00:47:52.990 --> 00:47:59.850
Well, we're trying to figure
out the next level, which
00:47:59.850 --> 00:48:04.050
is 7/2 h bar omega.
00:48:04.050 --> 00:48:11.690
If I say this is built
by two l equals 1's, I
00:48:11.690 --> 00:48:17.350
would have to put two things
here, and that's wrong.
00:48:17.350 --> 00:48:22.320
There cannot be two
multiplates at the same energy.
00:48:22.320 --> 00:48:27.490
So even though it looks like you
could build it with two l equal
00:48:27.490 --> 00:48:29.270
1's, you cannot.
00:48:29.270 --> 00:48:33.470
So it must be an l equals
2 and an l equals 0.
00:48:33.470 --> 00:48:39.810
So l equals 2 plus l equals
0, this one giving you
00:48:39.810 --> 00:48:43.240
five states and this
giving you one state.
00:48:43.240 --> 00:48:48.600
So at the next level,
this cannot be,
00:48:48.600 --> 00:48:52.170
but what you get
instead, l equals 2.
00:48:52.170 --> 00:48:56.590
You get one state here
and one state there.
00:49:05.330 --> 00:49:10.440
This is already something a
little strange and unexpected.
00:49:10.440 --> 00:49:13.380
For the first time,
you've got things
00:49:13.380 --> 00:49:17.000
in different columns that
are matching together.
00:49:17.000 --> 00:49:19.840
Why would these ones
match with these ones?
00:49:19.840 --> 00:49:22.580
That requires an explanation.
00:49:22.580 --> 00:49:26.140
You will see that explanation
a little later in the course,
00:49:26.140 --> 00:49:29.530
and that's something
we need to understand.
00:49:29.530 --> 00:49:31.060
So far, so good.
00:49:31.060 --> 00:49:33.530
We seem to be making
good progress.
00:49:33.530 --> 00:49:35.680
Let's do one more.
00:49:35.680 --> 00:49:39.074
In fact, we need to
do maybe a couple more
00:49:39.074 --> 00:49:40.115
to see the whole pattern.
00:49:55.650 --> 00:49:59.055
Let's do the next
one, N total equals 3.
00:50:02.570 --> 00:50:11.110
And now you have-- I'll be very
brief-- ax cubed, ay cubed, az
00:50:11.110 --> 00:50:28.094
cubed, ax squared times ay or
az, ay squared times ax or az,
00:50:28.094 --> 00:50:41.080
and az squared times ay or ax,
and ax ay az, all different.
00:50:41.080 --> 00:50:48.340
And that builds for three
states here, two states here,
00:50:48.340 --> 00:50:52.480
two states here, two states
here, and one state here.
00:50:52.480 --> 00:50:54.595
So that's 10 states.
00:51:01.820 --> 00:51:04.238
Yes?
00:51:04.238 --> 00:51:05.660
AUDIENCE: [INAUDIBLE]?
00:51:05.660 --> 00:51:06.493
BARTON ZWIEBACH: No.
00:51:06.493 --> 00:51:08.540
It's just laziness.
00:51:08.540 --> 00:51:10.890
I just should have
put ax squared
00:51:10.890 --> 00:51:14.461
ay dagger or ax
squared az squared.
00:51:14.461 --> 00:51:15.377
AUDIENCE: [INAUDIBLE]?
00:51:18.510 --> 00:51:20.710
BARTON ZWIEBACH:
No, this is a sum.
00:51:20.710 --> 00:51:25.740
This is what we used to call
the direct sum of vector spaces.
00:51:25.740 --> 00:51:28.840
This is not the product.
00:51:28.840 --> 00:51:31.630
That's pretty important.
00:51:31.630 --> 00:51:33.060
Here, it's a sum.
00:51:33.060 --> 00:51:38.310
We're saying 6 is 5
plus 1, basically.
00:51:38.310 --> 00:51:42.990
Six states are five
vectors plus one vector.
00:51:42.990 --> 00:51:44.790
Now, it can seem
a little confusing
00:51:44.790 --> 00:51:47.750
because-- well,
it's not confusing.
00:51:47.750 --> 00:51:51.760
If it would be a product, it
would be 1 times 5, which is 5.
00:51:51.760 --> 00:51:53.230
So here, it's 6.
00:51:53.230 --> 00:51:54.540
It's a direct sum.
00:51:54.540 --> 00:51:59.250
It's saying the space
of states at this level
00:51:59.250 --> 00:52:00.750
is six dimensional.
00:52:00.750 --> 00:52:03.290
This is a five
dimensional vector space,
00:52:03.290 --> 00:52:05.650
this is a one
dimensional vector space.
00:52:05.650 --> 00:52:09.270
This is a direct sum,
something we defined
00:52:09.270 --> 00:52:14.220
a month ago or two
months ago, direct sums.
00:52:14.220 --> 00:52:17.100
So this is funny how
this is happening.
00:52:17.100 --> 00:52:21.750
This tensor product is giving
you direct sums of states.
00:52:21.750 --> 00:52:25.550
Anyway, 10 states here.
00:52:25.550 --> 00:52:31.690
And now it does look like we
finally have an ambiguity.
00:52:31.690 --> 00:52:37.970
We could have l equals 4, which
is nine states, plus l equals
00:52:37.970 --> 00:52:38.470
0.
00:52:41.750 --> 00:52:45.100
You cannot use any
one more than once.
00:52:45.100 --> 00:52:48.075
We've learned that
for any energy level,
00:52:48.075 --> 00:52:52.050
we cannot have some l appear
more than once because it would
00:52:52.050 --> 00:52:53.320
imply degeneracy.
00:52:53.320 --> 00:52:57.180
So I cannot build
this with 10 singlets,
00:52:57.180 --> 00:53:01.860
or three l equal 1's
and one l equals 0.
00:53:01.860 --> 00:53:03.930
I have to build it
with different things,
00:53:03.930 --> 00:53:08.390
but I can build it as 9
plus 1, or I can build it
00:53:08.390 --> 00:53:13.190
as l equals 3 plus l equals 1.
