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PROFESSOR: All right.
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Hi, everyone.
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AUDIENCE: Hi.
00:00:30.276 --> 00:00:32.650
PROFESSOR: We're getting
towards the end of the semester.
00:00:32.650 --> 00:00:37.000
Things are starting to
cohere and come together.
00:00:37.000 --> 00:00:40.540
We have one more midterm exam.
00:00:40.540 --> 00:00:44.880
So there is an exam
next Thursday, the 18th.
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OK?
00:00:45.380 --> 00:00:47.730
There will be a problem set due.
00:00:47.730 --> 00:00:50.310
It'll be posted later today,
and it will be due next week
00:00:50.310 --> 00:00:51.920
on Tuesday as usual.
00:00:51.920 --> 00:00:55.840
Of course, next week on Tuesday
is a holiday technically,
00:00:55.840 --> 00:00:58.530
so we'll actually make the
due date be on Wednesday.
00:00:58.530 --> 00:01:01.280
So on Wednesday at 10 o'clock.
00:01:01.280 --> 00:01:02.870
You should think
of this problem set
00:01:02.870 --> 00:01:04.930
as part of the
review for the exam.
00:01:04.930 --> 00:01:09.820
Material that is covered
today and on Thursday
00:01:09.820 --> 00:01:11.900
will be fair game for the exam.
00:01:14.560 --> 00:01:16.070
So the format of
the exam is going
00:01:16.070 --> 00:01:17.722
to be much more canonical.
00:01:17.722 --> 00:01:19.430
It's going to be a
series of short answer
00:01:19.430 --> 00:01:21.450
plus a series of computations.
00:01:21.450 --> 00:01:25.070
They'll be, roughly speaking,
at the level of the problem
00:01:25.070 --> 00:01:26.990
sets, and a practice
exam will be
00:01:26.990 --> 00:01:28.650
posted in the next
couple of days.
00:01:32.130 --> 00:01:35.350
The practice exam is going
to be of the same general
00:01:35.350 --> 00:01:36.920
intellectual
difficulty, but it's
00:01:36.920 --> 00:01:40.140
going to be considerably
longer than the actual exam.
00:01:40.140 --> 00:01:42.560
So as a check for yourself
here's what I would recommend.
00:01:42.560 --> 00:01:44.820
I would recommend sitting
down and giving yourself
00:01:44.820 --> 00:01:48.660
an hour and a half or two
hours with the practice exam
00:01:48.660 --> 00:01:51.510
and see how that goes.
00:01:51.510 --> 00:01:53.380
OK?
00:01:53.380 --> 00:01:55.290
Try to give yourself
the time constraint.
00:01:55.290 --> 00:01:59.240
And here's one of the
things that-- there's
00:01:59.240 --> 00:02:01.810
a very strong correlation
between having
00:02:01.810 --> 00:02:05.540
nailed the problem sets, and
worked through all the practice
00:02:05.540 --> 00:02:07.780
exams, and just done
a lot of problems,
00:02:07.780 --> 00:02:09.130
and doing well on exams.
00:02:09.130 --> 00:02:10.694
You'll be more confident.
00:02:10.694 --> 00:02:12.610
The only way to study
for these things is just
00:02:12.610 --> 00:02:13.660
do a lot of problems.
00:02:13.660 --> 00:02:17.380
So on Stellar, for example,
are a bunch of previous problem
00:02:17.380 --> 00:02:19.780
sets, and previous
exams and practice
00:02:19.780 --> 00:02:22.080
exams from previous years.
00:02:22.080 --> 00:02:25.700
I would encourage you to
look at those also on OCW
00:02:25.700 --> 00:02:27.240
and use those as practice.
00:02:27.240 --> 00:02:28.794
OK, any practical
questions before we
00:02:28.794 --> 00:02:29.710
get on to the physics.
00:02:29.710 --> 00:02:30.417
Yeah?
00:02:30.417 --> 00:02:32.802
AUDIENCE: Will the exam
cover all the material,
00:02:32.802 --> 00:02:34.427
or all the material [INAUDIBLE]?
00:02:34.427 --> 00:02:36.010
PROFESSOR: This is
a cumulative topic.
00:02:36.010 --> 00:02:37.570
So the question is does it
cover everything, or just
00:02:37.570 --> 00:02:38.560
last couple of weeks.
00:02:38.560 --> 00:02:40.800
And it's a cumulative topic,
so the entire semester
00:02:40.800 --> 00:02:44.820
is necessary in order
to answer the problems.
00:02:44.820 --> 00:02:47.620
Other questions?
00:02:47.620 --> 00:02:48.621
OK.
00:02:48.621 --> 00:02:50.980
Anything else?
00:02:50.980 --> 00:02:57.350
So quick couple of
review questions
00:02:57.350 --> 00:03:00.234
before we launch into the
main material of today.
00:03:00.234 --> 00:03:01.900
These are going to
turn out to be useful
00:03:01.900 --> 00:03:04.970
reminders later on
in today's lecture.
00:03:04.970 --> 00:03:06.890
So first, suppose
I tell you I have
00:03:06.890 --> 00:03:11.700
a system with energy operator
E, which has an operator A such
00:03:11.700 --> 00:03:18.380
that the commutator of E with
A would say plus h bar A. OK?
00:03:18.380 --> 00:03:21.120
What does that tell you about
the spectrum of the energy
00:03:21.120 --> 00:03:21.620
operator?
00:03:24.876 --> 00:03:26.064
AUDIENCE: It's a ladder.
00:03:26.064 --> 00:03:27.480
PROFESSOR: It's a
ladder, exactly.
00:03:27.480 --> 00:03:30.400
So this tells you that the
spectrum of E, or the energy
00:03:30.400 --> 00:03:34.060
eigenvalues En,
are evenly spaced--
00:03:34.060 --> 00:03:40.030
and we need a dimensional
constant here,
00:03:40.030 --> 00:03:49.050
omega-- evenly spaced
by h bar omega.
00:03:49.050 --> 00:03:54.530
And more precisely, that
given a state phi E,
00:03:54.530 --> 00:03:58.030
we can act on it
with the operator A
00:03:58.030 --> 00:04:02.310
to give us a new state, which
is also an energy eigenstate,
00:04:02.310 --> 00:04:06.250
with energy E plus h bar omega.
00:04:06.250 --> 00:04:07.900
right?
00:04:07.900 --> 00:04:10.460
So any time you see that
commutation relation,
00:04:10.460 --> 00:04:13.610
you know this fact to be true.
00:04:13.610 --> 00:04:14.540
Second statement.
00:04:14.540 --> 00:04:17.220
Suppose I have an
operator B which
00:04:17.220 --> 00:04:19.860
commutes with the
energy operator.
00:04:19.860 --> 00:04:20.360
OK?
00:04:20.360 --> 00:04:22.547
So that the commutator vanishes.
00:04:22.547 --> 00:04:24.255
What does that tell
you about the system?
00:04:27.814 --> 00:04:30.130
AUDIENCE: Simultaneous
eigenfunctions [INAUDIBLE].
00:04:30.130 --> 00:04:31.005
PROFESSOR: Excellent.
00:04:31.005 --> 00:04:32.970
So one one consequence
is that there
00:04:32.970 --> 00:04:37.100
exists simultaneous
eigenfunctions phi sub E, B,
00:04:37.100 --> 00:04:46.576
which are simultaneous
eigenfunctions of both E and B.
00:04:46.576 --> 00:04:47.700
What else does it tell you?
00:04:52.090 --> 00:04:56.557
Well, notice the following.
00:04:56.557 --> 00:04:58.890
Notice that if we took this
computation relation and set
00:04:58.890 --> 00:05:01.670
omega to 0, we get this
commutation relation.
00:05:01.670 --> 00:05:04.440
So this commutation relation is
of the form of this commutation
00:05:04.440 --> 00:05:06.770
relation with omega equals 0.
00:05:06.770 --> 00:05:09.890
So what does that tell
you about the states?
00:05:09.890 --> 00:05:11.210
About the energy eigenvalues?
00:05:14.490 --> 00:05:17.740
What happens if I
take a state phi sub E
00:05:17.740 --> 00:05:20.060
and I act on it with B?
00:05:20.060 --> 00:05:20.780
What do I get?
00:05:25.819 --> 00:05:27.360
What can you say
about this function?
00:05:31.300 --> 00:05:34.287
Well, what is its
eigenvalue under E?
00:05:34.287 --> 00:05:36.370
It's E. It's the same
thing, because they commute.
00:05:36.370 --> 00:05:38.300
You can pull the E
through, multiply it,
00:05:38.300 --> 00:05:40.760
you get the constant E,
you pull it back out.
00:05:40.760 --> 00:05:44.770
This is still an
eigenfunction of phi
00:05:44.770 --> 00:05:47.345
of the energy operator
with the same eigenvalue.
00:05:47.345 --> 00:05:49.220
But is it necessarily
the same eigenfunction?
00:05:52.610 --> 00:05:54.590
No, because B may act
on phi and just give you
00:05:54.590 --> 00:05:55.340
a different state.
00:05:55.340 --> 00:06:00.520
So this is some eigenfunction
with the same energy,
00:06:00.520 --> 00:06:03.994
but it may be a
different eigenfunction.
00:06:03.994 --> 00:06:04.669
OK?
00:06:04.669 --> 00:06:06.210
So let's think of
an example of this.
00:06:06.210 --> 00:06:08.300
An example of this is
consider a free particle.
00:06:11.150 --> 00:06:13.210
In the case of a
free particle, we
00:06:13.210 --> 00:06:18.590
have E is equal to
P squared upon 2m.
00:06:18.590 --> 00:06:21.500
And as a consequence
E and P commute.
00:06:26.510 --> 00:06:29.160
Everyone agree with that?
00:06:29.160 --> 00:06:30.630
Everyone happy with
that statement?
00:06:30.630 --> 00:06:31.420
They commute.
00:06:31.420 --> 00:06:32.950
So plus 0 if you will.
00:06:35.750 --> 00:06:38.600
And here what I wanted to say
is when you have an operator
00:06:38.600 --> 00:06:41.990
that commutes with
the energy, then
00:06:41.990 --> 00:06:45.860
there can be multiple states
with the same energy which
00:06:45.860 --> 00:06:48.200
are different states, right?
00:06:48.200 --> 00:06:49.430
Different states entirely.
00:06:49.430 --> 00:06:51.230
So for example,
in this case, what
00:06:51.230 --> 00:06:51.880
are the energy eigenfunctions?
00:06:51.880 --> 00:06:52.850
Well, e to the ikx.
00:06:55.718 --> 00:07:00.391
And this has energy h bar
squared k squared upon 2m.
00:07:00.391 --> 00:07:02.890
But there's another state, which
is a different state, which
00:07:02.890 --> 00:07:05.325
has the same energy.
00:07:05.325 --> 00:07:08.340
e to the minus ikx.
00:07:08.340 --> 00:07:09.610
OK?
00:07:09.610 --> 00:07:13.320
So when you have an operator
that commutes with the energy
00:07:13.320 --> 00:07:19.770
operator, you can have
simultaneous eigenfunctions.
00:07:19.770 --> 00:07:24.950
And you can also have
multiple eigenfunctions
00:07:24.950 --> 00:07:26.900
that have the same
energy eigenvalue,
00:07:26.900 --> 00:07:28.080
but are different functions.
00:07:28.080 --> 00:07:29.920
For example, this.
00:07:29.920 --> 00:07:31.390
Everyone cool with that?
00:07:31.390 --> 00:07:34.040
Now, just to make clear
that it's actually
00:07:34.040 --> 00:07:36.320
the commuting that
matters, imagine
00:07:36.320 --> 00:07:39.020
we took not the free particle,
but the harmonic oscillator.
00:07:42.940 --> 00:07:51.700
E is p squared upon 2m plus m
omega squared upon 2 x squared.
00:07:51.700 --> 00:07:57.170
Is it true that E-- I'll call
this harmonic oscillator--
00:07:57.170 --> 00:08:00.869
does E harmonic
oscillator commute with P?
00:08:00.869 --> 00:08:02.410
No, you're going to
get a term that's
00:08:02.410 --> 00:08:06.282
m omega squared x
from the commutator.
00:08:06.282 --> 00:08:08.240
The potential is not
translationally invariant.
00:08:08.240 --> 00:08:09.780
It does not commute
with momentum.
00:08:09.780 --> 00:08:11.860
So this is not equal to 0.
00:08:11.860 --> 00:08:14.865
So that suggests that this
degeneracy, two states having
00:08:14.865 --> 00:08:16.490
the same energy,
should not be present.
00:08:16.490 --> 00:08:18.490
And indeed, are the
states degenerate
00:08:18.490 --> 00:08:20.530
for the harmonic oscillator.
00:08:20.530 --> 00:08:22.100
No.
00:08:22.100 --> 00:08:22.800
No degeneracy.
00:08:28.310 --> 00:08:28.930
Yeah?
00:08:28.930 --> 00:08:31.798
AUDIENCE: We did
something [INAUDIBLE]
00:08:31.798 --> 00:08:34.409
where I think it said that the
eigenfunctions were complete.
00:08:34.409 --> 00:08:35.034
PROFESSOR: Yes.
00:08:35.034 --> 00:08:36.020
AUDIENCE: What does that mean?
00:08:36.020 --> 00:08:37.340
PROFESSOR: What does it
mean for the eigenfunctions
00:08:37.340 --> 00:08:38.110
to be complete?
00:08:38.110 --> 00:08:39.860
What that means is
that they form a basis.
00:08:42.369 --> 00:08:44.119
AUDIENCE: So the basis
doesn't necessarily
00:08:44.119 --> 00:08:46.674
mean not [INAUDIBLE].
00:08:46.674 --> 00:08:49.340
PROFESSOR: Yeah, no one told you
the basis had to be degenerate,
00:08:49.340 --> 00:08:50.700
and in particular,
that's a excellent--
00:08:50.700 --> 00:08:52.324
so the question here
is, wait a minute,
00:08:52.324 --> 00:08:54.726
I thought a basis had to be
a complete set-- if you had
00:08:54.726 --> 00:08:57.100
an energy operator and you
constructed the energy eigen--
00:08:57.100 --> 00:08:57.970
this is a very good question.
00:08:57.970 --> 00:08:58.620
Thank you.
00:08:58.620 --> 00:09:01.220
If I have the energy operator
and I construct it's energy
00:09:01.220 --> 00:09:03.220
eigenfunctions, then those
energy eigenfunctions
00:09:03.220 --> 00:09:06.060
form a complete basis for
any arbitrary function.
00:09:06.060 --> 00:09:07.927
Any function can be
expanded in it, right?
00:09:07.927 --> 00:09:09.510
So for example, for
the free particle,
00:09:09.510 --> 00:09:11.301
the energy eigenfunctions
are e to the ikx,
00:09:11.301 --> 00:09:13.890
the momentum eigenfunctions
for any value of k.
00:09:13.890 --> 00:09:15.800
But wait, how can there
be multiple states
00:09:15.800 --> 00:09:17.804
with the same energy?
00:09:17.804 --> 00:09:19.470
Isn't that double
counting or something?
00:09:19.470 --> 00:09:21.680
And the important thing is
those guys have the same energy,
00:09:21.680 --> 00:09:23.150
but they have different momenta.
00:09:23.150 --> 00:09:24.630
They're different states.
00:09:24.630 --> 00:09:26.390
One has momentum h bar k.
00:09:26.390 --> 00:09:28.037
One has momentum minus h bar k.
00:09:28.037 --> 00:09:29.620
And you know that
that has to be true,
00:09:29.620 --> 00:09:31.400
because the Fourier
theorem tells you,
00:09:31.400 --> 00:09:33.830
in order to form a complete
basis, you need all of them,
00:09:33.830 --> 00:09:36.080
all possible values of k.
00:09:36.080 --> 00:09:38.340
So there's no problem with
being a complete basis
00:09:38.340 --> 00:09:43.120
and having states have the
same energy eigenvalue, OK?
00:09:43.120 --> 00:09:44.720
It's a good question.
00:09:44.720 --> 00:09:45.920
Other questions?
00:09:45.920 --> 00:09:46.420
Yeah?
00:09:46.420 --> 00:09:49.300
AUDIENCE: So if we have a
potential that only admits
00:09:49.300 --> 00:09:51.940
bound states, we'll never
have this commutation
00:09:51.940 --> 00:09:52.670
happen basically?
00:09:52.670 --> 00:09:53.550
PROFESSOR: Yeah, exactly.
00:09:53.550 --> 00:09:54.050
Excellent.
00:09:54.050 --> 00:09:55.360
So the observation is this.
00:09:55.360 --> 00:09:58.810
Look, imagine I have a
potential that's not trivial.
00:09:58.810 --> 00:10:02.200
It's not 0, OK?
00:10:02.200 --> 00:10:05.860
Will the momentum commute
with the energy operator.
00:10:05.860 --> 00:10:08.029
No, because it's got
a potential that's
00:10:08.029 --> 00:10:10.570
going to be acted upon by P, so
you'll get a derivative term.
00:10:10.570 --> 00:10:14.940
But more precisely, if I have
a system with bound states,
00:10:14.940 --> 00:10:17.980
I have to have a
potential, right?
00:10:17.980 --> 00:10:21.360
And then I can't have P
commuting with the energy
00:10:21.360 --> 00:10:24.160
operator, which means I
can't have degeneracies.
00:10:24.160 --> 00:10:25.770
So indeed, if you
have bound states,
00:10:25.770 --> 00:10:27.626
you cannot have degeneracies.
00:10:27.626 --> 00:10:28.500
That's exactly right.
00:10:28.500 --> 00:10:29.102
Yeah?
00:10:29.102 --> 00:10:31.560
AUDIENCE: But doesn't this
break down in higher dimensions?
00:10:31.560 --> 00:10:33.060
PROFESSOR: Excellent,
so we're going
00:10:33.060 --> 00:10:35.150
to come back to higher
dimensions later.
00:10:35.150 --> 00:10:37.069
So the question
predicts what's going
00:10:37.069 --> 00:10:38.610
to happen in the
rest of the lecture.
