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PROFESSOR: All right, so
for the next six lectures,

00:00:28.980 --> 00:00:33.250
including today, we're
going to finish off

00:00:33.250 --> 00:00:36.680
the course with the
application of everything

00:00:36.680 --> 00:00:39.560
we've studied so far
to a couple of ideas.

00:00:39.560 --> 00:00:42.820
The first being the existence
of solids-- why we have solids

00:00:42.820 --> 00:00:45.290
and why we have conductivity
in solids, which

00:00:45.290 --> 00:00:48.020
is basic properties
of materials.

00:00:48.020 --> 00:00:51.340
In particular, the story I want
you guys to leave the course

00:00:51.340 --> 00:00:54.860
with is an understanding of
why diamonds are transparent

00:00:54.860 --> 00:00:58.400
and why copper
isn't, which is sort

00:00:58.400 --> 00:00:59.797
of a crude fact about the world.

00:00:59.797 --> 00:01:01.630
But we can explain it
from first principles,

00:01:01.630 --> 00:01:03.840
which is pretty awesome.

00:01:03.840 --> 00:01:07.550
The second is we're going to
come back to this idea of spin

00:01:07.550 --> 00:01:09.330
and of 1/2 integral
angular momentum.

00:01:09.330 --> 00:01:11.510
This intrinsic angular
momentum of the electron.

00:01:11.510 --> 00:01:14.339
And we're going to use that
to motivate a couple of ideas.

00:01:14.339 --> 00:01:16.630
First off, we're going to
tie back to Bell's inequality

00:01:16.630 --> 00:01:17.730
from the very beginning,
and then we're

00:01:17.730 --> 00:01:19.730
going to do a little bit
on quantum computation.

00:01:19.730 --> 00:01:21.220
That will be the
last two lectures.

00:01:21.220 --> 00:01:25.817
So before I get started
though with today, two things.

00:01:25.817 --> 00:01:27.400
First, I'm going to
ask for questions.

00:01:27.400 --> 00:01:30.430
But the second is I got
some really good questions

00:01:30.430 --> 00:01:32.080
about hydrogen, so
I want to wrap up

00:01:32.080 --> 00:01:35.570
with one last
comment on hydrogen

00:01:35.570 --> 00:01:37.970
because it's entertaining.

00:01:37.970 --> 00:01:40.762
But before I get started
on that, questions

00:01:40.762 --> 00:01:41.970
about everything up till now?

00:01:49.560 --> 00:01:50.425
Yeah.

00:01:50.425 --> 00:01:52.365
AUDIENCE: So in the
[INAUDIBLE] there's

00:01:52.365 --> 00:01:55.980
someting called spirit orbit
couplings, another edition,

00:01:55.980 --> 00:02:00.180
the [INAUDIBLE], that
we haven't talked about,

00:02:00.180 --> 00:02:02.538
in addition throughout
this egression.

00:02:02.538 --> 00:02:04.923
Is that something we can
do with our knowledge now?

00:02:04.923 --> 00:02:06.134
Or is something [INAUDIBLE]?

00:02:06.134 --> 00:02:07.300
PROFESSOR: It absolutely is.

00:02:07.300 --> 00:02:09.591
So the question is-- and this
is a very good question--

00:02:09.591 --> 00:02:12.590
the question is, look,
if you look on Wikipedia

00:02:12.590 --> 00:02:15.880
about-- seriously, this
is a rational thing to do.

00:02:15.880 --> 00:02:17.940
If you look at Wikipedia
to learn something

00:02:17.940 --> 00:02:20.290
about hydrogen,
what you discover

00:02:20.290 --> 00:02:21.860
is everything
we've talked about,

00:02:21.860 --> 00:02:24.500
including the fine structure,
including the Zeeman effect.

00:02:24.500 --> 00:02:26.708
But you see that there are
a couple of other effects.

00:02:26.708 --> 00:02:29.460
So one effect, for example,
is when you look at the Zeeman

00:02:29.460 --> 00:02:31.300
effect a little more
carefully than we did,

00:02:31.300 --> 00:02:34.100
there's a second term in
the Zeeman effect, which

00:02:34.100 --> 00:02:36.760
is an induced magnetic moment.