00:53:17.880 --> 00:53:20.570
And the question
is, which one is it?
00:53:24.674 --> 00:53:25.890
AUDIENCE: [INAUDIBLE].
00:53:25.890 --> 00:53:28.570
BARTON ZWIEBACH: 3
and 1, is that right?
00:53:28.570 --> 00:53:31.190
How would you see that?
00:53:31.190 --> 00:53:34.100
AUDIENCE: Because the
lowest energy with l3
00:53:34.100 --> 00:53:37.565
has to be lower than the
lowest energy with l4.
00:53:37.565 --> 00:53:38.440
BARTON ZWIEBACH: Yes.
00:53:41.180 --> 00:53:43.310
Indeed, it would
be very strange.
00:53:43.310 --> 00:53:45.220
It shouldn't happen.
00:53:45.220 --> 00:53:47.170
The energies are
sort of in units,
00:53:47.170 --> 00:53:53.150
so here is l3 and
here is l equals 4.
00:53:53.150 --> 00:53:56.880
If l4 would be here,
where could be l3?
00:53:56.880 --> 00:53:59.350
It cannot be at a lower energy.
00:53:59.350 --> 00:54:01.650
We've accounted
for all of those.
00:54:01.650 --> 00:54:07.295
This is terribly unlikely,
and it must be this.
00:54:10.060 --> 00:54:19.180
And therefore, you found here
next level, 9/2 h bar omega,
00:54:19.180 --> 00:54:25.950
you got l equals 3, l equals 1.
00:54:25.950 --> 00:54:28.710
It's possible to count.
00:54:28.710 --> 00:54:31.810
You start to get bored
counting these things.
00:54:31.810 --> 00:54:35.100
So if you had to count,
for example, the number
00:54:35.100 --> 00:54:40.680
of states with 4, how would
you count them a little easier?
00:54:40.680 --> 00:54:50.500
Well, you say, I need ax dagger
to the nx, ay dagger to the ny,
00:54:50.500 --> 00:54:54.860
and az dagger to the nz.
00:54:54.860 --> 00:54:55.980
That's the state.
00:54:55.980 --> 00:55:03.410
And you must have nx
plus ny plus nz equals 4.
00:55:03.410 --> 00:55:06.820
And you can plot this,
make a little diagram
00:55:06.820 --> 00:55:12.890
like this, in which
you put nx, ny, and nz.
00:55:12.890 --> 00:55:16.470
And you say, well, this
can be as far as 4,
00:55:16.470 --> 00:55:20.310
this can be as high as 4,
this can be as high as 4,
00:55:20.310 --> 00:55:23.090
so you have triangle,
but you only
00:55:23.090 --> 00:55:25.430
have the integer solutions.
00:55:25.430 --> 00:55:31.200
nx plus ny plus nz equals 4 is
that whole hyperplane, but only
00:55:31.200 --> 00:55:33.320
integers and positive one.
00:55:33.320 --> 00:55:36.970
So you have here, for
example, a solution.
00:55:36.970 --> 00:55:41.500
This line is when nz
plus ny is equal to 4.
00:55:41.500 --> 00:55:46.595
So here's nz equals 4,
nz equals 3, 2, 1, 0.
00:55:49.450 --> 00:55:51.150
These are solutions.
00:55:51.150 --> 00:55:54.300
Here, you have
just one solution.
00:55:54.300 --> 00:55:57.750
Then you would have
two solutions here,
00:55:57.750 --> 00:56:03.050
three solutions here,
four here, and five there.
00:56:03.050 --> 00:56:06.910
So the number of states
is actually 1 plus 2
00:56:06.910 --> 00:56:10.930
plus 3 plus 4 plus 5.
00:56:10.930 --> 00:56:14.590
The number of states is
1 plus 2 plus 3 plus 4
00:56:14.590 --> 00:56:18.480
plus 5, which is 15.
00:56:21.030 --> 00:56:24.790
And you don't have
to write them.
00:56:24.790 --> 00:56:30.720
So 15 states, what could it be?
00:56:30.720 --> 00:56:33.120
Well, you go through
the numerology
00:56:33.120 --> 00:56:36.390
and there seem to be several
options, but not too many
00:56:36.390 --> 00:56:37.240
that make sense.
00:56:45.450 --> 00:56:47.780
You could have something
with l equals 5,
00:56:47.780 --> 00:56:50.990
but by the same
argument, it's unlikely.
00:56:50.990 --> 00:56:53.790
But you could have
something with l equals 4
00:56:53.790 --> 00:56:54.920
and begin with it.
00:56:54.920 --> 00:57:00.160
So it must be an l equals 4,
which gives me already nine
00:57:00.160 --> 00:57:04.550
states, and there are
left with six states.
00:57:04.550 --> 00:57:07.670
But you know that with
six states, pretty much
00:57:07.670 --> 00:57:12.900
the only thing you can do is
l equals 2 and l equals 0,
00:57:12.900 --> 00:57:15.860
so that must be it.
00:57:15.860 --> 00:57:20.390
The next state here,
l equals 4, is here.
00:57:20.390 --> 00:57:30.240
This was 11/2 h bar omega,
and then it goes 4, 2, 0.
00:57:30.240 --> 00:57:34.180
Enough to see the
pattern, I think.
00:57:34.180 --> 00:57:36.620
You could do the next one.
00:57:36.620 --> 00:57:40.420
Now it's quick because you
just need to add 6 here.
00:57:40.420 --> 00:57:43.310
It adds one more,
so it's 21 states,
00:57:43.310 --> 00:57:45.620
and you can see
what can you build.
00:57:45.620 --> 00:57:49.220
But it does look like
you have this, this,
00:57:49.220 --> 00:57:52.940
and that you jump by two units.
00:57:52.940 --> 00:57:56.750
So you have 0, then
1, and nothing.
00:57:56.750 --> 00:57:59.970
Then 2, and you
jump the next to 0.
00:57:59.970 --> 00:58:04.250
And then 3 is the next one, and
then you jump 2, and that's it.