00:10:38.610 --> 00:10:39.810
What we're going to
do in just a minute
00:10:39.810 --> 00:10:41.060
is we're going to
start working in three
00:10:41.060 --> 00:10:42.330
dimensions for the first time.
00:10:42.330 --> 00:10:43.840
We're going to leave 1D behind.
00:10:43.840 --> 00:10:45.690
We're going to take our
tripped out tricycle
00:10:45.690 --> 00:10:49.020
and replace it with a Yamaha.
00:10:49.020 --> 00:10:52.170
As you'll see, it has the
same basic physics driving
00:10:52.170 --> 00:10:54.800
its, well, self.
00:10:54.800 --> 00:10:56.312
It's the same dynamics.
00:10:56.312 --> 00:10:58.145
But I want to emphasize
a couple things that
00:10:58.145 --> 00:10:59.360
are going to show up.
00:10:59.360 --> 00:11:01.130
So the question is,
isn't this story
00:11:01.130 --> 00:11:02.590
different in three dimensions?
00:11:02.590 --> 00:11:06.192
And we shall see exactly what
happens in higher dimensions.
00:11:06.192 --> 00:11:07.400
We'll work in two dimensions.
00:11:07.400 --> 00:11:08.691
We'll work in three dimensions.
00:11:08.691 --> 00:11:10.882
We'll work in more.
00:11:10.882 --> 00:11:13.090
Doesn't really matter how
many dimensions we work in.
00:11:13.090 --> 00:11:14.900
You'll see it.
00:11:14.900 --> 00:11:16.394
OK, third thing.
00:11:16.394 --> 00:11:18.560
You studied this in some
detail in your problem set.
00:11:18.560 --> 00:11:20.040
Suppose I have an
energy operator
00:11:20.040 --> 00:11:23.240
that commutes with a
unitary operator, U, OK?
00:11:23.240 --> 00:11:25.800
So it commutes to 0.
00:11:25.800 --> 00:11:30.540
And U is unitary, so U dagger
U is one, is the identity.
00:11:30.540 --> 00:11:34.100
So what does this tell you?
00:11:34.100 --> 00:11:38.420
Well, first off
from these guys it
00:11:38.420 --> 00:11:48.510
tells us that we can have
simultaneous eigenfunctions.
00:11:55.400 --> 00:12:00.190
It also tells us too that
if we take our state phi
00:12:00.190 --> 00:12:02.860
and we act on it with
U, this could give us
00:12:02.860 --> 00:12:09.310
a new state, phi tilde sub
E, which will necessarily
00:12:09.310 --> 00:12:12.210
have the same energy eigenvalue,
because U and E commute.
00:12:12.210 --> 00:12:13.930
But it may be a different state.
00:12:16.510 --> 00:12:18.472
We'll come to this
in more detail later.
00:12:18.472 --> 00:12:20.555
But the third thing, and
I want to emphasize this,
00:12:20.555 --> 00:12:23.116
is this tells us, look, we
have a unitary operator.
00:12:23.116 --> 00:12:24.990
We can always write the
unitary operator as e
00:12:24.990 --> 00:12:26.720
to the i of a
Hermitian operator.
00:12:31.490 --> 00:12:34.410
So what is the meaning of
the Hermitian operator?
00:12:34.410 --> 00:12:35.530
What is this guy?
00:12:35.530 --> 00:12:37.180
So in your problem
set, you looked
00:12:37.180 --> 00:12:40.660
at what unitary operators are.
00:12:40.660 --> 00:12:43.365
And in the problem set, it's
discussed in some detail
00:12:43.365 --> 00:12:45.230
that there's a
relationship between
00:12:45.230 --> 00:12:47.630
a unitary transformation,
or unitary operator,
00:12:47.630 --> 00:12:49.455
and the symmetry.
00:12:49.455 --> 00:12:51.080
A symmetry is when
you take your system
00:12:51.080 --> 00:12:55.207
and you do something to it,
like a rotation or translation,
00:12:55.207 --> 00:12:57.290
and it's a symmetry if it
doesn't change anything,
00:12:57.290 --> 00:12:59.034
if the energy remains invariant.
00:12:59.034 --> 00:13:01.450
So if the energy doesn't change
under this transformation,
00:13:01.450 --> 00:13:03.880
we call that a symmetry.
00:13:03.880 --> 00:13:07.080
And we also showed that
symmetries, or translations,
00:13:07.080 --> 00:13:08.850
are generated by
unitary operators.
00:13:08.850 --> 00:13:12.350
For example, my favorite
examples are the translate by L
00:13:12.350 --> 00:13:14.350
operator, which is unitary.
00:13:14.350 --> 00:13:18.060
And also e to the minus dxl.
00:13:20.980 --> 00:13:25.890
And the boost by
q operator, which
00:13:25.890 --> 00:13:28.460
similarly is e to the minus qdp.
00:13:37.760 --> 00:13:40.610
And the time translation
operator, U sub t,
00:13:40.610 --> 00:13:46.690
which is equal to e to the minus
i t over h bar energy operator.
00:13:46.690 --> 00:13:47.310
OK?
00:13:47.310 --> 00:13:50.470
So these are
transformation operators.
00:13:50.470 --> 00:13:53.420
These are symmetry operators,
which translate you by L,
00:13:53.420 --> 00:13:55.900
boost or speed you
up by momentum q,
00:13:55.900 --> 00:13:57.510
evolve you forward in time t.
00:13:57.510 --> 00:14:00.920
And they can all be expressed
as e to the unitary operator.
00:14:00.920 --> 00:14:11.250
So this in particular
is l i over h bar p.
00:14:11.250 --> 00:14:14.530
And this is similarly x.
00:14:14.530 --> 00:14:19.402
And earlier we understood
the role of momentum
00:14:19.402 --> 00:14:21.360
having to do translations
in the following way.
00:14:21.360 --> 00:14:23.060
There's a beautiful
theorem about this.
00:14:23.060 --> 00:14:25.900
If you take a system and
its translation invariant,
00:14:25.900 --> 00:14:27.890
the classical statement
of Noether's theorem
00:14:27.890 --> 00:14:30.670
is that there's a conserved
quantity associated
00:14:30.670 --> 00:14:31.780
with that translation.
00:14:31.780 --> 00:14:34.260
That conserved quantity
is the momentum.
00:14:34.260 --> 00:14:36.100
And quantum mechanically,
the generator
00:14:36.100 --> 00:14:38.770
of that transformation
the Hermitian operator
00:14:38.770 --> 00:14:40.770
that goes upstairs
in the unitary
00:14:40.770 --> 00:14:43.260
is the operator associated
to that conserved quantity,
00:14:43.260 --> 00:14:45.280
associated to that observable.
00:14:45.280 --> 00:14:47.010
You have translations.
00:14:47.010 --> 00:14:49.011
There's a conserved
quantity, which is momentum.
00:14:49.011 --> 00:14:51.510
And the thing that generates
translations, the operator that
00:14:51.510 --> 00:14:54.300
generates translations, is the
operator representing momentum.
00:15:00.637 --> 00:15:02.845
So each of these are going
to come up later in today,
00:15:02.845 --> 00:15:06.250
and I just wanted to flag
them down before the moment.
00:15:06.250 --> 00:15:10.345
OK, questions before we move on?
00:15:10.345 --> 00:15:11.315
Yeah?
00:15:11.315 --> 00:15:14.952
AUDIENCE: So you made the claim
that every unitary operator can
00:15:14.952 --> 00:15:17.195
be expressed as p to
the eigenfunction.
00:15:17.195 --> 00:15:19.570
PROFESSOR: OK, I should be a
little bit careful, but yes.
00:15:19.570 --> 00:15:21.910
That's right.
00:15:21.910 --> 00:15:25.134
AUDIENCE: But if I
take the [INAUDIBLE] I
00:15:25.134 --> 00:15:26.800
should be able to
figure out what it is,
00:15:26.800 --> 00:15:28.230
but you can't take
the [INAUDIBLE]
00:15:28.230 --> 00:15:29.300
PROFESSOR: The more
precise statement
00:15:29.300 --> 00:15:31.140
is that any unitary--
any one parameter
00:15:31.140 --> 00:15:34.142
family of unitary operators
can be expressed in that form.
00:15:34.142 --> 00:15:35.600
And then you can
take a derivative.
00:15:35.600 --> 00:15:38.060
And that's the theory
of [INAUDIBLE],
00:15:38.060 --> 00:15:40.350
which is beyond the scope.
00:15:43.080 --> 00:15:45.350
Let me make a very
specific statement, which
00:15:45.350 --> 00:15:47.190
is that one parameter
of [INAUDIBLE]
00:15:47.190 --> 00:15:48.148
unitary transformation.
00:15:48.148 --> 00:15:50.240
So translations by l,
where you can vary l,
00:15:50.240 --> 00:15:52.150
can be expressed in that form.
00:15:52.150 --> 00:15:53.900
And that's a very
general statement.
00:15:53.900 --> 00:15:58.535
OK, so with all that as
prelude, let's go back to 3D.
00:16:03.360 --> 00:16:07.540
So in 3D, the energy operator--
so what's going to change?
00:16:07.540 --> 00:16:13.180
Now instead of just having
position and its momentum,
00:16:13.180 --> 00:16:16.090
we now also have-- I'll call
this P sub x-- we can also
00:16:16.090 --> 00:16:19.090
have a y-coordinate and
we have a z-coordinate.
00:16:19.090 --> 00:16:20.820
And each of them
has its momentum.
00:16:20.820 --> 00:16:23.610
P sub z and P sub y.
00:16:26.290 --> 00:16:28.740
And here's just a quick
practical question.
00:16:28.740 --> 00:16:36.020
We know that x with Px
is equal to i h bar.
00:16:44.170 --> 00:16:46.750
So what do you expect
to be true of x with y?
00:16:50.089 --> 00:16:51.050
AUDIENCE: 0.
00:16:51.050 --> 00:16:52.729
PROFESSOR: Why?
00:16:52.729 --> 00:16:53.645
AUDIENCE: [INAUDIBLE].
00:16:56.967 --> 00:16:58.800
PROFESSOR: What does
this equation tell you?
00:16:58.800 --> 00:17:00.008
What is its physical content?
00:17:02.170 --> 00:17:03.670
Well, that they
don't commute, good.
00:17:03.670 --> 00:17:05.140
What does that tell
you physically?
00:17:05.140 --> 00:17:05.479
Yes?
00:17:05.479 --> 00:17:07.437
AUDIENCE: That there's
an uncertainty principle
00:17:07.437 --> 00:17:08.420
connecting the two.
00:17:08.420 --> 00:17:08.720
PROFESSOR: Excellent.
00:17:08.720 --> 00:17:09.550
So that's one statement.
00:17:09.550 --> 00:17:11.000
So the consequence of
this is that there's
00:17:11.000 --> 00:17:12.041
an uncertainty principle.
00:17:12.041 --> 00:17:16.130
Delta x delta Px must be greater
than or equal to h bar upon 2.
00:17:16.130 --> 00:17:17.550
What's another way
of saying this?
00:17:22.349 --> 00:17:24.999
Do there exist simultaneous
eigenfunctions of x and P?
00:17:24.999 --> 00:17:25.540
AUDIENCE: No.
00:17:25.540 --> 00:17:26.123
PROFESSOR: No.
00:17:26.123 --> 00:17:28.400
No simultaneous eigenfunctions.
00:17:28.400 --> 00:17:32.280
OK, so you can't
have a definite value
00:17:32.280 --> 00:17:34.234
of x and a definite value
of P simultaneously.
00:17:34.234 --> 00:17:35.150
There's no such state.
00:17:35.150 --> 00:17:36.358
It's not that you can't know.
00:17:36.358 --> 00:17:39.180
It's that there's no such state.
00:17:39.180 --> 00:17:42.416
Do you expect to be able
to know the position in x
00:17:42.416 --> 00:17:45.710
and the position in
y simultaneously?
00:17:45.710 --> 00:17:46.240
Sure.
00:17:46.240 --> 00:17:48.551
OK, so this turns out to be 0.
00:17:48.551 --> 00:17:50.050
And in some sense,
you can take that
00:17:50.050 --> 00:17:51.850
as a definition of
quantum mechanics.
00:17:51.850 --> 00:17:54.500
x and y need to be 0.
00:17:54.500 --> 00:18:00.780
And similarly, Px
and Py commute.
00:18:00.780 --> 00:18:02.340
The momenta are independent.
00:18:02.340 --> 00:18:10.890
However, Py and y should
be equal to minus i h bar.
00:18:10.890 --> 00:18:11.390
Good.
00:18:11.390 --> 00:18:11.740
Exactly.
00:18:11.740 --> 00:18:13.290
So the commutators
work out exactly
00:18:13.290 --> 00:18:15.600
as you'd naively expect.
00:18:15.600 --> 00:18:20.450
Every pair of position and its
momenta commute canonically
00:18:20.450 --> 00:18:21.920
to i h bar.
00:18:21.920 --> 00:18:25.480
And every pair of
coordinates commute to 0.
00:18:25.480 --> 00:18:27.670
Every pair of
momenta commute to 0.
00:18:27.670 --> 00:18:28.170
Cool?
00:18:33.240 --> 00:18:36.835
So what kind of systems are
we going to interested in?
00:18:36.835 --> 00:18:38.710
Well, we're going to be
interested in systems
00:18:38.710 --> 00:18:41.430
where the energy
operator is equal to P
00:18:41.430 --> 00:18:49.790
vector hat squared upon 2m plus
U of x and y and z, hat, hat.
00:18:49.790 --> 00:18:57.770
You can see why dropping
the hats becomes almost /
00:18:57.770 --> 00:19:03.710
So in this language, we can
write the Schrodinger equation.
00:19:03.710 --> 00:19:07.220
This is just a direct extension
of the 1D Schrodinger equation.
00:19:07.220 --> 00:19:09.040
i h bar dt of psi.
00:19:09.040 --> 00:19:12.770
Now our wave function is a
function of x and y and z.
00:19:12.770 --> 00:19:14.350
There's some finite
probability then
00:19:14.350 --> 00:19:15.850
to find a particle
at some position.
00:19:15.850 --> 00:19:18.850
That position is labeled
by the three coordinates.
00:19:18.850 --> 00:19:21.630
Is equal to-- and of t.
00:19:24.890 --> 00:19:26.767
Is equal to-- well,
I'm actually write
00:19:26.767 --> 00:19:28.100
this in slightly different form.
00:19:28.100 --> 00:19:30.510
This is going to be easier
if I use vector notation.
00:19:30.510 --> 00:19:33.750
So I'm going to write
this as psi of r and t,
00:19:33.750 --> 00:19:36.260
where r denotes the
position vector,
00:19:36.260 --> 00:19:39.460
is equal to the energy
operator acting on it.
00:19:39.460 --> 00:19:46.790
And P is just equal to
minus i h bar the gradient.
00:19:49.830 --> 00:19:53.780
So this is minus i h bar
squared, or minus h bar squared
00:19:53.780 --> 00:20:04.960
upon 2m gradient squared plus
u of x or now u of r psi of r
00:20:04.960 --> 00:20:05.950
and t.
00:20:11.770 --> 00:20:13.520
Quick question, what
are the units or what
00:20:13.520 --> 00:20:18.940
are the dimensions
of psi of r in 3D?
00:20:22.692 --> 00:20:24.570
AUDIENCE: [INAUDIBLE].
00:20:24.570 --> 00:20:29.170
PROFESSOR: Yeah, one over
length to the root three halves.
00:20:29.170 --> 00:20:32.240
And the reason is this norm
squared gives us a probability
00:20:32.240 --> 00:20:34.500
density, something
that when we integrated
00:20:34.500 --> 00:20:38.660
over all positions in a
region integral d 3x is going
00:20:38.660 --> 00:20:40.620
to give us a number,
a probability.
00:20:40.620 --> 00:20:42.290
So its actual
magnitude must be--
00:20:42.290 --> 00:20:45.670
or its dimension must
be 1 over L to the 3/2.
00:20:45.670 --> 00:20:47.580
Just the cube of
what it was in 1D.
00:20:50.230 --> 00:20:54.415
And as you'll see
on the problems set
00:20:54.415 --> 00:20:57.580
and as we'll do in a couple
lectures down the road,
00:20:57.580 --> 00:21:02.430
it's convenient sometimes
to work in Cartesian,
00:21:02.430 --> 00:21:04.395
but it's also
sometimes convenient
00:21:04.395 --> 00:21:05.770
to work in spherical
coordinates.
00:21:09.632 --> 00:21:10.590
And it does not matter.
00:21:10.590 --> 00:21:12.006
And here's a really
deep statement
00:21:12.006 --> 00:21:13.750
that goes way beyond
quantum mechanics.
00:21:13.750 --> 00:21:15.170
It does not matter which
coordinates you work in.
00:21:15.170 --> 00:21:17.020
You cannot possibly get a
different answer by using
00:21:17.020 --> 00:21:18.140
different coordinates.
00:21:18.140 --> 00:21:20.760
So we're going to be ruthless
in exploiting coordinates that
00:21:20.760 --> 00:21:22.870
will simplify our problem
throughout the rest
00:21:22.870 --> 00:21:24.600
of this course.
00:21:24.600 --> 00:21:27.800
In the notes is it a short
discussion of the form
00:21:27.800 --> 00:21:30.540
of the Laplacian,
or the gradient
00:21:30.540 --> 00:21:34.532
squared in Cartesian spherical
and cylindrical coordinates.
00:21:34.532 --> 00:21:36.740
You should feel free to use
any coordinate system you
00:21:36.740 --> 00:21:37.630
want at any point.
00:21:37.630 --> 00:21:39.690
You just have to be
consistent about it.
00:21:42.630 --> 00:21:44.345
So let's work out a
couple of examples.
00:21:49.160 --> 00:21:51.010
And here are all
we're going to do
00:21:51.010 --> 00:21:53.120
is apply exactly the same
logic that we see over
00:21:53.120 --> 00:21:56.410
and over in 1D to
our 3D problems.
00:21:56.410 --> 00:22:07.070
So the first example is
a free particle in 3D.