00:02:36.760 --> 00:02:38.660
So even when the
angular momentum

00:02:38.660 --> 00:02:43.280
is smaller or even
zero, there's still

00:02:43.280 --> 00:02:46.740
a contribution to the energy
from the externally imposed

00:02:46.740 --> 00:02:48.136
magnetic field.

00:02:48.136 --> 00:02:49.760
So there are lots of
other corrections.

00:02:49.760 --> 00:02:51.556
One of them in
particular is something

00:02:51.556 --> 00:02:52.680
called spin orbit coupling.

00:02:56.990 --> 00:02:59.180
And so the question
was, what is that?

00:02:59.180 --> 00:03:02.450
And do we know enough to explain
what spin orbit coupling is?

00:03:02.450 --> 00:03:05.230
So let me give you a very
short answer to that question.

00:03:05.230 --> 00:03:08.320
So we know that if we
write down the energy

00:03:08.320 --> 00:03:14.230
operator for our system, we
have the energy operator--

00:03:14.230 --> 00:03:20.230
so the full energy operator
for hydrogen, I'll say,

00:03:20.230 --> 00:03:23.780
is the energy
operator for coulomb

00:03:23.780 --> 00:03:25.240
plus a bunch of corrections.

00:03:25.240 --> 00:03:27.670
So for example, we had the
relativistic correction

00:03:27.670 --> 00:03:29.350
plus a term that was
a coefficient times

00:03:29.350 --> 00:03:30.420
p to the fourth.

00:03:30.420 --> 00:03:32.567
And this is on your problem set.

00:03:32.567 --> 00:03:34.400
And this came from
relativistic corrections.

00:03:34.400 --> 00:03:35.400
I shouldn't call this c.

00:03:35.400 --> 00:03:37.590
I'll call it beta, maybe.

00:03:37.590 --> 00:03:39.566
Some coefficient.

00:03:39.566 --> 00:03:41.690
And there are a whole bunch
of other contributions.

00:03:41.690 --> 00:03:43.170
For example, there's
the term we studied

00:03:43.170 --> 00:03:44.870
that contributed to
the Zeeman effect.

00:03:44.870 --> 00:03:46.700
If there's an external
magnetic field,

00:03:46.700 --> 00:03:52.780
there's a B dot-- magnetic
moment of the system.

00:03:52.780 --> 00:03:55.800
And if the electron is in
a state with finite angular

00:03:55.800 --> 00:03:59.410
momentum, the magnetic moment is
some dimensionful coefficient,

00:03:59.410 --> 00:04:01.290
which we usually call
the Bohr magneton just

00:04:01.290 --> 00:04:03.390
because-- whatever--
we liked the guy.

00:04:03.390 --> 00:04:06.210
Times the angular momentum.

00:04:06.210 --> 00:04:08.930
But there can be
additional contributions.

00:04:08.930 --> 00:04:11.310
There are a whole bunch
of additional terms

00:04:11.310 --> 00:04:14.910
in the true potential
for hydrogen.

00:04:14.910 --> 00:04:16.810
One of them takes
the following form.

00:04:16.810 --> 00:04:25.280
It's a constant plus-- maybe
I'll call it kappa times

00:04:25.280 --> 00:04:33.200
spin dotted into the angular
momentum of the electron,

00:04:33.200 --> 00:04:36.360
of the electron bound
to the hydrogen system.

00:04:36.360 --> 00:04:39.997
So where this comes from is
a sort of beautiful story.

00:04:39.997 --> 00:04:41.330
So first off, could it be there?

00:04:41.330 --> 00:04:41.980
Sure.

00:04:41.980 --> 00:04:45.200
This is a term that could
exist with some coefficient.

00:04:45.200 --> 00:04:48.134
Why not?

00:04:48.134 --> 00:04:50.050
It turns out that you
can derive its existence

00:04:50.050 --> 00:04:55.280
from a study of the relativistic
version of the hydrogen system.

00:04:55.280 --> 00:04:58.790
We're not going to study
that in any detail in 804,

00:04:58.790 --> 00:05:01.560
but the basic idea
and what this does

00:05:01.560 --> 00:05:04.860
is totally amenable
to 804-level analysis.