00:58:04.250 --> 00:58:07.340
And here, jump 2 and jump 2.
00:58:07.340 --> 00:58:12.750
So in jumps of 2, you go to the
angular momentum that you need.
00:58:12.750 --> 00:58:16.390
So how can you
understand a little more
00:58:16.390 --> 00:58:17.840
of what's going on here?
00:58:17.840 --> 00:58:19.950
Why these things?
00:58:19.950 --> 00:58:28.820
Well, as you may recall, we
used to have this discussion
00:58:28.820 --> 00:58:32.970
in which you have an a x and ay.
00:58:32.970 --> 00:58:39.550
You could trade for
a right and a left.
00:58:39.550 --> 00:58:43.710
And with those, the angular
momentum in the z-direction
00:58:43.710 --> 00:58:48.750
was h bar N right minus N left.
00:58:48.750 --> 00:58:51.140
This is for a
two-dimensional oscillator,
00:58:51.140 --> 00:58:55.750
but the x and y of the
three-dimensional oscillator
00:58:55.750 --> 00:58:57.830
works exactly the same way.
00:58:57.830 --> 00:59:05.660
So Lz is nicely written in
terms of these variables.
00:59:05.660 --> 00:59:10.360
And it takes a little more
work to get the other-- the Lx
00:59:10.360 --> 00:59:13.760
and Ly, but they
can be calculated.
00:59:13.760 --> 00:59:16.260
And they correspond
to the true angular
00:59:16.260 --> 00:59:18.460
momentum of this particle.
00:59:18.460 --> 00:59:20.800
It's the real angular momentum.
00:59:20.800 --> 00:59:22.770
It's not the angular
momentum that you
00:59:22.770 --> 00:59:26.150
found for the two-dimensional
harmonic oscillator.
00:59:26.150 --> 00:59:27.280
It's the real one.
00:59:27.280 --> 00:59:33.670
So here we go with
a little analysis.
00:59:33.670 --> 00:59:38.310
How would you build now
states in this language?
00:59:38.310 --> 00:59:40.960
You can understand things
better in this case
00:59:40.960 --> 00:59:45.520
because, for example,
for N equals 1,
00:59:45.520 --> 00:59:49.520
you could have a
state a right dagger
00:59:49.520 --> 01:00:00.230
on the vacuum, a z dagger,
a left dagger on the vacuum.
01:00:00.230 --> 01:00:04.220
And then you can say, what
is the Lz of this state?
01:00:06.920 --> 01:00:14.030
Well, a right dagger on the
vacuum has Lz equal h bar.
01:00:14.030 --> 01:00:17.750
This has 0 because
Lz doesn't depend
01:00:17.750 --> 01:00:21.130
on the z-component
of the oscillator.
01:00:21.130 --> 01:00:24.020
And this has minus h bar.
01:00:24.020 --> 01:00:27.740
So here you see actually,
the whole structure of the L
01:00:27.740 --> 01:00:29.610
equal 1 multiplet.
01:00:29.610 --> 01:00:33.180
We said that we have at
this level L equals 1.
01:00:33.180 --> 01:00:36.640
And indeed, for L equals
1, you expect the state
01:00:36.640 --> 01:00:40.640
with Lz equal plus
1, 0, and minus 1.
01:00:40.640 --> 01:00:42.890
So you see the whole thing.
01:00:42.890 --> 01:00:48.260
For n equals 2, what do you get?
01:00:48.260 --> 01:00:54.570
Well, you see a state, for
example, of a right dagger
01:00:54.570 --> 01:00:58.800
a right dagger on the vacuum.
01:00:58.800 --> 01:01:07.740
And that has Lz equals 2 h bar.
01:01:07.740 --> 01:01:12.900
And therefore, you must have--
since you cannot have states
01:01:12.900 --> 01:01:16.880
with higher Lz, you cannot
have a state, for example,
01:01:16.880 --> 01:01:21.460
here with Lz equal 3.
01:01:21.460 --> 01:01:23.950
So you cannot have an L equal 3.
01:01:23.950 --> 01:01:27.300
In fact, for any N
that you build states,
01:01:27.300 --> 01:01:31.440
you can only get
states with whatever
01:01:31.440 --> 01:01:35.540
N is is the maximum
value that Lz can have,
01:01:35.540 --> 01:01:38.540
which is something I want to
illustrate just generically
01:01:38.540 --> 01:01:39.980
for a second.
01:01:39.980 --> 01:01:48.790
So in order to show that, let
me go back here and exhibit
01:01:48.790 --> 01:01:51.395
for you a little of
the general structure.
01:01:54.820 --> 01:02:01.450
So suppose you're building
now with N equal n.
01:02:01.450 --> 01:02:05.170
The total number
is N. So you have
01:02:05.170 --> 01:02:10.820
a state with a right dagger
to the n on the vacuum.
01:02:10.820 --> 01:02:15.240
And this is the state
with highest possible Lz
01:02:15.240 --> 01:02:19.840
because all the
oscillators are aR dagger.
01:02:19.840 --> 01:02:21.590
So Lz is the highest.
01:02:26.980 --> 01:02:32.965
And highest Lz is,
in fact, n h bar.
01:02:35.510 --> 01:02:40.150
Now, let's try to build a state
with a little bit less Lz.
01:02:40.150 --> 01:02:43.470
You see, if this is
a multiplet, this
01:02:43.470 --> 01:02:47.960
has to be a multiplet with some
amount of angular momentum.
01:02:47.960 --> 01:02:52.260
So it's going to go from
Lz equal n, n minus 1,
01:02:52.260 --> 01:02:53.900
up to minus n.
01:02:53.900 --> 01:02:59.190
There are going to be 2n plus
1 states of this much angular
01:02:59.190 --> 01:03:01.560
momentum because this
has to be a multiplet.
01:03:01.560 --> 01:03:05.190
So here you have a
state with one unit
01:03:05.190 --> 01:03:11.450
less of angular momentum, a
right dagger to the n minus 1,
01:03:11.450 --> 01:03:13.000
times an az dagger.