00:22:07.070 --> 00:22:09.192
So before I get started
on this, any questions?
00:22:09.192 --> 00:22:10.400
Just in general 3D questions?
00:22:13.150 --> 00:22:14.150
OK.
00:22:14.150 --> 00:22:15.447
So this stuff starts off easy.
00:22:15.447 --> 00:22:17.405
And I'm going to work in
Cartesian coordinates.
00:22:21.220 --> 00:22:24.990
And a fun problem is
to repeat this analysis
00:22:24.990 --> 00:22:28.740
in spherical coordinates,
and we'll do that later on.
00:22:28.740 --> 00:22:30.420
OK, so free particle
in 3D, so what
00:22:30.420 --> 00:22:32.378
is the energy eigenfunction
equation look like?
00:22:32.378 --> 00:22:34.480
We want to find-- the
Schrodinger equation has
00:22:34.480 --> 00:22:36.790
exactly the same
structure as before.
00:22:36.790 --> 00:22:38.840
It's a linear
differential equation.
00:22:38.840 --> 00:22:41.800
So if we find the eigenfunctions
of the energy operator,
00:22:41.800 --> 00:22:45.030
we can use superposition to
construct the general solution,
00:22:45.030 --> 00:22:46.160
right?
00:22:46.160 --> 00:22:49.980
So exactly as in 1D, I'm going
to construct first the energy
00:22:49.980 --> 00:22:52.700
eigenfunctions , and then use
them in superposition to find
00:22:52.700 --> 00:22:54.960
a general solution to
the Schrodinger equation.
00:22:54.960 --> 00:22:55.640
OK?
00:22:55.640 --> 00:22:57.910
So let's construct the
energy eigenfunctions.
00:22:57.910 --> 00:23:00.540
So what is the energy
eigenvalue equation look like?
00:23:00.540 --> 00:23:06.000
Well, E on psi is equal to
minus h bar squared upon 2m.
00:23:06.000 --> 00:23:08.860
And in Cartesian, the
Laplacian is derivative respect
00:23:08.860 --> 00:23:10.700
to x squared plus
derivative with respect
00:23:10.700 --> 00:23:15.280
to y squared plus derivative
with respect to z squared.
00:23:15.280 --> 00:23:17.904
And we have no potential,
so this is just psi.
00:23:17.904 --> 00:23:19.820
So that would be energy
operator acting on it.
00:23:19.820 --> 00:23:21.694
And the eigenvalue
equation is at a constant,
00:23:21.694 --> 00:23:24.886
the energy E on psi
satisfies this equation.
00:23:24.886 --> 00:23:26.260
I'm going to write
this phi sub e
00:23:26.260 --> 00:23:28.560
to continue with
our notation of phi
00:23:28.560 --> 00:23:30.070
being the energy eigenfunctions.
00:23:30.070 --> 00:23:31.903
It of course, doesn't
matter what I call it,
00:23:31.903 --> 00:23:36.270
but just for consistency I'm
going to use the letter phi.
00:23:36.270 --> 00:23:39.240
So this is a very easy
equation to solve.
00:23:41.770 --> 00:23:45.240
In particular, it has
a lovely property,
00:23:45.240 --> 00:23:46.560
which is that it's separable.
00:23:50.600 --> 00:23:51.150
OK?
00:23:51.150 --> 00:23:52.525
So separable means
the following.
00:23:52.525 --> 00:23:58.350
It means, look, I note, I just
observed that this differential
00:23:58.350 --> 00:24:00.749
equation can be written
as a sum of terms
00:24:00.749 --> 00:24:03.290
where there's a derivative with
respect to only one variable.
00:24:03.290 --> 00:24:04.390
There's a differential
operator with respect
00:24:04.390 --> 00:24:06.640
to only one variable and
another differential operator
00:24:06.640 --> 00:24:08.930
with respect to only one
variable added together.
00:24:08.930 --> 00:24:12.230
And when you see that,
you can separate.
00:24:12.230 --> 00:24:14.140
And here's what I
mean by separate.
00:24:14.140 --> 00:24:15.550
I'm going to just construct.
00:24:15.550 --> 00:24:17.250
I'm not going to say that
this is a general solution.
00:24:17.250 --> 00:24:18.666
I'm just going to
try to construct
00:24:18.666 --> 00:24:20.470
a set of solutions of
the following form.
00:24:20.470 --> 00:24:24.610
Psi E of x, y, and
z is equal to psi
00:24:24.610 --> 00:24:33.800
E-- I will call this psi
sub x of x times phi sub
00:24:33.800 --> 00:24:38.678
y of y times phi sub z of z.
00:24:41.460 --> 00:24:44.250
And if we take this and we plug
it in, let's see what we get.
00:24:44.250 --> 00:24:48.940
This gives us that E on phi E
is equal to minus h bar squared
00:24:48.940 --> 00:24:49.440
upon 2m.
00:24:52.210 --> 00:24:55.240
Well the dx squared
acting on phi sub e
00:24:55.240 --> 00:24:56.610
is only going to hit this guy.
00:24:56.610 --> 00:25:03.580
So I'm going to get phi x
prime prime phi y phi z.
00:25:03.580 --> 00:25:08.870
Plus from the next term phi
y prime prime phi x phi z.
00:25:08.870 --> 00:25:13.301
And from the next term phi
z prime prime phi x phi y.
00:25:15.817 --> 00:25:18.400
But now I can do a sneaky thing
and divide the entire equation
00:25:18.400 --> 00:25:19.470
by phi sub e.
00:25:19.470 --> 00:25:21.800
Phi sub e is phi x phi y phi z.
00:25:21.800 --> 00:25:28.420
So if I do so, I lose the phi
y phi z, and I divide by phi x.
00:25:28.420 --> 00:25:30.740
And in this term, when I
divide by phi x phi y phi z,
00:25:30.740 --> 00:25:36.100
I lose the x and z, and I have
a left over phi sub y, or phi y.
00:25:36.100 --> 00:25:38.240
And similarly here,
phi sub z upon phi z.
00:25:42.994 --> 00:25:44.780
Everyone cool with that?
00:25:44.780 --> 00:25:46.036
Yeah?
00:25:46.036 --> 00:25:50.350
AUDIENCE: Can we also
lose the [INAUDIBLE]?
00:25:50.350 --> 00:25:51.790
PROFESSOR: I don't think so.
00:25:51.790 --> 00:25:54.980
Minus h bar squared over 2m
just hangs out for the ride.
00:25:58.870 --> 00:26:01.390
So when I take the
derivatives, I get these guys.
00:26:01.390 --> 00:26:04.080
And I have E times the function.
00:26:04.080 --> 00:26:06.836
We could certainly write
this as 2m over h bar squared
00:26:06.836 --> 00:26:07.710
and put it over here.
00:26:07.710 --> 00:26:10.630
That's fine.
00:26:10.630 --> 00:26:13.460
OK, so this is the form
of the equation we have,
00:26:13.460 --> 00:26:15.680
and what does this give us?
00:26:15.680 --> 00:26:17.580
What content does this give us?
00:26:17.580 --> 00:26:19.120
Well, note the following.
00:26:19.120 --> 00:26:20.480
This is a funny system.
00:26:20.480 --> 00:26:24.020
This is a function of x.
00:26:24.020 --> 00:26:28.080
This is a function of y, g
of y only, and not of x or z.
00:26:28.080 --> 00:26:33.280
And this is a function only
of z, and not of x or y.
00:26:33.280 --> 00:26:35.750
Yeah?
00:26:35.750 --> 00:26:38.440
So we have that E,
and let's put this
00:26:38.440 --> 00:26:42.160
as minus 2m over
h bar squared is
00:26:42.160 --> 00:26:46.300
equal to a function of
x plus a function of y
00:26:46.300 --> 00:26:47.310
plus a function of h.
00:26:51.100 --> 00:26:52.120
What does this tell you?
00:26:52.120 --> 00:26:53.570
AUDIENCE: They're all constant.
00:26:53.570 --> 00:26:54.930
PROFESSOR: They're
all constant, right.
00:26:54.930 --> 00:26:56.320
So the important
thing is this equation
00:26:56.320 --> 00:26:58.260
has to be true for every
value of x, y, and z.
00:26:58.260 --> 00:26:59.130
It's a differential equation.
00:26:59.130 --> 00:27:00.005
It's true everywhere.
00:27:00.005 --> 00:27:00.650
It's true here.
00:27:00.650 --> 00:27:01.316
It's true there.
00:27:01.316 --> 00:27:03.288
It's true at every point.
00:27:03.288 --> 00:27:04.100
Yeah?
00:27:04.100 --> 00:27:07.450
So for any value of x, y, and
z, this equation must be true.
00:27:07.450 --> 00:27:10.117
So now imagine I have a
particular solution g at h.
00:27:10.117 --> 00:27:12.200
I'm going to fix y and z
to some particular point.
00:27:12.200 --> 00:27:13.630
I'm going to look right here.
00:27:13.630 --> 00:27:15.570
And here that fixes y and z.
00:27:15.570 --> 00:27:17.510
So these are just some numbers.
00:27:17.510 --> 00:27:19.970
And suppose we
satisfy this equation.
00:27:19.970 --> 00:27:23.280
Then there's a very x
leaving y and z fixed.
00:27:23.280 --> 00:27:27.054
What must be true of f of x?
00:27:27.054 --> 00:27:27.930
AUDIENCE: Constant.
00:27:27.930 --> 00:27:29.763
PROFESSOR: It's got to
be constant, exactly.
00:27:29.763 --> 00:27:31.870
So this tells us that
f of x is a constant.
00:27:31.870 --> 00:27:32.620
I.e.
00:27:32.620 --> 00:27:36.920
phi x prime prime of
over phi x is a constant.
00:27:36.920 --> 00:27:38.100
I'll call it epsilon x.
00:27:40.910 --> 00:27:46.560
And phi y prime prime
of over phi y and this
00:27:46.560 --> 00:27:51.950
is a function of y of x
is equal to epsilon y.
00:27:51.950 --> 00:27:56.840
And actually for--
yeah, that's fine.
00:27:56.840 --> 00:27:58.340
For fun I'm going
to put in a minus.
00:27:58.340 --> 00:28:00.298
It doesn't matter what
I call this coefficient.
00:28:00.298 --> 00:28:04.674
And similarly for phi z
prime prime z over phi z
00:28:04.674 --> 00:28:05.840
is equal to minus epsilon z.
00:28:08.410 --> 00:28:13.910
So this tells us that
minus 2m upon h bar squared
00:28:13.910 --> 00:28:19.400
e is equal to minus epsilon x
minus epsilon y minus epsilon
00:28:19.400 --> 00:28:21.080
z.
00:28:21.080 --> 00:28:23.160
And any solutions
of these equations
00:28:23.160 --> 00:28:25.316
with some constant value
of epsilon x, epsilon
00:28:25.316 --> 00:28:26.880
y, and epsilon z
is going to give me
00:28:26.880 --> 00:28:29.880
a solution of my original energy
eigenvalue equation, where
00:28:29.880 --> 00:28:33.000
the value of capital
E is equal to the sum.
00:28:33.000 --> 00:28:37.700
And I can take the minus
signs make this plus plus.
00:28:37.700 --> 00:28:38.200
Yeah?
00:28:41.539 --> 00:28:46.360
AUDIENCE: Can
epsilon x, epsilon y,
00:28:46.360 --> 00:28:50.495
and epsilon z-- can
one of them be negative
00:28:50.495 --> 00:28:52.942
if the other's are sufficiently
positive or vice versa?
00:28:52.942 --> 00:28:53.900
Or is that [INAUDIBLE]?
00:28:53.900 --> 00:28:54.890
PROFESSOR: Let's check.
00:28:54.890 --> 00:28:55.570
Let's check.
00:28:55.570 --> 00:28:57.640
So what are the solutions
of this equation?
00:29:03.170 --> 00:29:03.670
Yeah.
00:29:03.670 --> 00:29:05.357
So solutions to
this equation phi
00:29:05.357 --> 00:29:07.940
double-- so let's write this in
a slightly more familiar form.
00:29:07.940 --> 00:29:12.140
This says that phi prime
prime plus epsilon x
00:29:12.140 --> 00:29:15.800
phi is equal to 0, OK?
00:29:15.800 --> 00:29:18.730
But this just tells you
that phi is exponential.
00:29:18.730 --> 00:29:25.670
Phi is equal to a e
to the Ikx kxx plus B
00:29:25.670 --> 00:29:34.580
e to the minus ikxx,
where k squared is
00:29:34.580 --> 00:29:37.609
equal to kx squared
is equal to epsilon x.
00:29:37.609 --> 00:29:38.108
OK?
00:29:41.840 --> 00:29:47.340
So this becomes-- and similarly
for epsilon y and epsilon z,
00:29:47.340 --> 00:29:50.270
each with their own
value of ky and kz
00:29:50.270 --> 00:29:54.260
who squares the
epsilon accordingly.
00:29:54.260 --> 00:29:57.900
So what can you say
about these epsilons?
00:29:57.900 --> 00:30:01.945
Well, the epsilons are
strictly positive numbers.
00:30:01.945 --> 00:30:03.070
So to answer your question.
00:30:03.070 --> 00:30:05.710
So the epsilons
have to be positive.
00:30:05.710 --> 00:30:08.325
OK, so this equation becomes,
though, E is equal to-- I'm
00:30:08.325 --> 00:30:10.908
going to put the h bar squared
over 2m back on the other side.
00:30:10.908 --> 00:30:13.150
h bar squared upon 2m.
00:30:13.150 --> 00:30:15.150
Epsilon x, epsilon y, epsilon z.
00:30:15.150 --> 00:30:17.000
But those are just
Kx squared plus Ky
00:30:17.000 --> 00:30:18.420
squared plus Kz squared.
00:30:18.420 --> 00:30:21.650
Kx squared plus
Ky squared plus Kz
00:30:21.650 --> 00:30:32.680
squared, also known as
h bar squared upon 2m k
00:30:32.680 --> 00:30:36.890
vector squared, where the
wave function, phi sub
00:30:36.890 --> 00:30:41.550
E is equal to some overall
normalization constant times,
00:30:41.550 --> 00:30:46.320
for the first function--
where did the definition go?
00:30:46.320 --> 00:30:46.970
Right here.
00:30:46.970 --> 00:30:49.430
So from here, phi x is
the exponential with Kx.
00:30:49.430 --> 00:30:50.920
This is an
exponential y with Ky.
00:30:50.920 --> 00:30:53.060
And then the exponential
in z with Kz.
00:30:53.060 --> 00:31:00.700
e to the ikx times x plus
Ky times y plus Kz times z.
00:31:04.510 --> 00:31:06.850
Let me write this as e to the i.
00:31:06.850 --> 00:31:09.070
e to the i.
00:31:09.070 --> 00:31:13.970
Also known as some normalization
constant e to the i k vector
00:31:13.970 --> 00:31:17.772
dot r vector.
00:31:17.772 --> 00:31:18.272
OK?
00:31:21.580 --> 00:31:25.090
So the energy, if we have
a free particle, the energy
00:31:25.090 --> 00:31:28.066
eigenfunctions can
be put in the form,
00:31:28.066 --> 00:31:30.770
or at least we can build energy
eigenfunctions of the form,
00:31:30.770 --> 00:31:36.520
plane waves with some 3D
momentum with energy E
00:31:36.520 --> 00:31:41.370
is equal to h bar squared
k squared upon 2m,
00:31:41.370 --> 00:31:44.286
just as we saw for the free
particle in one dimension.
00:31:44.286 --> 00:31:45.910
And the actual wave
function is nothing
00:31:45.910 --> 00:31:48.580
but a product of
wave functions in 1D.
00:31:48.580 --> 00:31:49.530
Yeah?
00:31:49.530 --> 00:31:53.751
AUDIENCE: What happened
to the minus ikx minus iky
00:31:53.751 --> 00:31:55.280
minus ikz terms?
00:31:55.280 --> 00:31:57.710
PROFESSOR: Good, so here
I just dropped these guys.
00:31:57.710 --> 00:32:02.682
So I just picked examples where
we just picked e to the ikz.
00:32:02.682 --> 00:32:03.890
That's an excellent question.
00:32:03.890 --> 00:32:05.750
So I've done something here.
00:32:05.750 --> 00:32:08.654
In particular, I looked
at a special case.
00:32:08.654 --> 00:32:10.570
And here's an important
lesson from the theory
00:32:10.570 --> 00:32:13.010
of the separable
equations, which
00:32:13.010 --> 00:32:22.750
is that once I separate-- so
if I have a separable equation
00:32:22.750 --> 00:32:25.010
and I find it
separated solution,
00:32:25.010 --> 00:32:33.740
phi E is equal to phi x of
x, phi y of y phi z of z,
00:32:33.740 --> 00:32:36.310
not all functions-- not all
solutions of the equation
00:32:36.310 --> 00:32:37.520
are of this form.
00:32:37.520 --> 00:32:38.020
They're not.
00:32:38.020 --> 00:32:39.269
I had to make that assumption.
00:32:39.269 --> 00:32:42.070
I said suppose it's of
this form right here.
00:32:42.070 --> 00:32:43.150
So this is an assumption.
00:32:46.840 --> 00:32:47.340
OK?
00:32:47.340 --> 00:32:51.520
However, these
form a good basis.
00:32:51.520 --> 00:32:54.390
By taking suitable
linear combinations
00:32:54.390 --> 00:32:56.140
of them, suitable
super positions,
00:32:56.140 --> 00:32:58.320
I can build a completely
general solution.
00:32:58.320 --> 00:33:02.360
For example, as was noted, the
true solution of this equation,
00:33:02.360 --> 00:33:04.690
even just focusing on
the x, is a superposition
00:33:04.690 --> 00:33:08.030
of plus and minus
waves, waves with
00:33:08.030 --> 00:33:09.650
plus positive negative momentum.
00:33:09.650 --> 00:33:10.920
So how do we get that?
00:33:10.920 --> 00:33:12.810
Well, we could
write that as phi is
00:33:12.810 --> 00:33:18.940
equal to e to the ikx
phi y phi z plus e
00:33:18.940 --> 00:33:29.550
to the minus ikx phi y phi z
with the same phi y and phi z.