00:05:04.860 --> 00:05:08.050
This is just saying that when
you have some orbital momentum,

00:05:08.050 --> 00:05:10.640
or when you have some spin
angular momentum, both of those

00:05:10.640 --> 00:05:13.000
corresponds to angular
momentum of an object that

00:05:13.000 --> 00:05:14.562
carries charge.

00:05:14.562 --> 00:05:16.270
And so it's not
unreasonable that there's

00:05:16.270 --> 00:05:18.103
going to be some
interaction between the two

00:05:18.103 --> 00:05:21.466
where the magnetic moments--
those two magnetic moments

00:05:21.466 --> 00:05:23.590
either want to be aligned
or anti-aligned depending

00:05:23.590 --> 00:05:25.930
on the sign of this coefficient.

00:05:25.930 --> 00:05:28.661
However, we haven't talked about
spin in any great detail yet.

00:05:28.661 --> 00:05:30.910
We're going to do that in
the last week of the course.

00:05:30.910 --> 00:05:33.641
So I'm going to defer talking
about this in any detail.

00:05:33.641 --> 00:05:35.390
And we're only going
to discuss it briefly

00:05:35.390 --> 00:05:37.310
at the end for a
couple of weeks.

00:05:37.310 --> 00:05:39.799
But absolutely, these
additional couplings

00:05:39.799 --> 00:05:41.590
are important for
hydrogen and they're also

00:05:41.590 --> 00:05:44.964
things you can study at
this level, at 804 level.

00:05:44.964 --> 00:05:46.380
I should also
emphasize that there

00:05:46.380 --> 00:05:48.004
are a whole bunch of
other corrections.

00:05:48.004 --> 00:05:50.980
There are lots of terms.

00:05:50.980 --> 00:05:53.020
One set of terms, an
infinite number of terms,

00:05:53.020 --> 00:05:55.930
are the further sub-leading
relativistic corrections

00:05:55.930 --> 00:05:57.400
of kinetic energy.

00:05:57.400 --> 00:05:58.610
And there are a whole
bunch of other corrections,

00:05:58.610 --> 00:06:00.568
so I'm going to leave
those out for the moment.

00:06:00.568 --> 00:06:04.440
But I want to emphasize
to everyone that we're

00:06:04.440 --> 00:06:07.280
building models of hydrogen,
and they're all approximations.

00:06:07.280 --> 00:06:08.562
Yeah.

00:06:08.562 --> 00:06:10.895
AUDIENCE: So it's a question
regarding this [INAUDIBLE].

00:06:13.330 --> 00:06:14.755
PROFESSOR: Yes.

00:06:14.755 --> 00:06:16.250
There is.

00:06:16.250 --> 00:06:19.360
So on the problem
set, you're asked

00:06:19.360 --> 00:06:22.440
to estimate the
correction to the energy.

00:06:22.440 --> 00:06:25.340
I wrote down the answer
in lecture the other day.

00:06:25.340 --> 00:06:30.220
You're asked to compute, in
particular, the l dependence.

00:06:30.220 --> 00:06:33.630
But to estimate the
magnitude or to estimate

00:06:33.630 --> 00:06:38.590
the value of the shift
in the energy eigenvalues

00:06:38.590 --> 00:06:42.145
to hydrogen from the first
relativistic correction

00:06:42.145 --> 00:06:44.670
is p to the fourth correction.

00:06:44.670 --> 00:06:47.260
And I expect that
this will be covered

00:06:47.260 --> 00:06:50.960
in your recitation in some
detail, but there the question

00:06:50.960 --> 00:06:54.994
was, is there a trick to
make this a little easier?

00:06:54.994 --> 00:06:56.910
There are a couple of
good physical arguments,

00:06:56.910 --> 00:06:57.980
which I'm not going to tell you.

00:06:57.980 --> 00:06:59.510
But there are nice
ways to do this.

00:06:59.510 --> 00:07:01.360
But at the end of
the day, you do not

00:07:01.360 --> 00:07:03.432
want to end up doing the
following computation.