01:03:15.700 --> 01:03:18.390
I claim that's the
only state that you
01:03:18.390 --> 01:03:22.020
can build with one
unit less of angular
01:03:22.020 --> 01:03:25.560
momentum in the
z-direction because I've
01:03:25.560 --> 01:03:32.640
traded this aR for an az.
01:03:32.640 --> 01:03:35.930
So this must be the second
state in the multiplet.
01:03:35.930 --> 01:03:39.720
This multiplet with
highest value of L,
01:03:39.720 --> 01:03:45.630
which is equal to n, corresponds
to an angular momentum l,
01:03:45.630 --> 01:03:47.400
little l, equals n.
01:03:50.190 --> 01:03:53.990
And then, it must have
this 2n plus 1 states.
01:03:53.990 --> 01:03:55.510
And here is the second state.
01:03:55.510 --> 01:04:00.090
So this is Lz equals nh bar.
01:04:00.090 --> 01:04:03.050
And here, n minus 1 h bar.
01:04:03.050 --> 01:04:05.680
And I don't think there's any
other state at that level.
01:04:05.680 --> 01:04:09.870
Let's lower the angular
momentum once more.
01:04:09.870 --> 01:04:12.720
So what do we get?
01:04:12.720 --> 01:04:20.073
a right dagger n minus
2 az dagger squared.
01:04:24.210 --> 01:04:30.780
That's another state with
one less angular momentum
01:04:30.780 --> 01:04:31.440
than this.
01:04:31.440 --> 01:04:35.220
This, in fact, has n
minus 2 times h bar.
01:04:35.220 --> 01:04:38.270
Now, is that the
unique state that I
01:04:38.270 --> 01:04:41.510
can have with two units
less of angular momentum?
01:04:41.510 --> 01:04:42.360
No.
01:04:42.360 --> 01:04:44.282
What is the other one?
01:04:44.282 --> 01:04:47.270
AUDIENCE: aR to
the n minus 1 a l?
01:04:47.270 --> 01:04:49.110
PROFESSOR: Correct,
that lowers it.
01:04:49.110 --> 01:04:59.820
third so here you have aR to the
n minus 1 a left on the vacuum.
01:04:59.820 --> 01:05:05.270
That's another state with two
units less of angular momentum.
01:05:05.270 --> 01:05:11.160
So in this situation, a
funny thing has happened.
01:05:11.160 --> 01:05:17.620
And here's why you
understand the jump of 2.
01:05:17.620 --> 01:05:19.540
This state, you
actually-- if you're
01:05:19.540 --> 01:05:22.240
trying to build this
multiplet, now you
01:05:22.240 --> 01:05:26.670
have two states that have
the same value of Lz.
01:05:26.670 --> 01:05:30.590
And you actually don't
know whether the next state
01:05:30.590 --> 01:05:32.650
in the multiplet
is this, or that,
01:05:32.650 --> 01:05:34.770
or some linear combination.
01:05:34.770 --> 01:05:38.420
It better be some
linear combination.
01:05:38.420 --> 01:05:43.220
But the fact is that at this
level, you found another state.
01:05:43.220 --> 01:05:46.720
So this multiplet
will go on and it
01:05:46.720 --> 01:05:48.880
will be some linear combination.
01:05:48.880 --> 01:05:51.360
Maybe this diagram
doesn't illustrate that.
01:05:51.360 --> 01:05:54.810
But then you will have
another state here.
01:05:54.810 --> 01:05:59.280
So some other linear combination
that builds another multiplet.
01:05:59.280 --> 01:06:04.840
And this multiplet has two
units less of angular momentum.
01:06:04.840 --> 01:06:10.422
And that explains why
this diagram always jumps.
01:06:10.422 --> 01:06:13.010
It always jumps 2.
01:06:13.010 --> 01:06:15.590
And you could do that here.
01:06:15.590 --> 01:06:19.170
If you tried to write
the next things here,
01:06:19.170 --> 01:06:21.750
you will find two states
that you can write.
01:06:21.750 --> 01:06:24.190
But if you go one lower,
you will find three states.
01:06:24.190 --> 01:06:26.400
Which means that
at the next level,
01:06:26.400 --> 01:06:30.550
you built another-- you
need another state with two
01:06:30.550 --> 01:06:34.820
units less of angular
momentum each time.
01:06:34.820 --> 01:06:37.190
So pretty much that's it.
01:06:37.190 --> 01:06:41.132
That illustrates how
this diagram happens.
01:06:41.132 --> 01:06:42.590
The only thing we
haven't answered,
01:06:42.590 --> 01:06:48.770
and you will see
that in an exercise,
01:06:48.770 --> 01:06:53.010
how could I have understood
from the beginning
01:06:53.010 --> 01:06:56.010
that this would have happened
rather than building it
01:06:56.010 --> 01:06:58.880
this way that
there's this thing?
01:06:58.880 --> 01:07:00.500
And what you will
find is that there's
01:07:00.500 --> 01:07:04.200
some operators that commute
with the Hamiltonian that
01:07:04.200 --> 01:07:06.220
move you from here to here.
01:07:06.220 --> 01:07:07.960
And that explains it all.
01:07:07.960 --> 01:07:10.420
Because if you have an
operator that commutes with
01:07:10.420 --> 01:07:13.040
the Hamiltonian, it
cannot change the energy.
01:07:13.040 --> 01:07:18.020
And if it changes the value of
L, it explains why it happened.
01:07:18.020 --> 01:07:20.770
So that's something that
you need to discover,
01:07:20.770 --> 01:07:22.730
what are these operators.
01:07:22.730 --> 01:07:24.670
I can give you a hint.
01:07:24.670 --> 01:07:32.960
An operator for the
type ax dagger ay
01:07:32.960 --> 01:07:37.680
destroys a y oscillator,
creates an x one.
01:07:37.680 --> 01:07:41.480
It doesn't change the energy
because it adds one thing
01:07:41.480 --> 01:07:42.740
and loses one.
01:07:42.740 --> 01:07:46.290
So this kind of thing must
commute with the Hamiltonian.