00:33:29.550 --> 00:33:30.930
So these are actually in there.
00:33:34.390 --> 00:33:34.950
Yeah?
00:33:34.950 --> 00:33:38.280
AUDIENCE: My other
question is so I still
00:33:38.280 --> 00:33:39.828
don't see why any
of the epsilons
00:33:39.828 --> 00:33:40.952
can't have a negative sign.
00:33:40.952 --> 00:33:44.482
You have an exponential,
a real exponential as one
00:33:44.482 --> 00:33:45.580
of your products.
00:33:45.580 --> 00:33:51.230
PROFESSOR: OK, so if we
had a negative epsilon,
00:33:51.230 --> 00:33:55.130
is that wave function
going to be normalizable?
00:33:55.130 --> 00:33:57.100
AUDIENCE: Oh, as r
goes to-- but can you
00:33:57.100 --> 00:33:59.440
just keep the minus term?
00:33:59.440 --> 00:34:01.417
PROFESSOR: In which direction?
00:34:01.417 --> 00:34:02.250
AUDIENCE: Oh, right.
00:34:02.250 --> 00:34:03.950
PROFESSOR: So if it's
converging in this direction,
00:34:03.950 --> 00:34:05.790
it's got to be growing
in this direction.
00:34:05.790 --> 00:34:07.590
And that's not going
to be normalizable.
00:34:07.590 --> 00:34:14.000
And so as usual
with the plane wave,
00:34:14.000 --> 00:34:17.830
we can pick the oscillating
solutions that are also
00:34:17.830 --> 00:34:19.760
not normalizable
to one, but they're
00:34:19.760 --> 00:34:20.929
delta function normalizable.
00:34:20.929 --> 00:34:21.902
And so that's what
we've done here.
00:34:21.902 --> 00:34:23.280
It's exactly the
same thing as in 1D.
00:34:23.280 --> 00:34:23.780
Yeah?
00:34:23.780 --> 00:34:27.560
AUDIENCE: So does this mean
that any superposition of plane
00:34:27.560 --> 00:34:30.913
waves with wave vector
equal in magnitude
00:34:30.913 --> 00:34:33.620
will also be an eigenfunction
of the same energy?
00:34:33.620 --> 00:34:34.850
PROFESSOR: Absolutely.
00:34:34.850 --> 00:34:35.380
Awesome.
00:34:35.380 --> 00:34:36.130
Great observation.
00:34:36.130 --> 00:34:37.530
So the observation is this.
00:34:37.530 --> 00:34:41.199
Suppose I take K,
I'll call it K1.
00:34:41.199 --> 00:34:48.530
So this is a vector
such that K1 squared
00:34:48.530 --> 00:34:55.320
times h bar squared
over 2m is equal to E.
00:34:55.320 --> 00:34:59.890
Now there are many vectors k
which have the same magnitude,
00:34:59.890 --> 00:35:01.540
but not the same direction.
00:35:01.540 --> 00:35:05.010
So we could also make this equal
to h bar squared K2 squared
00:35:05.010 --> 00:35:12.320
vector squared over 2m where K2
is not equal to K1 as a vector,
00:35:12.320 --> 00:35:14.125
although they share
the same magnitude.
00:35:16.660 --> 00:35:17.940
So that's interesting.
00:35:17.940 --> 00:35:19.400
So that looks a lot like before.
00:35:19.400 --> 00:35:22.555
In 1D, we saw that if
we have k or minus k,
00:35:22.555 --> 00:35:23.680
these have the same energy.
00:35:26.750 --> 00:35:28.000
All right?
00:35:28.000 --> 00:35:32.800
Now if we have any K,
K1-- so this is 1D.
00:35:32.800 --> 00:35:44.920
In 3D, if we have K1 and
K2 with the same magnitude
00:35:44.920 --> 00:35:51.730
and the same energy,
they're degenerate.
00:35:57.010 --> 00:35:58.730
That's interesting.
00:35:58.730 --> 00:35:59.656
Why?
00:35:59.656 --> 00:36:01.962
Why do we have this
gigantic degeneracy
00:36:01.962 --> 00:36:04.420
of the energy eigenfunctions
for the free particle in three
00:36:04.420 --> 00:36:04.919
dimensions?
00:36:07.450 --> 00:36:07.950
Yeah?
00:36:07.950 --> 00:36:09.600
AUDIENCE: Well, there
are an infinite number
00:36:09.600 --> 00:36:11.910
of directions it could be going
in with the same momentum.
00:36:11.910 --> 00:36:12.370
PROFESSOR: Awesome.
00:36:12.370 --> 00:36:14.744
So this is clearly true that
there are an infinite number
00:36:14.744 --> 00:36:17.330
of momenta with
the same magnitude.
00:36:17.330 --> 00:36:19.935
So there are many,
many, but why?
00:36:19.935 --> 00:36:22.480
Why do they have
the same energy?
00:36:22.480 --> 00:36:24.390
Couldn't they have
different energy?
00:36:24.390 --> 00:36:26.784
Couldn't this one have
a different energy?
00:36:26.784 --> 00:36:27.700
AUDIENCE: [INAUDIBLE].
00:36:27.700 --> 00:36:28.740
PROFESSOR: Excellent,
it's symmetric.
00:36:28.740 --> 00:36:30.360
The system is invariant
under rotation.
00:36:30.360 --> 00:36:32.151
Who are you to say this
is the x direction?
00:36:32.151 --> 00:36:33.470
I call it y.
00:36:33.470 --> 00:36:34.910
Right?
00:36:34.910 --> 00:36:38.907
So the system has a symmetry.
00:36:38.907 --> 00:36:39.990
The symmetry is rotations.
00:36:42.560 --> 00:36:44.710
And when we have
a symmetry, that
00:36:44.710 --> 00:36:47.070
means there's an operator,
a unitary operator, which
00:36:47.070 --> 00:36:50.119
affects that rotation,
the rotation operator.
00:36:50.119 --> 00:36:51.660
It's the operator
that takes a vector
00:36:51.660 --> 00:36:54.250
and rotates in some
particular way.
00:36:54.250 --> 00:36:56.040
We have a unitary
operator that's
00:36:56.040 --> 00:36:59.070
a symmetry that means it
commutes with the energy
00:36:59.070 --> 00:36:59.807
operator.
00:36:59.807 --> 00:37:01.640
But if it commutes with
the energy operator,
00:37:01.640 --> 00:37:03.040
we get can degeneracies.
00:37:03.040 --> 00:37:06.004
We can get states that are
different states mapped
00:37:06.004 --> 00:37:08.545
to each other under our unitary
operator, under our rotation.
00:37:08.545 --> 00:37:11.749
We get states which are
different states manifestly.
00:37:11.749 --> 00:37:13.290
But which have the
same energy, which
00:37:13.290 --> 00:37:16.060
are shared energy eigenvalues.
00:37:16.060 --> 00:37:17.160
Cool?
00:37:17.160 --> 00:37:21.210
And this is a really lovely
example, both in 1D and 3D,
00:37:21.210 --> 00:37:25.230
that when you have a symmetry,
you get degeneracies.
00:37:29.570 --> 00:37:32.107
And when you have
a degeneracy, you
00:37:32.107 --> 00:37:33.690
should be very
suspicious that there's
00:37:33.690 --> 00:37:39.110
a symmetry hanging around,
lurking around ensuring it, OK?
00:37:39.110 --> 00:37:40.770
And this is an
important general lesson
00:37:40.770 --> 00:37:45.300
that goes way beyond the
specifics of the free particle.
00:37:45.300 --> 00:37:46.210
Yeah?
00:37:46.210 --> 00:37:48.370
AUDIENCE: So that
occurs in systems
00:37:48.370 --> 00:37:50.439
with bound states [INAUDIBLE]?
00:37:50.439 --> 00:37:52.730
PROFESSOR: Yeah, it occurs
in systems with bound states
00:37:52.730 --> 00:37:54.146
and systems with
non bound states.
00:37:54.146 --> 00:37:56.460
So here we're talking
about a free particle.
00:37:56.460 --> 00:37:57.600
Certainly not bound.
00:37:57.600 --> 00:37:59.430
And its true.
00:37:59.430 --> 00:38:01.320
For bound states, we'll
also see that there
00:38:01.320 --> 00:38:03.612
will be a degeneracy
associated with symmetry.
00:38:03.612 --> 00:38:05.570
Now your question is a
really, really good one,
00:38:05.570 --> 00:38:07.778
because what we found-- let
me rephrase the question.
00:38:07.778 --> 00:38:10.790
The question is, look, in
1D when we had bound states,
00:38:10.790 --> 00:38:12.150
there was no degeneracy.
00:38:12.150 --> 00:38:13.858
Didn't matter what
you did to the system.
00:38:13.858 --> 00:38:16.980
When you had bound states, bound
states were non degenerate.
00:38:16.980 --> 00:38:19.339
In 3D, we see that when
you have a free particle,
00:38:19.339 --> 00:38:20.380
you again get degeneracy.
00:38:20.380 --> 00:38:22.260
In fact, you get a heck
of a lot more degeneracy.
00:38:22.260 --> 00:38:23.140
You get a sphere's worth.
00:38:23.140 --> 00:38:25.290
Although actually, that's a
sphere in 0 dimension, right?
00:38:25.290 --> 00:38:26.962
It's a 0 dimensional
sphere, two points.
00:38:26.962 --> 00:38:29.270
So you get a sphere's
worth of degenerate states
00:38:29.270 --> 00:38:29.850
for the free particle.
00:38:29.850 --> 00:38:31.100
Well, what about bound states?
00:38:31.100 --> 00:38:34.110
Are bound states non
degenerate still?
00:38:34.110 --> 00:38:35.840
Fantastic question.
00:38:35.840 --> 00:38:36.910
Let's find out.
00:38:36.910 --> 00:38:38.410
So let's do the
harmonic oscillator.
00:38:38.410 --> 00:38:40.310
Let's do the 3D harmonic
oscillator to check.
00:38:44.990 --> 00:38:48.240
So the 3D harmonic oscillator,
the potential is h bar squared.
00:38:48.240 --> 00:38:52.020
And let's pick for fun the
rotationally symmetric 3D
00:38:52.020 --> 00:38:53.575
harmonic oscillator.
00:38:53.575 --> 00:39:01.050
m omega squared upon 2 x squared
plus y squared plus z squared.
00:39:01.050 --> 00:39:08.845
This could also be written m
omega squared upon 2 r squared.
00:39:11.287 --> 00:39:13.120
So I could write it in
spherical coordinates
00:39:13.120 --> 00:39:16.310
or in Cartesian coordinates.
00:39:16.310 --> 00:39:18.256
This is really in
vector notation.
00:39:18.256 --> 00:39:18.880
Doesn't matter.
00:39:18.880 --> 00:39:19.713
It's the same thing.
00:39:19.713 --> 00:39:22.780
3D harmonic oscillator.
00:39:22.780 --> 00:39:25.250
So what you immediately deduce
about the form of the energy
00:39:25.250 --> 00:39:25.875
eigenfunctions?
00:39:28.590 --> 00:39:33.400
Well, we have that E phi E
is equal to P squared over
00:39:33.400 --> 00:39:35.830
minus h bar squared
over 2m-- so here's
00:39:35.830 --> 00:39:42.430
the energy eigenvalue
equation-- times
00:39:42.430 --> 00:39:50.680
dx squared plus dy squared plus
dz squared plus m omega squared
00:39:50.680 --> 00:40:03.440
upon 2 times x squared plus y
squared plus z squared phi E.
00:40:03.440 --> 00:40:05.610
But I can put this
together in a nice way.
00:40:05.610 --> 00:40:11.330
This is minus h bar
squared upon 2m dx squared
00:40:11.330 --> 00:40:18.620
plus m omega squared upon 2
x squared plus ditto for y
00:40:18.620 --> 00:40:26.940
plus ditto for z phi E. Yeah?
00:40:26.940 --> 00:40:28.241
Everyone agree?
00:40:28.241 --> 00:40:30.490
So this is differential
operator that only involves x.
00:40:30.490 --> 00:40:31.930
Doesn't involve y or z.
00:40:31.930 --> 00:40:33.480
Ditto y, but no x or z.
00:40:33.480 --> 00:40:35.710
And ditto z, but no x or y.
00:40:35.710 --> 00:40:39.597
Aha, this is separable
just as before.
00:40:39.597 --> 00:40:41.180
So now we have a
nice separable system
00:40:41.180 --> 00:40:44.180
where I want to solve the
equations 3 times, once
00:40:44.180 --> 00:40:44.960
for x, y, and z.
00:40:44.960 --> 00:40:46.626
And I'm just going
to write it for x, y,
00:40:46.626 --> 00:40:51.530
and z epsilon sub x phi x is
equal to minus h bar squared
00:40:51.530 --> 00:40:57.680
over 2m dx squared plus
m omega squared upon 2 x
00:40:57.680 --> 00:41:02.000
squared phi sub x.
00:41:02.000 --> 00:41:04.180
And ditto for x, y, z.
00:41:04.180 --> 00:41:08.400
And then phi E is going
to be equal to phi
00:41:08.400 --> 00:41:15.600
x of x, phi y of y,
and phi z of z, OK?
00:41:15.600 --> 00:41:21.797
Where E is equal to epsilon x
plus epsilon y plus epsilon z.
00:41:21.797 --> 00:41:24.380
Note that I've used a slightly
different definition of epsilon
00:41:24.380 --> 00:41:25.230
here as before.
00:41:25.230 --> 00:41:28.715
Here it's explicitly
the energy eigenvalue.
00:41:28.715 --> 00:41:30.090
So what this is
telling is, look,
00:41:30.090 --> 00:41:31.130
we know what this equation is.
00:41:31.130 --> 00:41:32.530
This equation is
the same equation
00:41:32.530 --> 00:41:34.280
we ran into in the 1D
harmonic oscillator.
00:41:34.280 --> 00:41:38.590
It's exactly the 1D
harmonic oscillator problem.
00:41:38.590 --> 00:41:41.140
So the solution to the 3D
harmonic oscillator problem
00:41:41.140 --> 00:41:43.010
can be written for
energy eigenfunctions,
00:41:43.010 --> 00:41:45.850
can be constructed
by taking a harmonic
00:41:45.850 --> 00:41:48.750
oscillator in the x direction--
and we know what those are.
00:41:48.750 --> 00:41:49.750
There's a tower of them.
00:41:49.750 --> 00:41:52.335
There's a ladder of them
created by the raising operator
00:41:52.335 --> 00:41:56.160
and lowered by the
lowering operator.
00:41:56.160 --> 00:41:59.862
And similarly for y,
and similarly for z.
00:41:59.862 --> 00:42:01.820
So I'm going to write
this slightly differently
00:42:01.820 --> 00:42:10.540
in the same place as phi sub
E is equal to phi sub nx of x.
00:42:10.540 --> 00:42:13.560
The state with energy E sub nx.
00:42:13.560 --> 00:42:16.530
Phi sub ny of y.
00:42:16.530 --> 00:42:18.820
Phi sub nz of z.
00:42:18.820 --> 00:42:21.030
Where these are all
the single dimension,
00:42:21.030 --> 00:42:23.970
one dimensional harmonic
oscillator eigenfunctions.
00:42:23.970 --> 00:42:34.658
And E is equal to Ex E sub nx
plus E sub ny plus E sub nz.
00:42:34.658 --> 00:42:35.158
OK?
00:42:43.380 --> 00:42:46.845
So let's look at the
consequences of this.
00:42:46.845 --> 00:42:48.720
So first off, does anyone
have any questions?
00:42:48.720 --> 00:42:50.870
I went through
the kind of quick.
00:42:50.870 --> 00:42:52.485
Any questions about that?
00:42:52.485 --> 00:42:54.610
If you're not comfortable
with separable equations,
00:42:54.610 --> 00:42:56.970
you need to become super
comfortable with separable
00:42:56.970 --> 00:42:58.084
equations.
00:42:58.084 --> 00:42:59.250
It's an important technique.
00:42:59.250 --> 00:43:01.450
We're going to use it a lot.
00:43:01.450 --> 00:43:04.099
So the upshot is that
if I write-- in fact,
00:43:04.099 --> 00:43:06.390
I'm going to write that in
slightly different notation.
00:43:06.390 --> 00:43:12.410
If I write phi E
is equal to phi n
00:43:12.410 --> 00:43:22.474
of x, where it's the-- and phi
l of y and phi m of z-- actually
00:43:22.474 --> 00:43:23.515
that's a stupid ordering.
00:43:23.515 --> 00:43:24.470
Let's try that again.
00:43:28.830 --> 00:43:30.670
l, m, n.
00:43:30.670 --> 00:43:32.350
That is the
alphabetical ordering.
00:43:32.350 --> 00:43:34.500
With the energy is
now equal to-- we
00:43:34.500 --> 00:43:37.610
know the energy is of a
state with of the harmonic
00:43:37.610 --> 00:43:39.970
oscillator with
excitation number l.
00:43:39.970 --> 00:43:47.480
It's h bar omega, the overall
omega, times l plus 1/2.
00:43:47.480 --> 00:43:50.840
But from this guy it's
got excitation number m,
00:43:50.840 --> 00:43:56.140
so energy of that is h bar
omega m plus 1/2 plus m.
00:43:56.140 --> 00:43:59.010
And so now that's plus 1.
00:43:59.010 --> 00:44:00.850
And for this guy
similarly, h bar omega
00:44:00.850 --> 00:44:04.475
n plus n and now plus
1/2 again plus 3/2.
00:44:10.380 --> 00:44:12.781
This is a basis of solutions
of the energy eigenfunction
00:44:12.781 --> 00:44:13.280
equations.
00:44:13.280 --> 00:44:20.520
These are the solutions of
the energy eigenfunctions
00:44:20.520 --> 00:44:24.104
for the 3D harmonic oscillator.
00:44:24.104 --> 00:44:25.270
And now here's the question.
00:44:25.270 --> 00:44:27.460
The question that
was asked is, look,
00:44:27.460 --> 00:44:31.180
there are no degeneracies
in bound states in 1D.