00:07:03.432 --> 00:07:05.473
You don't want to take
the expectation value of p

00:07:05.473 --> 00:07:07.980
to the fourth and
write this as the sum

00:07:07.980 --> 00:07:11.660
integral of psi
complex conjugate

00:07:11.660 --> 00:07:14.540
derivative to the fourth psi.

00:07:14.540 --> 00:07:17.807
If you attempt to do this
integral, you will weep.

00:07:17.807 --> 00:07:19.390
This is not the way
you want to do it.

00:07:19.390 --> 00:07:20.880
And here's what you want to do.

00:07:20.880 --> 00:07:22.490
And I'm not going to
tell you how to get here,

00:07:22.490 --> 00:07:24.906
but you want to reduce the
calculation of this expectation

00:07:24.906 --> 00:07:28.920
value to the calculation of
the expectation value of 1

00:07:28.920 --> 00:07:33.570
over r and of 1 over r squared.

00:07:33.570 --> 00:07:36.100
And it turns out that if you
know these expectation values,

00:07:36.100 --> 00:07:38.730
that entirely suffices to
compute the expectation

00:07:38.730 --> 00:07:41.960
value of p to the fourth if
you take into account the fact

00:07:41.960 --> 00:07:46.130
that the energy eigenfunctions
satisfy the original energy

00:07:46.130 --> 00:07:47.660
eigenvalue equation.

00:07:47.660 --> 00:07:51.540
So p squared upon 2m
plus v of x of v of r

00:07:51.540 --> 00:07:55.720
is equal to e when acting
on those wave functions.

00:07:55.720 --> 00:07:57.930
So for computing
these guys, you could

00:07:57.930 --> 00:08:01.840
try to do this brute force,
not unrelated to problem,

00:08:01.840 --> 00:08:05.170
I believe, two on Rydberg
atoms on the problem set.

00:08:05.170 --> 00:08:06.670
But there are
actually sneakier ways

00:08:06.670 --> 00:08:11.580
to do these expectation
values as well.

00:08:11.580 --> 00:08:16.690
And I will leave that to you.

00:08:16.690 --> 00:08:19.810
But let me emphasize that you
can do a direct brute force

00:08:19.810 --> 00:08:22.780
calculation, but
you don't need to.

00:08:22.780 --> 00:08:24.580
And I would encourage
you to try to find

00:08:24.580 --> 00:08:29.375
an efficient, indirect way
to do these calculations.

00:08:29.375 --> 00:08:30.790
Did that answer your question?

00:08:30.790 --> 00:08:32.500
OK.

00:08:32.500 --> 00:08:34.409
Anything else?

00:08:34.409 --> 00:08:36.255
Yeah.

00:08:36.255 --> 00:08:38.183
AUDIENCE: So is [INAUDIBLE]
a time dependence

00:08:38.183 --> 00:08:45.368
to an operator with the
[INAUDIBLE] being [INAUDIBLE]?

00:08:45.368 --> 00:08:47.600
PROFESSOR: Does
adding-- well, it

00:08:47.600 --> 00:08:51.309
depends on how you introduce
the time dependents.

00:08:51.309 --> 00:08:52.850
AUDIENCE: Can a
operator [INAUDIBLE]?

00:08:56.780 --> 00:08:57.860
PROFESSOR: Yes.

00:08:57.860 --> 00:08:59.910
Yeah, absolutely.

00:08:59.910 --> 00:09:02.492
What's the reason
for the question?

00:09:02.492 --> 00:09:04.239
AUDIENCE: One of
the problems here.

00:09:04.239 --> 00:09:06.322
PROFESSOR: One of the
problems in the problem set?

00:09:06.322 --> 00:09:06.947
AUDIENCE: Yeah.

00:09:06.947 --> 00:09:08.740
It's like the last part of four.

00:09:08.740 --> 00:09:10.365
PROFESSOR: Remind me
which problem this

00:09:10.365 --> 00:09:11.445
is, I don't remember.

00:09:11.445 --> 00:09:13.007
AUDIENCE: [INAUDIBLE].

00:09:13.007 --> 00:09:13.590
PROFESSOR: Oh.

00:09:13.590 --> 00:09:14.727
Oh, oh!