01:07:46.290 --> 01:07:48.370
And these are the
kind of objects--
01:07:48.370 --> 01:07:49.880
there are lots of them.
01:07:49.880 --> 01:07:53.720
So surprising new things that
commute with the Hamiltonian.
01:07:53.720 --> 01:07:57.220
And there's a whole
hidden symmetry in here
01:07:57.220 --> 01:08:00.370
that is generated by
operators of this form.
01:08:00.370 --> 01:08:03.160
So it's something you will see.
01:08:03.160 --> 01:08:05.960
Now, the last 15 minutes
I want to show you
01:08:05.960 --> 01:08:09.780
what happens with hydrogen.
01:08:09.780 --> 01:08:12.000
There's a similar
phenomenon there
01:08:12.000 --> 01:08:15.670
that we're going to try
to explain in detail.
01:08:15.670 --> 01:08:20.210
So a couple of things
about hydrogen.
01:08:24.000 --> 01:08:32.290
So hydrogen H is
equal to p squared
01:08:32.290 --> 01:08:38.450
over 2m minus e squared over r.
01:08:38.450 --> 01:08:43.979
There's a natural length scale
that people many times struggle
01:08:43.979 --> 01:08:47.569
to find it, the Bohr radius.
01:08:47.569 --> 01:08:48.740
This does it come from here?
01:08:48.740 --> 01:08:52.149
How do you see immediately
what is the Bohr radius?
01:08:52.149 --> 01:08:56.260
Well, the Bohr radius can
be estimated by saying,
01:08:56.260 --> 01:09:01.350
well, this is an energy but it
must be similar to this energy.
01:09:01.350 --> 01:09:08.040
So p is like h over a distance.
01:09:08.040 --> 01:09:12.050
So let's call it a0.
01:09:12.050 --> 01:09:14.050
So that's p squared.
01:09:14.050 --> 01:09:20.740
m is the reduced mass of the
proton electron system roughly
01:09:20.740 --> 01:09:23.080
equal to the electron mass.
01:09:23.080 --> 01:09:26.300
And then you set it
equal to this one
01:09:26.300 --> 01:09:29.200
because you're just doing units.
01:09:29.200 --> 01:09:32.359
You want to find the quantity
that has units of length
01:09:32.359 --> 01:09:33.910
and there you got it.
01:09:33.910 --> 01:09:38.560
That's the famous Bohr radius.
01:09:38.560 --> 01:09:42.069
p is h bar over a
distance, therefore
01:09:42.069 --> 01:09:43.899
this thing must be an energy.
01:09:43.899 --> 01:09:45.510
It must be equal to this.
01:09:45.510 --> 01:09:49.750
And from this one,
you can solve for a0.
01:09:49.750 --> 01:09:54.710
It's h squared over m e squared.
01:09:54.710 --> 01:09:58.970
The 1 over e squared is
very famous and important.
01:09:58.970 --> 01:10:02.170
It reflects the
fact that if you had
01:10:02.170 --> 01:10:03.970
the interaction
between the electron
01:10:03.970 --> 01:10:08.140
and the proton go to 0, the
radius would be infinite.
01:10:08.140 --> 01:10:10.660
As it becomes weaker and
weaker the interaction,
01:10:10.660 --> 01:10:13.890
the radius of the
hydrogen atom blows up.
01:10:13.890 --> 01:10:20.330
So this is about
0.529 angstroms,
01:10:20.330 --> 01:10:23.210
where an angstrom is 10
to the minus 10 meters.
01:10:25.750 --> 01:10:28.016
And what is the energy scale?
01:10:34.980 --> 01:10:43.820
Well, e squared over a0 is
the energy scale, roughly.
01:10:43.820 --> 01:10:48.740
And in fact, e squared
over 2a 0, if you wish,
01:10:48.740 --> 01:10:49.885
is a famous number.
01:10:49.885 --> 01:10:57.302
It's about 13.6 ev.
01:10:57.302 --> 01:11:00.440
So how about the spectrum?
01:11:00.440 --> 01:11:03.600
And how do you find that?
01:11:03.600 --> 01:11:08.090
Well, there's one
nice way of doing
01:11:08.090 --> 01:11:10.570
this, which you will
see in the problem,
01:11:10.570 --> 01:11:12.670
to find at least
the ground state.
01:11:12.670 --> 01:11:18.330
And it's a very elegant
way based on factorization.
01:11:18.330 --> 01:11:20.670
Let we mention it.
01:11:20.670 --> 01:11:22.340
It is called Hamiltonian.
01:11:22.340 --> 01:11:26.580
It can be written as a
constant gamma plus 1
01:11:26.580 --> 01:11:42.645
over 2m sum over k pk plus i
beta xk over r times pk minus i
01:11:42.645 --> 01:11:48.194
beta xk over r.
01:11:48.194 --> 01:11:52.130
It's a factorized version of the
Hamiltonian of a hydrogen atom.
01:11:52.130 --> 01:11:55.710
Apparently, not a
well-known result.
01:11:55.710 --> 01:11:57.910
Professor [? Jackiw ?]
mentioned it to me.
01:11:57.910 --> 01:12:00.990
I don't think I've
seen it in any book.
01:12:04.200 --> 01:12:08.350
So there's a constant
beta and a constant gamma
01:12:08.350 --> 01:12:11.370
for which this becomes
exactly that one.
01:12:11.370 --> 01:12:13.410
So gamma and beta
to be determined.
01:12:20.100 --> 01:12:22.720
And you have to be a
little careful here
01:12:22.720 --> 01:12:28.060
when you expand that this term
and this term don't commute.
01:12:28.060 --> 01:12:29.920
And this and this don't commute.
01:12:29.920 --> 01:12:33.230
But after you're
done, it comes out.
01:12:33.230 --> 01:12:37.440
And then, the ground
state wave function
01:12:37.440 --> 01:12:41.940
is-- since this is an
operator and here it's dagger,
01:12:41.940 --> 01:12:46.690
the ground state wave function
is-- the lowest possible energy
01:12:46.690 --> 01:12:48.580
wave function is
one in which this
01:12:48.580 --> 01:12:50.380
would kill the wave function.