00:44:31.180 --> 00:44:33.970
Here we have manifestly
a 3D bound state system.
00:44:33.970 --> 00:44:35.299
Are there degeneracies?
00:44:35.299 --> 00:44:35.882
AUDIENCE: Yes.
00:44:35.882 --> 00:44:37.610
PROFESSOR: Yeah,
obviously, right?
00:44:37.610 --> 00:44:40.205
So for example, if I call
l1 this 0 and this 0.
00:44:40.205 --> 00:44:44.780
Or if I call this 010 or 001,
those all have the same energy.
00:44:44.780 --> 00:44:47.610
They have the energy h bar
omega 0 times 1 plus 3/2 or 5/2.
00:44:50.320 --> 00:44:53.970
So let's look at this
in a little more detail.
00:44:53.970 --> 00:44:57.394
Let's write a list
of the degeneracies
00:44:57.394 --> 00:44:58.560
as a function of the energy.
00:45:03.120 --> 00:45:06.660
So at energy what's
the ground state energy
00:45:06.660 --> 00:45:09.520
for the 3D harmonic oscillator?
00:45:09.520 --> 00:45:10.466
3 halves h bar omega.
00:45:10.466 --> 00:45:11.840
It's three times
the ground state
00:45:11.840 --> 00:45:13.830
energy for the single
1D harmonic oscillator.
00:45:13.830 --> 00:45:15.990
So 3/2 h bar omega.
00:45:19.420 --> 00:45:19.974
Yeah?
00:45:19.974 --> 00:45:20.890
AUDIENCE: [INAUDIBLE].
00:45:23.830 --> 00:45:26.150
PROFESSOR: Good, the way
we arrived that this was we
00:45:26.150 --> 00:45:31.300
found that the energy-- so
the energy operator acting
00:45:31.300 --> 00:45:34.040
on the 3D wave
function is what I
00:45:34.040 --> 00:45:36.870
get by taking the
energy operator in 1D
00:45:36.870 --> 00:45:41.186
and acting on the wave function,
and the energy operator in y
00:45:41.186 --> 00:45:43.685
acting on the wave function,
the energy operator in z acting
00:45:43.685 --> 00:45:45.000
on the wave function, where
the energy operator for each
00:45:45.000 --> 00:45:46.660
of those is the 1D
harmonic oscillator
00:45:46.660 --> 00:45:48.560
with the same frequency.
00:45:48.560 --> 00:45:49.150
OK?
00:45:49.150 --> 00:45:50.120
And then I separated.
00:45:50.120 --> 00:45:51.600
I said look, let
the wave function,
00:45:51.600 --> 00:45:55.400
the 3D wave function, since
I know this is separable
00:45:55.400 --> 00:45:58.560
and each separated part
of the wave function
00:45:58.560 --> 00:46:00.990
satisfies the 1D harmonic
oscillator equation,
00:46:00.990 --> 00:46:04.380
I know what the eigenfunctions
of the 1D harmonic oscillator
00:46:04.380 --> 00:46:05.790
energy eigenvalue problem are.
00:46:05.790 --> 00:46:08.780
They are the phi n's.
00:46:08.780 --> 00:46:10.840
And so I can just take
my 3D wave function.
00:46:10.840 --> 00:46:13.510
I can say the 3D wave function
is going to be the separated
00:46:13.510 --> 00:46:15.180
form, the product
of 1D wave function
00:46:15.180 --> 00:46:19.610
in x, a 1D wave function in
y, and 1D wave function in z.
00:46:19.610 --> 00:46:20.240
Cool?
00:46:20.240 --> 00:46:22.220
So now let's look back
at what let's think
00:46:22.220 --> 00:46:23.740
about what happens when I
take the energy operator
00:46:23.740 --> 00:46:25.270
and I act on that [INAUDIBLE].
00:46:25.270 --> 00:46:27.160
When I take the 3D
energy operator, which
00:46:27.160 --> 00:46:29.209
is the sum of the three
1D energy operators,
00:46:29.209 --> 00:46:30.750
harmonic oscillator
energy operators.
00:46:30.750 --> 00:46:32.541
When thinking I'm going
to act on this guy,
00:46:32.541 --> 00:46:35.150
the first one, which only
knows about the x direction,
00:46:35.150 --> 00:46:38.060
sees the y and z
parts as constants.
00:46:38.060 --> 00:46:40.210
And it's a phi x, and
what does it give us back?
00:46:40.210 --> 00:46:44.440
E on phi x is just h bar
omega n plus one half.
00:46:44.440 --> 00:46:45.654
Ditto for this guy.
00:46:45.654 --> 00:46:47.070
And then the energy
operator in 3D
00:46:47.070 --> 00:46:49.640
is the sum of the three
1D energy operators.
00:46:49.640 --> 00:46:52.240
So that tells us the energy is
the sum of the three energies.
00:46:52.240 --> 00:46:53.840
Is that cool?
00:46:53.840 --> 00:46:54.460
OK good.
00:46:54.460 --> 00:46:55.370
Other questions?
00:46:55.370 --> 00:46:56.193
Yeah?
00:46:56.193 --> 00:46:58.508
AUDIENCE: Is the number of
degeneracies essentially
00:46:58.508 --> 00:47:01.760
[INAUDIBLE] of number theory.
00:47:01.760 --> 00:47:03.230
PROFESSOR: Ask me
that after class.
00:47:07.120 --> 00:47:08.810
So let's look at
the degeneracies
00:47:08.810 --> 00:47:11.530
as a function of the energy.
00:47:11.530 --> 00:47:15.160
So at the lowest possible
energy, 3/2 omega, what states
00:47:15.160 --> 00:47:17.584
can I possibly have?
00:47:17.584 --> 00:47:20.250
I'm going to label the states by
the three numbers, l, m, and n.
00:47:20.250 --> 00:47:22.670
So this is just the
ground state 0 0 0.
00:47:22.670 --> 00:47:24.740
So is that degenerate?
00:47:24.740 --> 00:47:27.720
No, because there's
just the one state.
00:47:27.720 --> 00:47:29.040
What about at the next level?
00:47:29.040 --> 00:47:30.727
What's the next allowed energy?
00:47:30.727 --> 00:47:31.310
AUDIENCE: 5/2.
00:47:31.310 --> 00:47:32.340
PROFESSOR: 5/2.
00:47:32.340 --> 00:47:33.660
OK, 5/2.
00:47:33.660 --> 00:47:35.660
So at 5/2 what
states do we have?
00:47:35.660 --> 00:47:36.620
Well, we have 1 0 0.
00:47:36.620 --> 00:47:38.520
But we also have 0, 1, 0.
00:47:38.520 --> 00:47:41.410
And we also 0 0 1.
00:47:41.410 --> 00:47:44.044
Aha, this is looking good.
00:47:44.044 --> 00:47:45.710
What does this
correspond to physically?
00:47:45.710 --> 00:47:48.860
This says you have
excitation, so you've
00:47:48.860 --> 00:47:51.110
got a node in the x direction.
00:47:51.110 --> 00:47:55.220
But your Gaussian in
the y and z directions.
00:47:55.220 --> 00:48:00.340
This one says your Gaussian
in the x direction.
00:48:00.340 --> 00:48:02.590
You have a node in the y
direction as a function of y,
00:48:02.590 --> 00:48:05.280
because it's phi 1 of y.
00:48:05.280 --> 00:48:07.440
And you're Gaussian
in the z directions.
00:48:07.440 --> 00:48:11.310
And this one says you have 1
excitation in the z direction.
00:48:11.310 --> 00:48:15.290
So they sound sort of
rotated from each other.
00:48:15.290 --> 00:48:18.500
That sounds promising.
00:48:18.500 --> 00:48:20.740
But in particular, what
we just discovered sort of
00:48:20.740 --> 00:48:22.370
by construction
is that there can
00:48:22.370 --> 00:48:35.580
be degeneracies among
bound states in 3D.
00:48:35.580 --> 00:48:38.310
This was not possible
in 1D, but it
00:48:38.310 --> 00:48:40.950
is possible in
3D, which is cool.
00:48:40.950 --> 00:48:42.360
But we've actually learned more.
00:48:42.360 --> 00:48:43.570
What's the form of
the degeneracies?
00:48:43.570 --> 00:48:46.028
So here it looks like they're
just rotations of each other.
00:48:46.028 --> 00:48:48.460
You call this x, I call it
y, someone else calls it z.
00:48:48.460 --> 00:48:50.510
These can't possibly
look different functions
00:48:50.510 --> 00:48:53.330
because they're just
rotations of each other.
00:48:53.330 --> 00:48:55.017
However, things get
a little more messy
00:48:55.017 --> 00:48:56.850
when you write, well,
what's the next level?
00:48:56.850 --> 00:48:59.100
What's the next energy
after 5/2 h bar omega?
00:48:59.100 --> 00:49:00.750
7/2 h bar omega, exactly.
00:49:00.750 --> 00:49:01.354
7/2.
00:49:01.354 --> 00:49:02.270
And that's not so bad.
00:49:02.270 --> 00:49:08.790
So for that one, we get
2 0 0, 0 2 0, 0 0 2.
00:49:08.790 --> 00:49:09.610
But is that it?
00:49:09.610 --> 00:49:10.370
AUDIENCE: No.
00:49:10.370 --> 00:49:12.410
PROFESSOR: What else do we get?
00:49:12.410 --> 00:49:17.560
1 1 0, 0 1 1, 1 0 1.
00:49:17.560 --> 00:49:19.200
So first off, let's
look at the number.
00:49:19.200 --> 00:49:21.270
The degeneracy number
here-- I'll call this d sub
00:49:21.270 --> 00:49:26.250
0-- the degeneracy of the
ground state is 1, OK?
00:49:26.250 --> 00:49:27.790
The degeneracy--
and in fact, I'm
00:49:27.790 --> 00:49:31.530
going to write this as a table--
the degeneracy as level n.
00:49:31.530 --> 00:49:35.150
So for d0 is equal to 1.
00:49:35.150 --> 00:49:37.990
d1 is equal to 3.
00:49:37.990 --> 00:49:44.150
And d2 is equal to 6.
00:49:44.150 --> 00:49:46.590
Now it's less clear
here what's going on,
00:49:46.590 --> 00:49:50.650
because is this just
this guy relabelled?
00:49:50.650 --> 00:49:52.100
No.
00:49:52.100 --> 00:49:53.870
So this is weird,
because we already
00:49:53.870 --> 00:50:00.280
said that the reason we expect
that there might be degeneracy,
00:50:00.280 --> 00:50:02.369
is because of
rotational symmetry.
00:50:02.369 --> 00:50:03.910
The system is
rotationally invariant.
00:50:03.910 --> 00:50:06.326
The potential, which is the
harmonic oscillator potential,
00:50:06.326 --> 00:50:15.600
doesn't care in what direction
the radial displacement
00:50:15.600 --> 00:50:17.380
vector is pointing.
00:50:17.380 --> 00:50:18.630
It's rotationally symmetrical.
00:50:18.630 --> 00:50:21.180
When we have symmetry--
on general grounds, when
00:50:21.180 --> 00:50:23.350
we have a symmetry, we
expect to have degeneracies.
00:50:26.422 --> 00:50:28.130
But this are kind of
weird, because these
00:50:28.130 --> 00:50:30.129
don't seem to be simple
rotations of each other,
00:50:30.129 --> 00:50:32.410
and yet they're degenerate.
00:50:32.410 --> 00:50:35.200
So what's up with that?
00:50:35.200 --> 00:50:35.711
Question?
00:50:35.711 --> 00:50:36.210
Yeah?
00:50:36.210 --> 00:50:40.230
AUDIENCE: [INAUDIBLE] Gaussian
in certain directions?
00:50:40.230 --> 00:50:41.350
PROFESSOR: Yeah, sure OK.
00:50:41.350 --> 00:50:44.420
So let me just explain what
this notation means again.
00:50:44.420 --> 00:50:50.260
So by 1 0 0, what I mean is
that the number l is equal to 1,
00:50:50.260 --> 00:50:52.095
the number m is equal
to 0, n is equal to 0.
00:50:52.095 --> 00:50:53.470
That means that
the wave function
00:50:53.470 --> 00:51:03.280
phi 3D is equal to phi 1 of
x, phi 0 of y, phi 0 of z.
00:51:03.280 --> 00:51:05.230
But what's phi 0 of z?
00:51:05.230 --> 00:51:07.350
What's a Gaussian
in the z direction?
00:51:07.350 --> 00:51:08.790
Phi 0 of y.
00:51:08.790 --> 00:51:10.550
That's the ground state
in the y direction
00:51:10.550 --> 00:51:11.260
of the harmonic oscillator.
00:51:11.260 --> 00:51:12.710
It's Gaussian in
the y direction.
00:51:12.710 --> 00:51:15.255
If I wanted x, that's
not the ground state.
00:51:15.255 --> 00:51:16.350
That's the excited state.
00:51:16.350 --> 00:51:19.450
And in particular, sort of being
a Gaussian it goes through 0.
00:51:19.450 --> 00:51:20.900
It has a node.
00:51:20.900 --> 00:51:24.250
So this wave function is
not rotationally invariant.
00:51:24.250 --> 00:51:26.890
It as a node in the x direction,
but no nodes and y and z
00:51:26.890 --> 00:51:29.000
direction.
00:51:29.000 --> 00:51:30.750
And similarly for these guys.
00:51:30.750 --> 00:51:32.280
Did that answer your questions?
00:51:32.280 --> 00:51:33.510
Great.
00:51:33.510 --> 00:51:38.596
OK, so we have
these degeneracies,
00:51:38.596 --> 00:51:40.160
and they beg an explanation.
00:51:40.160 --> 00:51:41.650
And if you look
at the next level,
00:51:41.650 --> 00:51:43.930
it turns out that d3-- and
you can do this quickly
00:51:43.930 --> 00:51:48.300
on a scrap of
paper-- d3 is 10, OK?
00:51:48.300 --> 00:51:49.440
And they go on.
00:51:49.440 --> 00:51:52.802
And if you keep
writing this list out,
00:51:52.802 --> 00:51:54.510
I guess it goes up--
what's the next one?
00:51:54.510 --> 00:51:57.260
15.
00:51:57.260 --> 00:51:58.870
21, yeah.
00:51:58.870 --> 00:52:01.260
So this has a simple
mathematical structure,
00:52:01.260 --> 00:52:03.150
and you can very quickly
convince yourself
00:52:03.150 --> 00:52:05.740
of the form of this degeneracy.
00:52:05.740 --> 00:52:09.680
dn is n n plus 1 over 2.
00:52:09.680 --> 00:52:12.090
So let's just make
sure that works.
00:52:12.090 --> 00:52:15.940
1 1 plus 1 over 2.
00:52:15.940 --> 00:52:17.606
Sorry I should
really call this 1.
00:52:20.868 --> 00:52:25.740
n plus 1, n plus 2
if I count from 0.
00:52:25.740 --> 00:52:30.620
So for 0, this is going to
give us 1 times 2 over 2.
00:52:30.620 --> 00:52:31.190
That's 1.
00:52:31.190 --> 00:52:31.940
That works.
00:52:31.940 --> 00:52:35.620
So for 1 that gives
2 times 3 over 2,
00:52:35.620 --> 00:52:38.594
which is 3, and so
on and so forth.
00:52:38.594 --> 00:52:39.760
So where did this come from?
00:52:39.760 --> 00:52:42.140
This is something we're
going to have to answer.
00:52:42.140 --> 00:52:43.950
Why that degeneracy?
00:52:43.950 --> 00:52:45.420
That seems important.
00:52:45.420 --> 00:52:46.630
Why is it that number?
00:52:46.630 --> 00:52:48.210
Why do we have that
much degeneracy?
00:52:48.210 --> 00:52:50.460
But the thing I really want
to emphasize at this point
00:52:50.460 --> 00:52:55.050
is that there's an absolutely
essential deep connection
00:52:55.050 --> 00:52:57.770
between symmetries
and degeneracies.
00:52:57.770 --> 00:53:00.140
If we didn't have symmetry,
we wouldn't have degeneracy,
00:53:00.140 --> 00:53:03.870
and we can see that
very easily here.
00:53:03.870 --> 00:53:07.640
Imagine that this potential
was not exactly symmetric.
00:53:07.640 --> 00:53:10.530
Imagine we made it slightly
different by adding
00:53:10.530 --> 00:53:17.200
a little bit of extra
frequency to z direction.
00:53:17.200 --> 00:53:19.300
Make the z frequency
slightly different.
00:53:19.300 --> 00:53:24.210
Plus m omega tilde
squared upon 2 z
00:53:24.210 --> 00:53:29.310
squared, where omega tilde
is not equal to omega 0.
00:53:29.310 --> 00:53:30.540
OK?
00:53:30.540 --> 00:53:33.230
The system is still
separable, but this guy
00:53:33.230 --> 00:53:34.982
has frequency omega 0.
00:53:34.982 --> 00:53:36.970
The x part has omega 0.
00:53:36.970 --> 00:53:38.040
This has omega 0.
00:53:38.040 --> 00:53:41.205
But this has omega tilde.
00:53:41.205 --> 00:53:42.690
OK?
00:53:42.690 --> 00:53:45.390
And so exactly the same
argument is going to go through,
00:53:45.390 --> 00:53:48.530
but the energy now is going
to have a different form.
00:53:48.530 --> 00:53:53.500
The energy is going to have h
bar omega-- h bar omega 0 times
00:53:53.500 --> 00:53:57.060
l plus m plus 1.
00:53:57.060 --> 00:53:59.890
But from the z part
it's going to have
00:53:59.890 --> 00:54:05.010
plus h bar omega tilde
times n plus 1/2.
00:54:09.861 --> 00:54:11.360
And now these
degeneracies are going
00:54:11.360 --> 00:54:13.420
to be broken, because
this state will not
00:54:13.420 --> 00:54:16.430
have the same
energy as these two.
00:54:16.430 --> 00:54:18.540
Everyone see that?
00:54:18.540 --> 00:54:20.810
When you have symmetry,
you get to degeneracy.