00:09:14.727 --> 00:09:15.227
Ha, ha, ha.

00:09:18.640 --> 00:09:20.880
Sorry, I really
like that problem.

00:09:27.010 --> 00:09:29.210
Let me rephrase your question
in the following way.

00:09:34.640 --> 00:09:37.090
Is it intrinsically
non-Hermitian

00:09:37.090 --> 00:09:39.040
to have time
dependence in a system?

00:09:41.700 --> 00:09:44.280
So no, you can have-- so
what does Hermitian mean?

00:09:44.280 --> 00:09:46.380
Physically, what does
it mean for the energy

00:09:46.380 --> 00:09:47.579
to be Hermitian for example?

00:09:47.579 --> 00:09:49.120
What it means for
the energy operator

00:09:49.120 --> 00:09:54.160
to be Hermitian is that time
evolution, which is represented

00:09:54.160 --> 00:09:56.644
by the oper-- so let
me phrase it this way.

00:09:56.644 --> 00:09:58.810
So if we know that the
energy operator is Hermitian,

00:09:58.810 --> 00:09:59.720
what is that telling us?

00:09:59.720 --> 00:10:00.550
Well, it tells us
a lot of things.

00:10:00.550 --> 00:10:02.581
It tells us the energy
eigenvalues are real.

00:10:02.581 --> 00:10:03.080
That's good.

00:10:03.080 --> 00:10:06.267
So E dagger is equal to E.
But it tell us something else.

00:10:06.267 --> 00:10:07.850
Remember that the
Schrodinger equation

00:10:07.850 --> 00:10:14.110
is that ih bar dt on psi
is equal to E on psi.

00:10:14.110 --> 00:10:18.171
And we use this to argue
that the general solution

00:10:18.171 --> 00:10:19.920
to the Schrodinger
equation can be written

00:10:19.920 --> 00:10:22.960
in the following
elegant way, psi of t

00:10:22.960 --> 00:10:37.010
is equal to e to the minus
i upon h bar tE on psi at 0,

00:10:37.010 --> 00:10:41.730
where this was the
evolution operator u sub t.

00:10:41.730 --> 00:10:42.600
Yeah?

00:10:42.600 --> 00:10:44.560
And this is a unitary operator.

00:10:44.560 --> 00:10:47.122
And the way they say its
unitary goes back to the exam.

00:10:47.122 --> 00:10:48.580
The way to say it's
unitary is this

00:10:48.580 --> 00:10:51.200
is e to the i
Hermitian operator.

00:10:51.200 --> 00:10:53.400
t is a real number,
h is a real number.

00:10:53.400 --> 00:10:56.160
So this is e to the
i Hermitian operator.

00:10:56.160 --> 00:10:58.990
And anything of the form e
to the i Hermitian operator

00:10:58.990 --> 00:10:59.640
is unitary.

00:10:59.640 --> 00:11:01.450
It's adjoint to its inverse.

00:11:01.450 --> 00:11:04.930
What that tells you is
that since this is unitary,

00:11:04.930 --> 00:11:08.005
it preserves the magnitude, or
the norm, of the wave function.

00:11:08.005 --> 00:11:09.860
So probability is conserved.

00:11:09.860 --> 00:11:12.360
What would it mean for
the energy operator

00:11:12.360 --> 00:11:14.740
to be not Hermitian?

00:11:14.740 --> 00:11:16.240
Well, it would mean
a lot of things.

00:11:16.240 --> 00:11:17.880
One thing it would mean is
that the energy eigenvalues

00:11:17.880 --> 00:11:19.114
are no longer real.

00:11:19.114 --> 00:11:20.030
That's a little weird.

00:11:20.030 --> 00:11:21.430
But the much more
troubling thing

00:11:21.430 --> 00:11:24.000
is that the energy would
no longer be-- sorry,

00:11:24.000 --> 00:11:26.490
the probability would
no longer be conserved.

00:11:26.490 --> 00:11:29.430
The evolution operator, which is
the solution to the Schrodinger

00:11:29.430 --> 00:11:32.020
equation, would no longer
be a unitary operator.