01:12:50.380 --> 01:12:58.540
So pk minus i beta xk over r
should kill the ground state
01:12:58.540 --> 01:13:01.270
wave function.
01:13:01.270 --> 01:13:04.570
And then the energy,
E ground, would
01:13:04.570 --> 01:13:08.850
be equal to precisely
this constant gamma.
01:13:08.850 --> 01:13:13.200
And you will show, in fact,
that yes, this has a solution.
01:13:13.200 --> 01:13:15.090
And that's the
ground state energy
01:13:15.090 --> 01:13:20.380
of the oscillator of the
hydrogen atom, of course.
01:13:20.380 --> 01:13:28.690
So this looks like three
equations, pk with k
01:13:28.690 --> 01:13:30.155
equals 1 to 3.
01:13:33.140 --> 01:13:37.690
But it reduces to 1 if the
state is spherically symmetric.
01:13:37.690 --> 01:13:43.070
So it's a nice thing and
it gives you the answer.
01:13:43.070 --> 01:13:46.280
Now, the whole spectrum
of the hydrogen atom
01:13:46.280 --> 01:13:51.240
is as interestingly
degenerate as one
01:13:51.240 --> 01:13:54.570
of the three-dimensional
harmonic oscillator.
01:13:54.570 --> 01:14:03.180
And a reminder of it is
that-- should I go here?
01:14:03.180 --> 01:14:05.880
Yes.
01:14:05.880 --> 01:14:10.510
You have here energies
and here l's. l
01:14:10.510 --> 01:14:13.310
equals 0 you have one state.
01:14:13.310 --> 01:14:17.730
l equals 1 you have
another state that's here.
01:14:17.730 --> 01:14:21.090
But actually, l equals 0
will have another state.
01:14:21.090 --> 01:14:24.270
And then it goes on like
that, another state here,
01:14:24.270 --> 01:14:28.240
state here, state
here for l equals 2.
01:14:31.000 --> 01:14:33.300
And the first state is here.
01:14:33.300 --> 01:14:35.690
The first state of this
one aligns with this one.
01:14:35.690 --> 01:14:38.180
The first state of
that aligns with that.
01:14:38.180 --> 01:14:44.170
So they go like
that, the states just
01:14:44.170 --> 01:14:48.750
continue to go exactly
with this symmetry.
01:14:48.750 --> 01:14:54.180
So let me use label
that is common,
01:14:54.180 --> 01:14:58.173
to call this the state nu
equals 0 for L equals 0.
01:14:58.173 --> 01:14:59.735
Nu equals 1.
01:14:59.735 --> 01:15:01.000
Nu equals 2.
01:15:01.000 --> 01:15:02.970
Nu equals 3.
01:15:02.970 --> 01:15:06.620
This is the first state
with L equals 1 is here.
01:15:06.620 --> 01:15:08.900
So we'll call it nu equals 0.
01:15:08.900 --> 01:15:10.460
Nu equals 1.
01:15:10.460 --> 01:15:12.210
New equals 2.
01:15:12.210 --> 01:15:18.070
The first state here is
nu equals 0, nu equals 1.
01:15:18.070 --> 01:15:21.190
And then the energies.
01:15:21.190 --> 01:15:28.170
You define n to be
nu plus l plus 1.
01:15:28.170 --> 01:15:33.170
Therefore, this
corresponds to n equals 1.
01:15:33.170 --> 01:15:36.000
This corresponds to n equals 2.
01:15:36.000 --> 01:15:40.920
That corresponds to nu can be
1 and l equals 0 or nu can be 0
01:15:40.920 --> 01:15:43.370
and l equals 1.
01:15:43.370 --> 01:15:47.850
This is n equal 3,
and things like that.
01:15:47.850 --> 01:15:54.380
And then the energies
of those states, nl
01:15:54.380 --> 01:15:58.450
is, in fact, minus z squared.
01:15:58.450 --> 01:16:00.840
Well, forget the z squared.
01:16:00.840 --> 01:16:05.145
e squared over 2 a0
1 over n squared.
01:16:09.960 --> 01:16:11.650
So the only thing
that happens is
01:16:11.650 --> 01:16:15.840
that there's a degeneracy,
complete degeneracy.
01:16:15.840 --> 01:16:18.200
Very powerful degeneracy.
01:16:18.200 --> 01:16:30.730
And then, l can only run up
to-- from 0, 1, up to n minus 1
01:16:30.730 --> 01:16:32.350
in these variables.
01:16:32.350 --> 01:16:35.450
So this is the
picture of hydrogen.
01:16:35.450 --> 01:16:40.320
So you've seen several pictures
already-- the square well,
01:16:40.320 --> 01:16:43.530
the three-dimensional
harmonic oscillator,
01:16:43.530 --> 01:16:44.860
and the hydrogen one.
01:16:44.860 --> 01:16:48.200
Each one has a
different picture.
01:16:48.200 --> 01:16:52.510
Now, in order to
understand this one--
01:16:52.510 --> 01:16:54.470
this one is not that difficult.
01:16:54.470 --> 01:16:57.790
But the one of the hydrogen
is really more interesting.
01:16:57.790 --> 01:17:00.780
It all originates
with the idea of what
01:17:00.780 --> 01:17:04.830
is called the
Runge-Lenz vector, which
01:17:04.830 --> 01:17:08.720
I'm going to use the last
five minutes to introduce.
01:17:08.720 --> 01:17:10.745
And think about it a little.
01:17:17.660 --> 01:17:20.680
So it comes from
classical mechanics.
01:17:20.680 --> 01:17:26.180
So we have an elliptical
orbit, orbits,
01:17:26.180 --> 01:17:28.910
and people figured out
there was something
01:17:28.910 --> 01:17:32.620
very funny about characterizing
elliptical orbit.
01:17:32.620 --> 01:17:35.600
So consider a
Hamiltonian, which is p
01:17:35.600 --> 01:17:40.780
squared over 2m plus
v of r, a potential.