00:54:20.810 --> 00:54:23.740
When you don't have a symmetry,
you do not get degeneracy.
00:54:23.740 --> 00:54:25.770
This connection is
extremely important,
00:54:25.770 --> 00:54:27.550
because it allows
you to do two things.
00:54:27.550 --> 00:54:29.504
It allows you to
first not solve things
00:54:29.504 --> 00:54:30.670
you don't need to solve for.
00:54:30.670 --> 00:54:33.253
If you know there's a symmetry,
solve it once and then compute
00:54:33.253 --> 00:54:34.820
the degeneracy and you're done.
00:54:34.820 --> 00:54:36.620
On the other hand,
if you have a system
00:54:36.620 --> 00:54:40.040
and you see just manifestly
you measure the energies,
00:54:40.040 --> 00:54:42.990
and you measure that the
energies are degenerate,
00:54:42.990 --> 00:54:46.279
you know there's a symmetry
protecting those degeneracies.
00:54:46.279 --> 00:54:47.820
You actually can't
be 100% confident,
00:54:47.820 --> 00:54:49.695
because I didn't prove
that these are related
00:54:49.695 --> 00:54:52.760
to each other, but you
should be highly suspicious.
00:54:52.760 --> 00:54:55.130
And in fact, this is an
incredibly powerful tool
00:54:55.130 --> 00:54:57.140
in building models
of physical systems.
00:54:57.140 --> 00:54:59.696
If you see a degeneracy or
an approximate degeneracy,
00:54:59.696 --> 00:55:02.070
you can exploit that to learn
things about the underlying
00:55:02.070 --> 00:55:02.680
system.
00:55:02.680 --> 00:55:03.263
Yeah?
00:55:03.263 --> 00:55:05.096
AUDIENCE: So we just
add the different omega
00:55:05.096 --> 00:55:08.180
to each omega
[INAUDIBLE] number there
00:55:08.180 --> 00:55:10.430
is still a possibility
to get a degeneracy.
00:55:10.430 --> 00:55:12.320
PROFESSOR: Exactly.
00:55:12.320 --> 00:55:20.120
So it's possible for these
omegas to be specially tuned
00:55:20.120 --> 00:55:24.010
so that rational combinations
of them give you a degeneracy.
00:55:24.010 --> 00:55:26.330
But it's extraordinarily
unlikely for that
00:55:26.330 --> 00:55:28.470
to happen accidentally,
because they
00:55:28.470 --> 00:55:30.700
have to be rationally
related to each other,
00:55:30.700 --> 00:55:34.220
and the rationals are a set
of measures 0 in the reels.
00:55:34.220 --> 00:55:36.560
So if you just randomly
pick some frequencies,
00:55:36.560 --> 00:55:37.380
they'll be totally
incommensurate,
00:55:37.380 --> 00:55:38.900
and you'll never
get a degeneracy.
00:55:38.900 --> 00:55:41.750
So it is possible to have
an accidental degeneracy.
00:55:41.750 --> 00:55:44.240
Whoops, just pure coincidence.
00:55:44.240 --> 00:55:45.960
But it's extraordinarily
unlikely.
00:55:45.960 --> 00:55:48.210
And as you'll see when you
get to perturbation theory,
00:55:48.210 --> 00:55:50.110
it's more than unlikely.
00:55:50.110 --> 00:55:51.682
It's almost impossible.
00:55:51.682 --> 00:55:53.890
So it's very rare that you
get accidental degeneracy.
00:55:53.890 --> 00:55:57.000
It happens, but it's rare.
00:55:57.000 --> 00:55:58.960
Other questions?
00:55:58.960 --> 00:56:00.950
OK, nothing?
00:56:00.950 --> 00:56:03.170
OK so here we're now
going to launch into--
00:56:03.170 --> 00:56:05.740
so this leads us into
a very simple question.
00:56:05.740 --> 00:56:07.620
At the end of the
day, the degeneracies
00:56:07.620 --> 00:56:11.420
that we see for the 3D
free particle, which
00:56:11.420 --> 00:56:13.610
is a whole sphere's
worth of degeneracy,
00:56:13.610 --> 00:56:16.130
and the degeneracy we see for
the 3D harmonic oscillator,
00:56:16.130 --> 00:56:18.496
the bound states,
which is discrete,
00:56:18.496 --> 00:56:19.870
but with more and
more degeneracy
00:56:19.870 --> 00:56:22.610
the higher and
higher energy you go.
00:56:22.610 --> 00:56:24.640
Those we're blaming,
at the moment,
00:56:24.640 --> 00:56:26.540
on a symmetry, on
rotational symmetry,
00:56:26.540 --> 00:56:29.050
rotational invariance.
00:56:29.050 --> 00:56:32.240
So it seems wise
to study rotations,
00:56:32.240 --> 00:56:35.220
to study rotational invariance
and rotational transformations
00:56:35.220 --> 00:56:36.110
in the first place.
00:56:36.110 --> 00:56:38.890
In the first part of the
course, in 1D quantum mechanics,
00:56:38.890 --> 00:56:42.000
we got an awful lot of juice
out of studying translations.
00:56:42.000 --> 00:56:44.354
And the generator of
translations was momentum.
00:56:44.354 --> 00:56:46.020
So we're going to do
the same thing now.
00:56:46.020 --> 00:56:48.370
We're going to study
rotations and the generators
00:56:48.370 --> 00:56:50.700
of rotations, which are the
angular momentum operators,
00:56:50.700 --> 00:56:52.824
and that's going to occupy
us for the rest of today
00:56:52.824 --> 00:56:53.430
and Thursday.
00:56:53.430 --> 00:56:53.930
Yeah?
00:56:53.930 --> 00:56:56.720
AUDIENCE: So your
rotational symmetry
00:56:56.720 --> 00:57:03.690
will explain a factor of three
in your degeneracy, right?
00:57:03.690 --> 00:57:07.500
But what's the symmetry that
explains the way this grows.
00:57:07.500 --> 00:57:10.150
Because this very
clearly appears
00:57:10.150 --> 00:57:12.835
there's 1 up to a
factor of three.
00:57:12.835 --> 00:57:14.626
And then there's 2 up
to a factor of three.
00:57:14.626 --> 00:57:18.300
And there's even more that's
not even a multiple of three.
00:57:18.300 --> 00:57:19.895
PROFESSOR: Right,
actually so here's
00:57:19.895 --> 00:57:21.270
a very tempting
bit of intuition.
00:57:21.270 --> 00:57:23.811
Very tempting bit of intuition
is going to say the following.
00:57:23.811 --> 00:57:26.010
Look, rotational invariance,
there's x, there's y,
00:57:26.010 --> 00:57:26.220
and there's z.
00:57:26.220 --> 00:57:28.386
It's going to explain
rotations amongst those three.
00:57:28.386 --> 00:57:31.130
So that could only possibly
give you a factor of three.
00:57:31.130 --> 00:57:33.415
But it's important
to keep in mind
00:57:33.415 --> 00:57:35.100
that that's not
correct intuition.
00:57:35.100 --> 00:57:36.580
It's tempting intuition,
but it's not correct.
00:57:36.580 --> 00:57:38.371
And an easy way to see
that its not correct
00:57:38.371 --> 00:57:41.540
is that for the
free particle, there
00:57:41.540 --> 00:57:44.340
is a continuum, a
whole sphere's worth
00:57:44.340 --> 00:57:46.600
of degenerate states
in any energy.
00:57:46.600 --> 00:57:49.550
And all of those are related to
each other by simple rotation
00:57:49.550 --> 00:57:52.430
of the k vector, of
the wave vector, right?
00:57:52.430 --> 00:57:54.850
So the rotational
symmetry is giving us
00:57:54.850 --> 00:57:56.400
a lot more than a
factor of three.
00:57:56.400 --> 00:57:58.150
And in fact, as
we'll see, it's going
00:57:58.150 --> 00:58:00.360
to explain exactly the n
plus 1 n plus 2 over 2.
00:58:02.890 --> 00:58:05.450
OK, so with that
motivation let's
00:58:05.450 --> 00:58:07.335
start talking about
angular momentum.
00:58:11.900 --> 00:58:19.790
So I found this topic
to be not obviously
00:58:19.790 --> 00:58:23.500
the most powerful or interesting
thing in the world when I first
00:58:23.500 --> 00:58:24.150
studied it.
00:58:24.150 --> 00:58:25.697
And my professor
was like no, no, no.
00:58:25.697 --> 00:58:26.780
This is the deepest thing.
00:58:26.780 --> 00:58:28.412
And recently I had a fun
conversation with one
00:58:28.412 --> 00:58:29.786
of my colleagues,
Frank Wilczeck,
00:58:29.786 --> 00:58:32.530
who said yeah, in intro
to quantum mechanics
00:58:32.530 --> 00:58:35.310
the single most interesting
thing is the angular momentum
00:58:35.310 --> 00:58:36.930
and the addition of
angular momentum.
00:58:36.930 --> 00:58:39.890
And something has happened to
me in the intervening 20 years
00:58:39.890 --> 00:58:41.470
that I totally agree with him.
00:58:41.470 --> 00:58:46.020
So I will attempt to convey to
you the awesomeness of this.
00:58:46.020 --> 00:58:48.050
But you have to buy in a little.
00:58:48.050 --> 00:58:50.740
So work with me in the math
at the beginning of this,
00:58:50.740 --> 00:58:53.380
and it has a great payoff.
00:58:53.380 --> 00:58:55.927
OK so the question is,
what is the operator.
00:58:55.927 --> 00:58:58.010
So we're going to talk
about angular momentum now.
00:59:01.770 --> 00:59:05.510
And I want to start with
the following question.
00:59:05.510 --> 00:59:07.170
In the same sense
as we started out
00:59:07.170 --> 00:59:08.640
by asking what
represents position
00:59:08.640 --> 00:59:11.970
and momentum, linear momentum,
in quantum mechanics, what
00:59:11.970 --> 00:59:17.020
represents what operator by
our first, second, or third
00:59:17.020 --> 00:59:19.020
postulate-- I don't even
remember the order now.
00:59:19.020 --> 00:59:26.060
What operator represents angular
momentum in quantum mechanics?
00:59:30.980 --> 00:59:33.370
And let's start by remembering
what angular momentum is
00:59:33.370 --> 00:59:35.550
in classical mechanics.
00:59:35.550 --> 00:59:41.070
So L in classical
mechanics is r cross p.
00:59:41.070 --> 00:59:43.702
In classical mechanics.
00:59:43.702 --> 00:59:44.660
So let's just try this.
00:59:44.660 --> 00:59:45.950
Let's construct that operator.
00:59:45.950 --> 00:59:48.450
This is not the world's most
beautiful way of deriving this,
00:59:48.450 --> 00:59:50.880
but let's just write
down natural guess.
00:59:50.880 --> 00:59:53.320
For in quantum mechanics
what's the operator we want?
00:59:53.320 --> 00:59:56.430
Well, we want a vectors
worth of operators,
00:59:56.430 --> 00:59:58.267
because angular
momentum is a vector.
00:59:58.267 --> 01:00:00.100
It's a vector of
operators, three operators.
01:00:00.100 --> 01:00:06.280
And I'm going to write these
as r vector the operators x, y,
01:00:06.280 --> 01:00:12.920
z cross p the vector
of momentum operators.
01:00:12.920 --> 01:00:14.670
So at this point, you
should really worry,
01:00:14.670 --> 01:00:18.050
because do r and p commute?
01:00:18.050 --> 01:00:19.770
Not so much, right?
01:00:19.770 --> 01:00:22.150
However, the situation is
better than it first appears.
01:00:22.150 --> 01:00:24.840
Let's write this out
in terms of components.
01:00:24.840 --> 01:00:27.387
So this is in components.
01:00:27.387 --> 01:00:28.970
And I'm going to
work, for the moment,
01:00:28.970 --> 01:00:30.560
in Cartesian coordinates.
01:00:30.560 --> 01:00:34.120
So Lx is equal to?
01:00:34.120 --> 01:00:37.400
Lx is equal to?
01:00:37.400 --> 01:00:38.840
You all took mechanics.
01:00:38.840 --> 01:00:41.332
Lx is equal to?
01:00:41.332 --> 01:00:43.575
AUDIENCE: [INAUDIBLE].
01:00:43.575 --> 01:00:44.450
PROFESSOR: Thank you.
01:00:44.450 --> 01:00:47.920
YPz minus ZPy.
01:00:47.920 --> 01:00:50.960
And that's the curl, the
x component of the curl.
01:00:50.960 --> 01:00:55.650
And similarly, the x component--
so the way to remember this
01:00:55.650 --> 01:00:56.740
is that its cyclic.
01:00:56.740 --> 01:00:58.644
x, y, z.
01:00:58.644 --> 01:01:00.470
y, z, x.
01:01:00.470 --> 01:01:02.400
So z, p, y.
01:01:05.450 --> 01:01:10.310
PX minus XPz.
01:01:10.310 --> 01:01:14.030
And then we have
z XPy minus YPx.
01:01:18.210 --> 01:01:21.200
So we were worried here about
maybe an ordering problem.
01:01:21.200 --> 01:01:24.930
Is there an ordering
problem here?
01:01:24.930 --> 01:01:27.923
Does it matter if
I write YPz or PZy?
01:01:27.923 --> 01:01:28.465
AUDIENCE: No.
01:01:28.465 --> 01:01:30.631
PROFESSOR: No, because they
commute with each other.
01:01:30.631 --> 01:01:32.570
PZ is momentum for
the z-coordinate, not
01:01:32.570 --> 01:01:34.700
the y-coordinate, and they
commute with each other.
01:01:34.700 --> 01:01:36.200
So there's no ambiguity.
01:01:36.200 --> 01:01:37.480
It's perfectly well defined.
01:01:37.480 --> 01:01:39.063
So we're just going
to take this to be
01:01:39.063 --> 01:01:41.430
the definition of the components
of the angular momentum
01:01:41.430 --> 01:01:44.710
operator Lx, Ly, and Lz.
01:01:50.460 --> 01:01:53.180
And just for fun, I
want to write this out.
01:01:53.180 --> 01:01:56.870
So because we know that Px,
Py, and Pz can be expressed
01:01:56.870 --> 01:01:59.600
in terms of derivatives
or differential operators,
01:01:59.600 --> 01:02:01.970
we can write the same operator
in Cartesian coordinates
01:02:01.970 --> 01:02:02.960
in the following way.
01:02:02.960 --> 01:02:06.880
So clearly we could
write this as Y d dx i
01:02:06.880 --> 01:02:09.120
upon h bar-- or
sorry, h bar upon i.
01:02:09.120 --> 01:02:12.232
And z h bar upon i d dy.
01:02:12.232 --> 01:02:14.190
So we could write that
in Cartesian coordinates
01:02:14.190 --> 01:02:16.320
as a differential operator.
01:02:16.320 --> 01:02:19.350
But we can also write this
in spherical coordinates.
01:02:19.350 --> 01:02:21.300
I'm just going to take
a quick side note just
01:02:21.300 --> 01:02:22.350
to write down what it is.
01:02:22.350 --> 01:02:24.391
If we did this
spherical coordinates,
01:02:24.391 --> 01:02:26.016
it's particularly
convenient to write--
01:02:26.016 --> 01:02:27.980
let me just write
Lz for the moment.
01:02:27.980 --> 01:02:31.890
This is equal to minus i h bar
derivative with respect to phi,
01:02:31.890 --> 01:02:33.430
where the coordinates,
[INAUDIBLE]
01:02:33.430 --> 01:02:36.010
coordinates and spherical
coordinates in this class
01:02:36.010 --> 01:02:38.670
is theta is going
to be the angle down
01:02:38.670 --> 01:02:40.550
from the vertical
axis, from the z-axis.
01:02:40.550 --> 01:02:43.370
And phi is going to be
the angle of a period 2pi
01:02:43.370 --> 01:02:44.860
that goes around
the equator, OK?
01:02:44.860 --> 01:02:45.860
Just to give you a name.
01:02:45.860 --> 01:02:47.510
This is typically what
physicists call them.
01:02:47.510 --> 01:02:49.676
This is typically not what
mathematicians call them.
01:02:49.676 --> 01:02:51.180
This leads to
enormous confusion.
01:02:51.180 --> 01:02:52.520
I apologize for my field.
01:02:52.520 --> 01:02:55.420
So here it is.
01:02:55.420 --> 01:02:57.450
I can also construct
the operator associated
01:02:57.450 --> 01:02:59.150
with the square of the momentum.
01:02:59.150 --> 01:03:01.640
And why would we care about
the square of the momentum?
01:03:01.640 --> 01:03:03.848
Well, that's what shows up
in the Hamiltonian, that's
01:03:03.848 --> 01:03:05.020
what shows up in the energy.
01:03:05.020 --> 01:03:07.603
So I can construct the operator
for the square of the momentum
01:03:07.603 --> 01:03:10.280
and write it, and it takes
a surprisingly simple form.
01:03:10.280 --> 01:03:12.780
When I see surprisingly simple,
you might disagree with me,
01:03:12.780 --> 01:03:14.750
but if you actually do
the derivation of this,
01:03:14.750 --> 01:03:16.190
it's much worse in between.
01:03:16.190 --> 01:03:21.340
1 over sine theta
d theta sine theta
01:03:21.340 --> 01:03:28.020
d theta plus 1 over sine
squared theta d phi.
01:03:30.620 --> 01:03:31.770
Couple quick things.
01:03:31.770 --> 01:03:33.615
What are the dimensions
of angular momentum?
01:03:36.890 --> 01:03:38.170
Length and momentum.
01:03:38.170 --> 01:03:40.880
What else has units
of angular momentum?
01:03:40.880 --> 01:03:41.960
AUDIENCE: h bar.
01:03:41.960 --> 01:03:44.226
PROFESSOR: Solid. h
bar, dimensionless.
01:03:44.226 --> 01:03:46.350
Angular momentum squared,
angular momentum squared.
01:03:46.350 --> 01:03:46.880
OK, great.
01:03:46.880 --> 01:03:48.463
So that's going to
be very convenient.