00:11:32.020 --> 00:11:34.810
And the probability, the
norm or the wave function,

00:11:34.810 --> 00:11:37.210
would no longer be conserved.

00:11:37.210 --> 00:11:40.310
So the question at the
end of-- the last question

00:11:40.310 --> 00:11:42.600
in problem four
is really asking,

00:11:42.600 --> 00:11:44.670
in the system you're
thinking about

00:11:44.670 --> 00:11:46.040
is, probability conserved?

00:11:49.080 --> 00:11:53.190
And that's the question that
you should be asking yourself

00:11:53.190 --> 00:11:56.350
when you finish up problem four.

00:11:56.350 --> 00:11:57.990
Good?

00:11:57.990 --> 00:11:59.410
Yeah.

00:11:59.410 --> 00:12:01.810
AUDIENCE: When we solve
the Schrodinger equation

00:12:01.810 --> 00:12:04.795
in that way, does this
solution come from the fact

00:12:04.795 --> 00:12:08.971
that E is Hermitian or that
E is like that nice little

00:12:08.971 --> 00:12:10.839
[INAUDIBLE], that E
is time dependent.

00:12:10.839 --> 00:12:11.889
And that if E was
time dependent,

00:12:11.889 --> 00:12:14.305
why couldn't the coefficients
for the probability also be?

00:12:14.305 --> 00:12:15.910
PROFESSOR: Absolutely.

00:12:15.910 --> 00:12:18.070
So here I was just focusing
on the Hermiticity.

00:12:18.070 --> 00:12:19.880
And in solving
this, I'm assuming

00:12:19.880 --> 00:12:21.380
that the energy is
time independent.

00:12:21.380 --> 00:12:23.000
If the energy is not
time independent,

00:12:23.000 --> 00:12:26.117
then this is not the right
answer as you're pointing out.

00:12:26.117 --> 00:12:27.700
So in fact, you have
to do an integral

00:12:27.700 --> 00:12:29.116
and you have to
time order things,

00:12:29.116 --> 00:12:30.980
and it's a complicated story.

00:12:30.980 --> 00:12:32.380
But this is not the solution.

00:12:32.380 --> 00:12:36.150
But even if we have a
time independent system,

00:12:36.150 --> 00:12:40.980
if the energy operator is not
Hermitian, this is not unitary.

00:12:40.980 --> 00:12:43.200
So indeed, you're absolutely
right that if the energy

00:12:43.200 --> 00:12:44.460
operator is time
dependent, the story's

00:12:44.460 --> 00:12:46.010
more complicated than just this.

00:12:46.010 --> 00:12:49.410
But we already see the problem
that's salient for problem four

00:12:49.410 --> 00:12:52.070
at this stage with a time
independent operator.

00:12:52.070 --> 00:12:52.692
Yeah.

00:12:52.692 --> 00:12:54.150
AUDIENCE: Just one
question please.

00:12:54.150 --> 00:12:57.380
So if I had some sort of
system which is [INAUDIBLE],

00:12:57.380 --> 00:12:59.534
some of my particles are
weak in the other process.

00:12:59.534 --> 00:13:00.470
PROFESSOR: Yeah.

00:13:00.470 --> 00:13:02.836
AUDIENCE: And now the
probability is not conserved.

00:13:02.836 --> 00:13:05.850
Saying that this thing is not
[INAUDIBLE], the issue is,

00:13:05.850 --> 00:13:08.548
I think, which is also extremely
disturbing for [INAUDIBLE]

00:13:08.548 --> 00:13:11.880
complex in energy,
[INAUDIBLE], what is that?

00:13:11.880 --> 00:13:12.957
PROFESSOR: Yeah.

00:13:12.957 --> 00:13:14.290
AUDIENCE: That seems even worse.

00:13:14.290 --> 00:13:16.289
PROFESSOR: Well, I guess
it's a matter of taste.

00:13:19.367 --> 00:13:21.700
I would say that they're the
same thing in the following

00:13:21.700 --> 00:13:22.080
sense.

00:13:22.080 --> 00:13:23.650
Suppose I have complex
energy eigenvalues.