01:17:40.780 --> 01:17:44.530
The force, classically,
would be minus the gradient
01:17:44.530 --> 01:17:49.320
of the potential, which is minus
the derivative of the potential
01:17:49.320 --> 01:17:52.445
with respect to r times
the r unit vector.
01:17:55.850 --> 01:18:00.385
Now classically-- this
all begins classically.
01:18:04.950 --> 01:18:09.260
Except for spin 1/2
systems, classical physics
01:18:09.260 --> 01:18:13.680
really tells you a lot
of what's going on.
01:18:13.680 --> 01:18:25.490
So classically, dp dt is
the force and it's minus v
01:18:25.490 --> 01:18:30.220
prime over r r vector over r.
01:18:30.220 --> 01:18:37.330
And dl dt, the angular momentum,
it's a central potential.
01:18:37.330 --> 01:18:41.040
The angular momentum is 0.
01:18:41.040 --> 01:18:42.720
It's rate of change is 0.
01:18:42.720 --> 01:18:48.710
There's no torque on the
particle, so this should be 0.
01:18:48.710 --> 01:18:51.560
Now, the interesting
thing that happens
01:18:51.560 --> 01:18:54.690
is that this doesn't
exhaust the kind of things
01:18:54.690 --> 01:18:58.490
that are, in fact, conserved.
01:18:58.490 --> 01:19:06.900
So there is something
more that is conserved.
01:19:06.900 --> 01:19:08.825
And it's a very
surprising quantity.
01:19:08.825 --> 01:19:11.780
It's so surprising
that people have
01:19:11.780 --> 01:19:16.960
a hard time
imagining what it is.
01:19:16.960 --> 01:19:25.430
I will write it down and show
you how it sort of happens.
01:19:25.430 --> 01:19:30.750
Well, you have to
begin with p cross L.
01:19:30.750 --> 01:19:34.570
Why you would think of p cross
l is a little bit of a mystery,
01:19:34.570 --> 01:19:38.530
but it's an interesting thing.
01:19:38.530 --> 01:19:42.900
Now, here is a computation
that will be in the notes
01:19:42.900 --> 01:19:45.030
that you can try doing.
01:19:45.030 --> 01:19:52.740
And it takes a little bit
of work, but it's algebra.
01:19:52.740 --> 01:19:57.490
If you compute this and
do a fair amount of work,
01:19:57.490 --> 01:20:01.070
like five, six lines--
I would suspect it's
01:20:01.070 --> 01:20:03.130
fairly non-trivial
to do it if you
01:20:03.130 --> 01:20:06.170
don't see how it's
being done, but it
01:20:06.170 --> 01:20:09.720
will be in the notes-- you
get the following thing.
01:20:13.980 --> 01:20:18.110
Just by manipulating the
time derivative of p cross L,
01:20:18.110 --> 01:20:21.280
you get this.
01:20:21.280 --> 01:20:25.420
Which is equal to m times
the potential differentiator
01:20:25.420 --> 01:20:28.040
times r squared times the
time derivative of this.
01:20:28.040 --> 01:20:30.410
So time derivative,
time derivative.
01:20:30.410 --> 01:20:35.840
You can get the conservation
if this is a constant.
01:20:35.840 --> 01:20:37.800
So when is this a constant?
01:20:37.800 --> 01:20:42.390
If this is some
constant, say, e squared,
01:20:42.390 --> 01:20:44.190
you would get a conservation.
01:20:44.190 --> 01:20:52.550
But what is v prime equals
e squared over r squared?
01:20:52.550 --> 01:20:57.180
It would give you that v
of r is essentially minus
01:20:57.180 --> 01:20:59.830
e squared over r.
01:20:59.830 --> 01:21:02.050
That's the potential
of hydrogen.
01:21:02.050 --> 01:21:08.870
Or the 1 over r potential, 1
over r squared force field.
01:21:08.870 --> 01:21:14.730
So in 1 over r potentials,
this is a number.
01:21:14.730 --> 01:21:19.360
And then you get an incredible
conservation law, d dt
01:21:19.360 --> 01:21:31.810
of p cross L minus m e squared
r hat over r is equal to 0.
01:21:31.810 --> 01:21:36.420
So something fairly unexpected
that something like this
01:21:36.420 --> 01:21:38.290
could be conserved.
01:21:38.290 --> 01:21:42.625
So actually, you can try
to figure out what this is.
01:21:45.990 --> 01:21:49.360
And there's two neat--
first, one thing
01:21:49.360 --> 01:21:52.920
that people do,
which is convenient,
01:21:52.920 --> 01:21:56.050
is to make this into
unit-free vector.
01:21:56.050 --> 01:22:05.450
So define R to be p
cross L over m e squared
01:22:05.450 --> 01:22:09.150
minus r vector over r.
01:22:09.150 --> 01:22:10.460
This has no units.
01:22:13.450 --> 01:22:14.990
And it's supposed
to be conserved.
01:22:21.230 --> 01:22:25.570
Now, one thing you will
check in the homework
01:22:25.570 --> 01:22:28.980
is that this is conserved
quantum mechanically as well.
01:22:28.980 --> 01:22:31.190
That is, this is
an operator that
01:22:31.190 --> 01:22:33.370
commutes with a Hamiltonian.
01:22:33.370 --> 01:22:36.800
Very interesting calculation.
01:22:36.800 --> 01:22:38.945
This is a Hermitian
operator, so you
01:22:38.945 --> 01:22:43.400
will have to Hermiticize
the p cross L to do that.
01:22:43.400 --> 01:22:46.020
But it will commute
with the Hamiltonian.
01:22:46.020 --> 01:22:49.870
But what I want to finish
now is with your intuition
01:22:49.870 --> 01:22:52.530
as to what this is.
01:22:52.530 --> 01:22:56.640
And this was a very interesting
discovery, this vector.
01:22:56.640 --> 01:22:59.170
In fact, people
didn't appreciate
01:22:59.170 --> 01:23:03.680
what was the role of this
vector for quite some time.