01:03:48.463 --> 01:03:52.540
h bars are just going to
float around willy nilly.
01:03:52.540 --> 01:03:57.800
OK, so suppose I ask you
the following-- bless you.
01:03:57.800 --> 01:03:59.680
Suppose I ask you the
following questions.
01:03:59.680 --> 01:04:02.950
I say look, here are the
operators of angular momentum.
01:04:02.950 --> 01:04:03.550
This is Lz.
01:04:03.550 --> 01:04:05.040
We could have written down
the same expression for Lx,
01:04:05.040 --> 01:04:06.490
and a Ly, and L squared.
01:04:06.490 --> 01:04:08.650
What are the eigenfunctions
of these operators?
01:04:08.650 --> 01:04:10.070
Suppose I ask you this question.
01:04:10.070 --> 01:04:12.270
You all know how to
answer this question.
01:04:12.270 --> 01:04:14.020
You take these operators--
so for example,
01:04:14.020 --> 01:04:17.360
if I ask you what are
the eigenfunctions of Lz?
01:04:17.360 --> 01:04:18.910
Well, that's not so bad, right?
01:04:18.910 --> 01:04:20.409
The eigenfunction
of Lz is something
01:04:20.409 --> 01:04:25.010
where Lz on phi--
I'll call little m--
01:04:25.010 --> 01:04:29.680
is equal to minus I h
bar d d theta phi sub m.
01:04:29.680 --> 01:04:31.550
But I want the
eigenvalue, so I'll
01:04:31.550 --> 01:04:35.130
call this h bar
times some number.
01:04:35.130 --> 01:04:37.920
Let's call it m, because
Lz is an angular momentum.
01:04:37.920 --> 01:04:40.230
It carries units of
h bar, and its h bar
01:04:40.230 --> 01:04:41.800
times some number which is
dimensional, so we'll call it
01:04:41.800 --> 01:04:42.350
m.
01:04:42.350 --> 01:04:44.510
And we all know the
solution of this equation.
01:04:44.510 --> 01:04:47.290
The derivative is equal
to a constant times--
01:04:47.290 --> 01:04:49.340
we can lose the h bar.
01:04:49.340 --> 01:04:51.335
We get a minus i,
so we pick up an i.
01:04:51.335 --> 01:04:55.640
So therefore phi sub m is
equal to some constant times e
01:04:55.640 --> 01:04:58.200
to the im phi.
01:04:58.200 --> 01:05:00.590
What can we say about m?
01:05:00.590 --> 01:05:02.570
Well, heres an important
thing-- oh shoot.
01:05:02.570 --> 01:05:05.139
I'm using phi in so many
different ways here.
01:05:05.139 --> 01:05:06.680
Let's call this not
phi, because it's
01:05:06.680 --> 01:05:08.180
going to confuse
the heck out of us.
01:05:08.180 --> 01:05:12.600
Let's call it Y. So
we'll call it Y sub m.
01:05:12.600 --> 01:05:13.200
Why not?
01:05:16.650 --> 01:05:17.970
It's not my joke.
01:05:17.970 --> 01:05:22.070
This goes back to a bunch of--
yeah well, it goes way back.
01:05:22.070 --> 01:05:23.380
So phi is the variable.
01:05:23.380 --> 01:05:26.670
Y is the deciding eigenfunction
of Lz, and that's great.
01:05:26.670 --> 01:05:30.170
But what can you say about m?
01:05:30.170 --> 01:05:32.970
Well phi is the variable
around the equator
01:05:32.970 --> 01:05:34.957
is periodic with period 2pi.
01:05:34.957 --> 01:05:37.040
And our wave function had
better be single valued.
01:05:37.040 --> 01:05:40.040
So what does that
tell you about m?
01:05:40.040 --> 01:05:41.850
Well, under phi goes
to phi plus 2pi.
01:05:41.850 --> 01:05:44.859
This shifts by im 2pi.
01:05:44.859 --> 01:05:46.650
And that's only one to
make a single valued
01:05:46.650 --> 01:05:48.707
if m is an integer.
01:05:48.707 --> 01:05:49.790
So m has to be an integer.
01:05:52.654 --> 01:05:53.320
m is an integer.
01:05:53.320 --> 01:05:55.050
Now we did that for Lz.
01:05:55.050 --> 01:05:56.500
We found the
eigenfunctions of Lz.
01:05:56.500 --> 01:05:59.310
What about finding the
eigenfunctions L squared?
01:05:59.310 --> 01:06:00.360
Exactly the same thing.
01:06:00.360 --> 01:06:02.234
We're going to solve
the eigenvalue equation,
01:06:02.234 --> 01:06:04.970
but it's going to be
horrible, horrible to find
01:06:04.970 --> 01:06:05.965
these functions, right?
01:06:05.965 --> 01:06:06.840
Because look at this.
01:06:06.840 --> 01:06:08.370
1 over sine squared d d phi.
01:06:08.370 --> 01:06:11.230
And then 1 over sine
d theta sine d theta.
01:06:11.230 --> 01:06:13.590
This is not going to
be a fun thing to do.
01:06:13.590 --> 01:06:17.020
So we could just brute
force this, but let's not.
01:06:17.020 --> 01:06:20.730
Let's all agree that
that's probably a bad idea.
01:06:20.730 --> 01:06:23.340
Let's find a better way to
construct the eigenfunctions
01:06:23.340 --> 01:06:25.720
of the angular
momentum operators.
01:06:25.720 --> 01:06:26.840
So let's do it.
01:06:26.840 --> 01:06:28.590
So we ran into a
situation like this
01:06:28.590 --> 01:06:31.625
before when we dealt with
the harmonic oscillator.
01:06:31.625 --> 01:06:33.000
There was a
differential equation
01:06:33.000 --> 01:06:34.290
that we wanted to solve.
01:06:34.290 --> 01:06:37.760
And OK, this one isn't nearly
as bad, not nearly as bad
01:06:37.760 --> 01:06:40.950
as that one would have been.
01:06:40.950 --> 01:06:44.740
But still it was more useful
to work with operator methods.
01:06:44.740 --> 01:06:48.270
So let's take a hint from that
and work with operator methods.
01:06:48.270 --> 01:06:53.090
So now we need to study the
operators of angular momentum.
01:06:53.090 --> 01:06:54.924
So let's study them in
a little more detail.
01:06:54.924 --> 01:06:57.131
So something you're going
to show in your problem set
01:06:57.131 --> 01:06:58.010
is the following.
01:06:58.010 --> 01:07:03.280
The commutator of Lx with Ly
takes a really simple form.
01:07:03.280 --> 01:07:07.409
This is equal to i h bar--
let's just do this out.
01:07:07.409 --> 01:07:08.450
Let's do this commutator.
01:07:08.450 --> 01:07:09.510
We're OK.
01:07:09.510 --> 01:07:12.210
So Lx with Ly, this is
equal to the commutator
01:07:12.210 --> 01:07:21.400
of YPz minus ZPy with
Ly ZPx minus XPz.
01:07:24.830 --> 01:07:26.630
So let's look at
these term by term.
01:07:26.630 --> 01:07:29.790
So the first one is YPz ZPx.
01:07:29.790 --> 01:07:33.130
YPz ZPx.
01:07:33.130 --> 01:07:34.310
That's a Px.
01:07:34.310 --> 01:07:34.810
Sorry, ZPy.
01:07:37.540 --> 01:07:40.230
ZPx, this term.
01:07:40.230 --> 01:07:41.900
X, good.
01:07:41.900 --> 01:07:43.360
That's the first commutator.
01:07:43.360 --> 01:07:51.310
The second commutator, it can
be YPz an XPz minus YPz XPz.
01:07:54.470 --> 01:07:56.665
And then these
commutators, the next two,
01:07:56.665 --> 01:08:03.400
are going to be ZPy with ZPx.
01:08:06.060 --> 01:08:08.390
And finally ZPy.
01:08:08.390 --> 01:08:10.260
And that's minus
and this is a plus.
01:08:10.260 --> 01:08:12.030
ZPy and XPz.
01:08:15.810 --> 01:08:19.240
OK, so let's look at these.
01:08:19.240 --> 01:08:21.180
These look kind
of scary at first.
01:08:21.180 --> 01:08:22.760
But in this one
notice the following.
01:08:22.760 --> 01:08:28.630
This is XPz ZPx minus ZPx YPz.
01:08:28.630 --> 01:08:33.779
But what you say about Y with
all these other operators?
01:08:33.779 --> 01:08:35.210
Y commutes with all of them.
01:08:35.210 --> 01:08:36.740
So in each term I
could just pull Y
01:08:36.740 --> 01:08:38.689
all the way out to one side.
01:08:38.689 --> 01:08:39.505
Yeah?
01:08:39.505 --> 01:08:43.460
So I could just pull out this
Y. So for the first term,
01:08:43.460 --> 01:08:48.582
I'm going to write this as
Y commutator PZ with ZPx.
01:08:48.582 --> 01:08:50.040
And let me just do
that explicitly.
01:08:50.040 --> 01:08:51.200
There's no reason to.
01:08:51.200 --> 01:09:02.580
So this is YPz
ZPx minus ZPx YPz.
01:09:02.580 --> 01:09:05.560
And I can pull the Y out front,
because this commutes with Px
01:09:05.560 --> 01:09:12.979
and with Z. So I can make this
Y times PZ ZPx minus ZPx Pz.
01:09:12.979 --> 01:09:15.260
But now note that I can
do exactly the same thing
01:09:15.260 --> 01:09:16.179
with the Px.
01:09:16.179 --> 01:09:16.970
Px commutes with z.
01:09:16.970 --> 01:09:18.240
Px commutes with Pz.
01:09:18.240 --> 01:09:19.529
And it commutes with y.
01:09:19.529 --> 01:09:22.310
So I can pull the px
from each term out.
01:09:22.310 --> 01:09:27.415
Px Y. And now I lose the Px.
01:09:27.415 --> 01:09:28.620
I lose the Px.
01:09:28.620 --> 01:09:30.569
But now this is looking good.
01:09:30.569 --> 01:09:38.210
This is Px times Y. And Pz
minus ZPz PZz minus ZPz.
01:09:38.210 --> 01:09:47.047
This is also known as PXy
times commutator of PZ with Z.
01:09:47.047 --> 01:09:48.130
And what is this equal to?
01:09:51.497 --> 01:09:52.580
Let's get our signs right.
01:09:52.580 --> 01:09:55.930
This is Px with Y
time-- OK, we all
01:09:55.930 --> 01:09:57.930
agree that this is going
to have an h bar in it.
01:09:57.930 --> 01:09:59.290
Let's write units.
01:09:59.290 --> 01:10:00.950
There's going to be an i.
01:10:00.950 --> 01:10:04.030
And is it a plus or minus?
01:10:04.030 --> 01:10:05.020
Minus.
01:10:05.020 --> 01:10:05.770
OK, good.
01:10:05.770 --> 01:10:08.730
So we get minus h bar XPy.
01:10:08.730 --> 01:10:17.060
So this term is going to give
us minus i h bar Py XPx times Y.
01:10:17.060 --> 01:10:18.480
And let's look at this term.
01:10:18.480 --> 01:10:24.440
OK, this is Y, and Y commutes
with PZx and PZ, right?
01:10:24.440 --> 01:10:26.080
So I could just pull out the y.
01:10:26.080 --> 01:10:28.700
And x commutes with everything,
so I can pull out the x.
01:10:28.700 --> 01:10:31.410
And I'm left with the
commutator of Pz with Pz.
01:10:31.410 --> 01:10:32.910
What's the commutator
of Pz with Pz?
01:10:32.910 --> 01:10:33.410
AUDIENCE: 0.
01:10:33.410 --> 01:10:33.951
PROFESSOR: 0.
01:10:33.951 --> 01:10:35.170
This term gives me a 0.
01:10:35.170 --> 01:10:37.150
Similarly here, Py
commutes with everything.
01:10:37.150 --> 01:10:38.696
Z ZPx.
01:10:38.696 --> 01:10:40.100
Px commutes with everything.
01:10:40.100 --> 01:10:41.590
Z ZPy.
01:10:41.590 --> 01:10:44.870
So I can pull out the Px Py,
and I get Z commutator Z,
01:10:44.870 --> 01:10:45.610
and what's that?
01:10:45.610 --> 01:10:46.159
AUDIENCE: 0.
01:10:46.159 --> 01:10:46.700
PROFESSOR: 0.
01:10:46.700 --> 01:10:47.710
So this gives me 0.
01:10:47.710 --> 01:10:51.480
And now this term, ZPy
XPz, the only two things
01:10:51.480 --> 01:10:54.600
that don't commute with each
other are the Z and the Pz.
01:10:54.600 --> 01:10:56.580
The Py and the X
I pull out, so I
01:10:56.580 --> 01:11:00.640
get it a term that's XPy and
the commutator of Z with Pz.
01:11:00.640 --> 01:11:03.190
And what is that
going to give me?
01:11:03.190 --> 01:11:05.250
PyX i h bar.
01:11:11.685 --> 01:11:14.870
Aha, look at what we got.
01:11:14.870 --> 01:11:22.050
This is equal to i h bar times--
did I screw up the signs?
01:11:22.050 --> 01:11:24.765
XPy minus YPz.
01:11:30.060 --> 01:11:33.130
Oh sorry, Px.
01:11:33.130 --> 01:11:35.036
And what is this equal to?
01:11:35.036 --> 01:11:36.380
AUDIENCE: [INAUDIBLE].
01:11:36.380 --> 01:11:37.801
PROFESSOR: Yeah, i h bar Lz.
01:11:43.580 --> 01:11:46.964
And more generally, as you'll
show on the problem set,
01:11:46.964 --> 01:11:48.380
you get the following
commutators.
01:11:48.380 --> 01:11:51.510
Once you've done this once, you
can do the rest very easily.
01:11:51.510 --> 01:11:57.980
Lx Ly is-- so Lx with
Ly is i h bar Lz.
01:12:03.010 --> 01:12:06.000
And then the rest can be
got from cyclic rotations
01:12:06.000 --> 01:12:16.970
Ly with Lz is i h bar Lx and
Lz with Lx is i h bar Ly.
01:12:21.460 --> 01:12:22.930
And now here's a fancier one.
01:12:22.930 --> 01:12:25.536
This is less obvious, but
exactly the same machinations
01:12:25.536 --> 01:12:27.660
will give you this result,
and you'll do this again
01:12:27.660 --> 01:12:28.580
on the problems set.
01:12:28.580 --> 01:12:31.840
If I take L squared,
k and L squared
01:12:31.840 --> 01:12:36.150
here is going to be
L squared, I just
01:12:36.150 --> 01:12:42.520
mean Lx squared plus Ly
squared plus Lz squared.
01:12:42.520 --> 01:12:46.130
This is the norm squared of
the vector, the operator form.
01:12:46.130 --> 01:12:49.470
L squared with Lz--
or sorry, with Lx.
01:12:49.470 --> 01:12:50.350
Yeah, fine.
01:12:50.350 --> 01:12:52.570
Lx is equal to 0.
01:12:52.570 --> 01:12:55.910
So Lx commutes with the
magnitude L squared.
01:12:55.910 --> 01:12:58.910
Similarly L squared, now just
by rotational invariance,
01:12:58.910 --> 01:13:00.814
if Lx commutes
with it, Ly and Lz
01:13:00.814 --> 01:13:02.480
had better also commute
with it, because
01:13:02.480 --> 01:13:06.180
of who you to see what's Lx.
01:13:06.180 --> 01:13:09.290
And L squared with Lz
must be equal to 0.
01:13:15.340 --> 01:13:15.840
Yeah?
01:13:15.840 --> 01:13:16.860
Everyone cool with that?
01:13:21.160 --> 01:13:23.320
And the thing is
we've just-- and I'm
01:13:23.320 --> 01:13:24.950
going to put big
flags around this.
01:13:24.950 --> 01:13:26.620
We've just learned
a tremendous amount
01:13:26.620 --> 01:13:29.520
about the eigenfunctions of
the angular momentum operators.
01:13:29.520 --> 01:13:31.411
Why?
01:13:31.411 --> 01:13:33.900
Let's leave that up.
01:13:33.900 --> 01:13:36.590
So what have we just learned
about the eigenfunctions
01:13:36.590 --> 01:13:39.221
of the angular momentum
operators Lx, Ly, Lz, and L
01:13:39.221 --> 01:13:39.720
squared?
01:13:44.890 --> 01:13:45.562
Anyone?
01:13:45.562 --> 01:13:47.770
We just learned something
totally awesome about them.
01:13:57.620 --> 01:13:58.750
Anyone?
01:13:58.750 --> 01:14:03.174
So can you find simultaneous
eigenfunctions of Lx and Ly?
01:14:03.174 --> 01:14:03.912
AUDIENCE: No.
01:14:03.912 --> 01:14:04.870
PROFESSOR: Not so much.
01:14:04.870 --> 01:14:06.000
They don't commute to 0.
01:14:06.000 --> 01:14:07.810
What about Ly and Lz?
01:14:07.810 --> 01:14:08.800
Nope.
01:14:08.800 --> 01:14:11.080
Lz Lx, nope.
01:14:11.080 --> 01:14:14.060
So you cannot find simultaneous
eigenfunctions of Lx and Ly.
01:14:17.010 --> 01:14:20.680
What about Lx and L squared?
01:14:20.680 --> 01:14:22.120
Yes.
01:14:22.120 --> 01:14:24.480
So we can find
simultaneously eigenfunctions
01:14:24.480 --> 01:14:29.740
of L squared and Lx.
01:14:29.740 --> 01:14:32.850
OK, what about can we find
simultaneous eigenfunctions
01:14:32.850 --> 01:14:35.370
of L squared and Ly?
01:14:35.370 --> 01:14:35.990
Yep.
01:14:35.990 --> 01:14:36.489
Exactly.
01:14:36.489 --> 01:14:39.840
What about L
squared, Ly, and Lz?
01:14:39.840 --> 01:14:40.580
Nope.
01:14:40.580 --> 01:14:42.530
No such luck.
01:14:42.530 --> 01:14:44.720
And this leads us to
the following idea.