01:23:03.680 --> 01:23:08.270
So apparently, it was discovered
and forgotten, and discovered
01:23:08.270 --> 01:23:11.490
and forgotten like
two or three times.
01:23:11.490 --> 01:23:15.700
And for us, it's going to
be quite crucial because I
01:23:15.700 --> 01:23:19.170
said to you that this operator
commutes with the Hamiltonian.
01:23:22.920 --> 01:23:25.660
So actually, you will
get conservation laws
01:23:25.660 --> 01:23:28.911
and will help us explain the
degeneracy of the hydrogen
01:23:28.911 --> 01:23:29.410
atom.
01:23:29.410 --> 01:23:32.390
So it will be very
important for us.
01:23:32.390 --> 01:23:35.850
Now, how does this look?
01:23:35.850 --> 01:23:39.750
First of all, if you
had a circular orbit,
01:23:39.750 --> 01:23:42.500
how does it work?
01:23:42.500 --> 01:23:45.050
Have a circular orbit.
01:23:45.050 --> 01:23:51.355
Let's see, p is here,
L is out of the board.
01:23:51.355 --> 01:23:59.400
p cross L is here
over m e squared.
01:23:59.400 --> 01:24:08.190
And the radial vector
is here, the hat vector.
01:24:08.190 --> 01:24:13.870
So the sum of these two vectors
p cross L and the radial vector
01:24:13.870 --> 01:24:14.685
must be conserved.
01:24:17.250 --> 01:24:19.820
But how could it be?
01:24:19.820 --> 01:24:24.950
If they don't cancel, it
either points in or points out.
01:24:24.950 --> 01:24:29.140
And then it would just rotate
and it would not be conserved.
01:24:29.140 --> 01:24:33.650
So actually, for a circular
orbit, you can calculate it.
01:24:33.650 --> 01:24:35.070
See the notes.
01:24:35.070 --> 01:24:36.880
Actually, it's an
easy calculation.
01:24:36.880 --> 01:24:40.960
And you can verify that this
vector is, in fact, precisely
01:24:40.960 --> 01:24:42.260
opposite this.
01:24:42.260 --> 01:24:44.080
And it's 0.
01:24:44.080 --> 01:24:45.160
So you say, great.
01:24:45.160 --> 01:24:48.495
You discover something that
is conserved, but it's 0.
01:24:48.495 --> 01:24:50.010
No.
01:24:50.010 --> 01:24:54.935
The thing is that this thing is
not 0 for an elliptical orbit.
01:24:57.970 --> 01:24:59.830
So how can you see that?
01:24:59.830 --> 01:25:05.330
Well here at this
point, p is up here.
01:25:05.330 --> 01:25:06.450
L is out.
01:25:06.450 --> 01:25:13.680
And p cross L, just like
before, is out and r hat is in.
01:25:16.662 --> 01:25:18.890
And you say, well, OK.
01:25:18.890 --> 01:25:20.420
Now the same problem.
01:25:20.420 --> 01:25:23.820
If they don't cancel,
it's going to be a vector
01:25:23.820 --> 01:25:25.160
and going to rotate.
01:25:25.160 --> 01:25:27.540
But it has to be conserved.
01:25:27.540 --> 01:25:31.840
So actually, let's
look at it here.
01:25:31.840 --> 01:25:33.910
Here, the main
thing of an ellipse,
01:25:33.910 --> 01:25:36.670
if you have the focus here,
is that this line is not--
01:25:36.670 --> 01:25:39.420
the tangent is not horizontal.
01:25:39.420 --> 01:25:42.480
So the momentum is here.
01:25:42.480 --> 01:25:48.120
L is out of the blackboard,
but p cross L now is like that.
01:25:53.930 --> 01:25:57.500
And the radial vector is here.
01:25:57.500 --> 01:26:00.120
And they don't cancel.
01:26:00.120 --> 01:26:02.870
So the only thing
that can happen--
01:26:02.870 --> 01:26:04.750
since this is vertical,
this is vertical.
01:26:04.750 --> 01:26:09.190
It's a little bit to the left--
is that the r vector must
01:26:09.190 --> 01:26:14.260
be a little vector
horizontal here.
01:26:14.260 --> 01:26:19.380
Because the sum of this
vector and this vector--
01:26:19.380 --> 01:26:20.900
it has to be horizontal.
01:26:20.900 --> 01:26:23.100
Here we don't know
if they can cancel.
01:26:23.100 --> 01:26:26.410
But if they don't cancel,
it's definitely horizontal.
01:26:26.410 --> 01:26:30.220
We know it's conserved, so
it must be horizontal here.
01:26:30.220 --> 01:26:32.010
So it points in.
01:26:32.010 --> 01:26:36.950
So the Runge-Lenz
vector r points in.
01:26:36.950 --> 01:26:40.820
And it's, in fact, that.
01:26:40.820 --> 01:26:46.520
So here you go, this is a
vector that is conserved.
01:26:46.520 --> 01:26:49.010
And its properties
that is really
01:26:49.010 --> 01:26:51.720
about the axis of the ellipse.
01:26:51.720 --> 01:26:54.500
It tells you where the axis is.
01:26:54.500 --> 01:26:59.470
In Einstein's theory of gravity,
the potential is not 1/r
01:26:59.470 --> 01:27:02.590
and the ellipsis [? precess ?]
and the Runge vector
01:27:02.590 --> 01:27:03.920
is not conserved.
01:27:03.920 --> 01:27:07.810
But in 1/r potentials,
it is conserved.
01:27:07.810 --> 01:27:11.540
The final thing-- sorry
for taking so long--
01:27:11.540 --> 01:27:14.940
is that the magnitude
of r is precisely
01:27:14.940 --> 01:27:17.730
the eccentricity of the orbit.
01:27:17.730 --> 01:27:21.110
So it's a really nice way
of characterizing the orbits
01:27:21.110 --> 01:27:24.930
and we'll be using it
in the next lecture.
01:27:24.930 --> 01:27:26.907
See you on Wednesday.