01:14:44.720 --> 01:14:51.130
The idea is a complete set
of commuting observables.
01:14:54.790 --> 01:14:58.180
And here's what this
idea is meant to contain.
01:14:58.180 --> 01:15:00.350
You can always write
down a lot of operators.
01:15:00.350 --> 01:15:02.710
So let me step back and
ask a classical question.
01:15:02.710 --> 01:15:04.730
Classically, suppose I have a
particle in three dimensions,
01:15:04.730 --> 01:15:07.063
a particle moving around in
this room, non relativistic,
01:15:07.063 --> 01:15:08.380
familiar 801.
01:15:08.380 --> 01:15:10.720
I have a particle moving
around in this room.
01:15:10.720 --> 01:15:12.835
How much data must
I specify to specify
01:15:12.835 --> 01:15:14.210
the configuration
of this system?
01:15:14.210 --> 01:15:16.290
Well, I have to tell you
where the particle is,
01:15:16.290 --> 01:15:18.530
and I have to tell you what
its momentum is, right?
01:15:18.530 --> 01:15:21.130
So I have to tell you the
three coordinates and the three
01:15:21.130 --> 01:15:22.230
momenta.
01:15:22.230 --> 01:15:26.480
If I give you five numbers,
that's not enough, right?
01:15:26.480 --> 01:15:29.902
I need to give you
six bits of data.
01:15:29.902 --> 01:15:31.860
On the other hand, if I
give you seven numbers,
01:15:31.860 --> 01:15:37.750
like I give x, y, z, Px, Py,
Pz, and e, that's over complete.
01:15:37.750 --> 01:15:38.250
Right?
01:15:38.250 --> 01:15:41.214
That was unnecessary.
01:15:41.214 --> 01:15:42.630
So in classical
mechanics, you can
01:15:42.630 --> 01:15:45.570
ask what data must you
specify to completely specify
01:15:45.570 --> 01:15:46.570
the state of the system.
01:15:46.570 --> 01:15:47.600
And that's usually pretty easy.
01:15:47.600 --> 01:15:48.620
You specify the
number coordinates
01:15:48.620 --> 01:15:49.802
and the number of momenta.
01:15:49.802 --> 01:15:52.260
In quantum mechanics, we ask
a slightly different question.
01:15:52.260 --> 01:15:54.970
We ask, in order to specify
the state of a system,
01:15:54.970 --> 01:15:57.860
we say we want to specify
which state it is.
01:15:57.860 --> 01:15:59.930
You can specify
that by saying which
01:15:59.930 --> 01:16:01.830
superposition in a
particular basis.
01:16:01.830 --> 01:16:04.440
So you pick a basis,
and you specify
01:16:04.440 --> 01:16:05.980
which particular
superposition, what
01:16:05.980 --> 01:16:09.970
are the eigenvalues of
the operators you've
01:16:09.970 --> 01:16:12.410
diagonalized in that basis?
01:16:12.410 --> 01:16:16.619
So another way to phrase
this is for a 1D problem,
01:16:16.619 --> 01:16:18.160
say we have just a
simple 1D problem.
01:16:18.160 --> 01:16:23.140
We have X and we have P. What
is a complete set of commuting
01:16:23.140 --> 01:16:23.780
operators?
01:16:23.780 --> 01:16:25.405
Well, you have to
have enough operators
01:16:25.405 --> 01:16:27.540
so that the eigenvalue
specifies a state.
01:16:27.540 --> 01:16:30.130
So X would be-- is that enough?
01:16:32.650 --> 01:16:35.520
So if I take the operator
X and I say look,
01:16:35.520 --> 01:16:37.881
my system is in the state
with an eigenvalue X not of X,
01:16:37.881 --> 01:16:39.630
does that specify the
state of the system?
01:16:42.817 --> 01:16:45.150
It tells you the particles
are in a delta function state
01:16:45.150 --> 01:16:46.590
right here.
01:16:46.590 --> 01:16:47.820
Does that specify the state?
01:16:47.820 --> 01:16:48.130
AUDIENCE: Yes.
01:16:48.130 --> 01:16:49.380
PROFESSOR: Fantastic, it does.
01:16:49.380 --> 01:16:51.030
Now what if I tell
you instead, oh it's
01:16:51.030 --> 01:16:55.460
in a state of definite P.
Does that specify the state?
01:16:55.460 --> 01:16:56.570
Absolutely.
01:16:56.570 --> 01:16:59.910
Can I say it's in a state with
definite X and definite P?
01:16:59.910 --> 01:17:00.730
AUDIENCE: No.
01:17:00.730 --> 01:17:01.480
PROFESSOR: No.
01:17:01.480 --> 01:17:03.839
These don't commute.
01:17:03.839 --> 01:17:06.130
So a complete set of commuting
observables in this case
01:17:06.130 --> 01:17:09.300
would be either X
or P, but not both.
01:17:09.300 --> 01:17:10.410
Yeah?
01:17:10.410 --> 01:17:15.090
Now if we're in three
dimensions, is x a complete set
01:17:15.090 --> 01:17:16.466
of commuting observables?
01:17:16.466 --> 01:17:17.205
AUDIENCE: No.
01:17:17.205 --> 01:17:18.830
PROFESSOR: No, because
it's not enough.
01:17:18.830 --> 01:17:20.880
You tell me that it's
X, that doesn't tell me
01:17:20.880 --> 01:17:22.255
what state it is
because it could
01:17:22.255 --> 01:17:24.420
have y dependence
or z dependence.
01:17:24.420 --> 01:17:30.760
So in 3D, we take, for
example, x, and y, and z.
01:17:30.760 --> 01:17:37.010
Or Px, and Py, and Pz.
01:17:37.010 --> 01:17:42.940
We could also pick
z, and Px, and y.
01:17:42.940 --> 01:17:45.710
Are these complete?
01:17:45.710 --> 01:17:47.690
If I tell you I have
definite position in z,
01:17:47.690 --> 01:17:50.050
definite position in y,
and definite momentum in x?
01:17:50.050 --> 01:17:51.380
Does that completely
specify my state?
01:17:51.380 --> 01:17:51.855
AUDIENCE: Yes.
01:17:51.855 --> 01:17:53.479
PROFESSOR: Yeah,
totally unambiguously.
01:17:53.479 --> 01:18:00.030
e to the ikx delta of y delta
of z totally fixes my function,
01:18:00.030 --> 01:18:01.030
completely specifies it.
01:18:01.030 --> 01:18:03.190
If I'd only picked
two of these, it
01:18:03.190 --> 01:18:04.712
would not have been complete.
01:18:04.712 --> 01:18:06.170
It would have told
me, for example,
01:18:06.170 --> 01:18:09.004
e to the ikx delta of y,
but it doesn't tell me
01:18:09.004 --> 01:18:11.850
how it depends on z, how
the wave functions on z.
01:18:11.850 --> 01:18:16.154
And if I added another
operator, for example, Py,
01:18:16.154 --> 01:18:17.320
this is no longer commuting.
01:18:17.320 --> 01:18:18.751
These two operators
don't commute.
01:18:18.751 --> 01:18:20.500
So a complete set of
commuting observables
01:18:20.500 --> 01:18:22.270
can be thought of as
the most operators
01:18:22.270 --> 01:18:25.020
you can write down that
all commute with each other
01:18:25.020 --> 01:18:28.060
and the minimum number
whose eigenvalues completely
01:18:28.060 --> 01:18:30.008
specify the state of the system.
01:18:30.008 --> 01:18:30.508
Cool?
01:18:30.508 --> 01:18:31.340
OK.
01:18:31.340 --> 01:18:33.750
So with all that said,
what is a complete set
01:18:33.750 --> 01:18:36.400
of commuting observables for
the angular momentum system?
01:18:40.730 --> 01:18:44.552
Well, it can't be any
two of Lx, Ly, and Lz.
01:18:44.552 --> 01:18:45.510
So let's just pick one.
01:18:45.510 --> 01:18:47.210
I'll call it Lz.
01:18:47.210 --> 01:18:48.220
I could've called it Lx.
01:18:48.220 --> 01:18:48.650
It doesn't matter.
01:18:48.650 --> 01:18:50.210
It's up to you
what axis is what.
01:18:50.210 --> 01:18:53.950
I'll just call it
Lz conventionally.
01:18:53.950 --> 01:18:56.280
And then L squared
also commutes,
01:18:56.280 --> 01:18:57.780
because L squared
commutes with Lx.
01:18:57.780 --> 01:18:59.420
It commutes Ly and with Lz.
01:18:59.420 --> 01:19:01.100
So this actually
forms a complete set
01:19:01.100 --> 01:19:06.010
of commuting observables for
the angular momentum system.
01:19:06.010 --> 01:19:08.290
Complete set of
commuting observables
01:19:08.290 --> 01:19:09.840
for angular momentum.
01:19:15.670 --> 01:19:21.000
So this idea will come
up more in the future.
01:19:21.000 --> 01:19:28.860
And here is going to
be the key [INAUDIBLE].
01:19:28.860 --> 01:19:31.320
So we'd like to use
the following fact.
01:19:31.320 --> 01:19:33.010
We want to construct
the eigenfunctions
01:19:33.010 --> 01:19:34.731
of our complete set
of-- yeah, question?
01:19:34.731 --> 01:19:36.314
AUDIENCE: Really
quick can you explain
01:19:36.314 --> 01:19:38.470
how you got that L
squared and Lx commute?
01:19:38.470 --> 01:19:39.970
PROFESSOR: Yeah, I
got it by knowing
01:19:39.970 --> 01:19:42.010
what you're going to write
on your solution set.
01:19:42.010 --> 01:19:43.530
So this is on your problem set.
01:19:43.530 --> 01:19:46.230
So the way it goes-- so there
are fancy ways of doing it,
01:19:46.230 --> 01:19:48.170
but the just direct
way of doing,
01:19:48.170 --> 01:19:49.930
how do you construct
these commutators?
01:19:49.930 --> 01:19:51.388
Is you know what
the operators are.
01:19:51.388 --> 01:19:52.802
You know what L squared is.
01:19:52.802 --> 01:19:54.510
And you know that L
squared is Lx squared
01:19:54.510 --> 01:19:55.950
plus Ly squared plus Lz squared.
01:19:55.950 --> 01:19:59.310
It's built in that
fashion out of x and Py.
01:19:59.310 --> 01:20:02.460
And then I literally just put
in the definitions of Lx, Ly,
01:20:02.460 --> 01:20:04.910
and Lz into that
expression for L squared
01:20:04.910 --> 01:20:06.910
and compute the
commutator with Lx, again,
01:20:06.910 --> 01:20:10.880
using the definition
in terms of Py and z.
01:20:10.880 --> 01:20:13.390
And then you just chug
through the commutators.
01:20:13.390 --> 01:20:14.390
Yeah, it just works out.
01:20:14.390 --> 01:20:17.700
So it's not obvious from
the way I just phrased it
01:20:17.700 --> 01:20:20.022
that it works out like that.
01:20:20.022 --> 01:20:21.980
Later on you'll probably
develop some intuition
01:20:21.980 --> 01:20:23.370
that it should be obvious.
01:20:23.370 --> 01:20:24.786
But for the moment,
I'm just going
01:20:24.786 --> 01:20:26.636
to call it a brute
force computation.
01:20:26.636 --> 01:20:29.010
And that's how you're going
to do it on your problem set.
01:20:29.010 --> 01:20:32.480
So let me tell you
where we're going next.
01:20:32.480 --> 01:20:35.310
So the question I
really want to deal with
01:20:35.310 --> 01:20:37.780
is what are the eigenfunctions?
01:20:37.780 --> 01:20:39.730
So this is where
we're leaving off.
01:20:42.270 --> 01:20:47.530
What are the eigenfunctions
of our complete set
01:20:47.530 --> 01:20:50.950
of commuting variable
L squared and Lz.
01:20:55.270 --> 01:20:56.120
What are these guys?
01:20:56.120 --> 01:20:57.190
And we know that
we could solve them
01:20:57.190 --> 01:20:58.960
by solving the differential
equations using
01:20:58.960 --> 01:21:00.520
those operators, but
that would be horrible.
01:21:00.520 --> 01:21:01.980
We'd like to do something
a little smarter.
01:21:01.980 --> 01:21:04.438
We'd like to use the commutation
relations and the algebra.
01:21:06.650 --> 01:21:11.690
And here a really beautiful
thing is going to happen.
01:21:11.690 --> 01:21:13.601
When you look at these
commutation relations,
01:21:13.601 --> 01:21:16.100
one thing they're telling us
is that we can't simultaneously
01:21:16.100 --> 01:21:20.030
have eigenfunctions
of Lx and Ly.
01:21:20.030 --> 01:21:24.919
However, the way that Lx and Ly
commute together is to form Lz.
01:21:24.919 --> 01:21:26.210
That gives us some information.
01:21:26.210 --> 01:21:29.910
That gives us some
magic, some power.
01:21:29.910 --> 01:21:32.339
And in particular, much like
that moment in the harmonic
01:21:32.339 --> 01:21:34.380
oscillator I said, well
look, we could write down
01:21:34.380 --> 01:21:37.040
these operators as a.
01:21:37.040 --> 01:21:40.170
Well look, we can write
down these operators,
01:21:40.170 --> 01:21:43.320
which I'm going to call
L plus and L minus.
01:21:43.320 --> 01:21:48.200
L plus is going to be
equal to Lx plus i Ly.
01:21:48.200 --> 01:21:53.390
And L minus is going to
be equal to Lx minus I Ly.
01:21:53.390 --> 01:21:55.880
Now Lx, Ly, those
are observables?
01:21:55.880 --> 01:21:59.464
What can you say about
them as operators?
01:21:59.464 --> 01:22:01.880
What kind of operators are
they since they're observables?
01:22:01.880 --> 01:22:03.180
Hermitian, exactly.
01:22:03.180 --> 01:22:07.320
So what's the Hermitian adjoint
of Hermitian plus i Hermitian?
01:22:07.320 --> 01:22:08.571
Hermitian minus i Hermitian.
01:22:08.571 --> 01:22:09.070
OK, good.
01:22:09.070 --> 01:22:11.120
So L minus is the
adjoint of L plus.
01:22:15.392 --> 01:22:17.100
So this is just going
to be a definition.
01:22:17.100 --> 01:22:19.308
Let's take these to be the
definitions of these guys.
01:22:22.902 --> 01:22:25.110
If we take their commutator,
something totally lovely
01:22:25.110 --> 01:22:26.080
happens.
01:22:26.080 --> 01:22:26.930
I'm not going to write
all the commutators.
01:22:26.930 --> 01:22:28.305
I'm just going to
write a couple.
01:22:28.305 --> 01:22:33.560
The first is if I take L squared
and I commute with L plus,
01:22:33.560 --> 01:22:36.010
well, L plus is Lx
plus Ly, and we already
01:22:36.010 --> 01:22:37.510
know that L squared
commutes with Lx
01:22:37.510 --> 01:22:38.510
and it commutes with Ly.
01:22:38.510 --> 01:22:42.600
So L squared
commutes with L plus.
01:22:42.600 --> 01:22:46.140
And similarly for L minus,
it commutes with each term.
01:22:46.140 --> 01:22:48.737
Question?
01:22:48.737 --> 01:22:49.320
Oh, I'm sorry.
01:22:49.320 --> 01:22:50.360
Lx.
01:22:50.360 --> 01:22:51.320
Shoot, that was an x.
01:22:51.320 --> 01:22:52.630
It just didn't look like it.
01:22:52.630 --> 01:22:54.000
Lx minus i Ly.
01:22:56.760 --> 01:22:59.796
So it commutes both
L plus and L minus.
01:22:59.796 --> 01:23:02.350
But here's the
real beauty of it.
01:23:02.350 --> 01:23:07.760
Lz with L plus is
equal to-- well,
01:23:07.760 --> 01:23:10.110
let's just do
dimensional analysis.
01:23:10.110 --> 01:23:12.070
L plus is Lx plus i Ly.
01:23:12.070 --> 01:23:14.930
We know that Lz with Lx
is something like Ly.
01:23:14.930 --> 01:23:17.930
And Lz with Ly is
something like Lx.
01:23:17.930 --> 01:23:19.560
With factors of i's an h bars.
01:23:22.340 --> 01:23:26.930
And when you work out the
commutator, which should only
01:23:26.930 --> 01:23:32.040
take you second, you
get i h bar L plus.
01:23:32.040 --> 01:23:37.020
And similarly, when we
construct Lz and L minus,
01:23:37.020 --> 01:23:42.630
we get minus h bar L minus.
01:23:42.630 --> 01:23:46.160
Are L plus an L minus Hermitian?
01:23:46.160 --> 01:23:49.310
No, they're each
other's adjoints.
01:23:49.310 --> 01:23:50.620
Lz is Hermitian.
01:23:53.360 --> 01:23:56.111
And look at this
commutation relation.
01:23:56.111 --> 01:23:57.110
What does that tell you?
01:24:00.520 --> 01:24:02.840
From the first observation,
if we have an energy
01:24:02.840 --> 01:24:05.070
or if we have an operator
e and an operator a
01:24:05.070 --> 01:24:06.850
that commute in this
fashion, then this
01:24:06.850 --> 01:24:09.010
tells you that the
eigenfunctions of this operator
01:24:09.010 --> 01:24:11.660
are staggered in a the
ladder spaced by h bar.
01:24:11.660 --> 01:24:16.680
The eigenvalues of Lz come
in a ladder spaced by h bar.
01:24:16.680 --> 01:24:20.030
We can raise with
L plus, and we can
01:24:20.030 --> 01:24:22.670
lower with L minus just like
in the harmonic oscillator
01:24:22.670 --> 01:24:25.440
problem.
01:24:25.440 --> 01:24:28.187
And we'll exploit the
rest of the-- we'll
01:24:28.187 --> 01:24:30.520
deduce the rest of the structure
of the angular momentum
01:24:30.520 --> 01:24:33.920
operator eigenfunctions next
time using this computation
01:24:33.920 --> 01:24:34.826
relation.
01:24:34.826 --> 01:24:37.580
See you next time.