00:13:23.650 --> 00:13:25.700
Well, I know that if I
have a stationary state,

00:13:25.700 --> 00:13:28.430
then that stationary state
as a function of time

00:13:28.430 --> 00:13:31.630
is equal to the stationary
state at time 0 times

00:13:31.630 --> 00:13:35.370
e-- so this is an energy
eigenfunction rather than-- is

00:13:35.370 --> 00:13:39.830
e to the minus i the
energy times t over h bar?

00:13:39.830 --> 00:13:42.750
Or this is the
energy eigenvalue.

00:13:42.750 --> 00:13:44.780
Now, imagine e is complex.

00:13:44.780 --> 00:13:47.390
Sorry, first imagine e is real.

00:13:47.390 --> 00:13:50.110
That's not hard to imagine
because it's usually the case.

00:13:50.110 --> 00:13:53.510
And if that's true, what
happens to the wave function

00:13:53.510 --> 00:13:54.670
as we evolve through time?

00:13:54.670 --> 00:13:56.090
It rotates by phase.

00:13:56.090 --> 00:14:01.171
Now, if e is complex, imagine
e is of the form e-- let e

00:14:01.171 --> 00:14:01.670
be complex.

00:14:01.670 --> 00:14:10.740
So e is going to be E real minus
i gamma, some imaginary piece.

00:14:10.740 --> 00:14:15.370
So what that's going to give me
is an e to the minus gamma t.

00:14:15.370 --> 00:14:18.800
And that's a decaying thing that
when you get to norm squared

00:14:18.800 --> 00:14:20.640
gives you a decaying
envelope over time.

00:14:20.640 --> 00:14:22.610
It's going to give you
an exponential decay.

00:14:22.610 --> 00:14:25.310
This is going to be equal
to e to the minus ie

00:14:25.310 --> 00:14:31.490
real t over h bar times
e to be minus gamma

00:14:31.490 --> 00:14:34.849
t over h bar phi of 0.

00:14:34.849 --> 00:14:37.140
And when we take the norm
squared, the phase goes away,

00:14:37.140 --> 00:14:38.570
but this doesn't.

00:14:38.570 --> 00:14:44.110
So having loss of probability
is having a complex energy

00:14:44.110 --> 00:14:45.586
eigenvalue.

00:14:45.586 --> 00:14:46.570
Yeah.

00:14:46.570 --> 00:14:50.506
AUDIENCE: Why are we
talking about time

00:14:50.506 --> 00:14:53.950
independent operators if
there's an electromagnetic field

00:14:53.950 --> 00:14:56.642
and stuff is clearly going
on between the avenue

00:14:56.642 --> 00:14:56.950
of the electromagnetic field?

00:14:56.950 --> 00:14:58.080
PROFESSOR: This is a
very good question.

00:14:58.080 --> 00:15:00.070
And that actually
leads me into the thing

00:15:00.070 --> 00:15:01.375
I wanted to talk about first.

00:15:01.375 --> 00:15:03.750
So I got a bunch of questions--
thanks for that question.

00:15:03.750 --> 00:15:05.791
I got a bunch of questions
over the past few days

00:15:05.791 --> 00:15:11.180
about the magnetic moment of
the hydrogen system and what

00:15:11.180 --> 00:15:12.260
a strange idea that is.

00:15:12.260 --> 00:15:13.980
So let me talk about
that for a second.

00:15:13.980 --> 00:15:16.390
And when I'm done with
this little spiel,

00:15:16.390 --> 00:15:19.730
tell me if I've
answered your question.

00:15:19.730 --> 00:15:22.140
Well, let's just leave this up.

00:15:22.140 --> 00:15:22.640
OK.

00:15:22.640 --> 00:15:26.370
So in particular, let's think
about this term for a moment.

00:15:28.880 --> 00:15:31.130
Where this term came from
last time was we said,

00:15:31.130 --> 00:15:34.160
look, we turn on an
external magnetic field.

00:15:34.160 --> 00:15:36.500
Someone turns on the
switch and current runs

00:15:36.500 --> 00:15:39.870
through an electromagnet and
we get a uniform magnetic field

00:15:39.870 --> 00:15:42.360
in our fiducial volume, where
we're doing the experiment.

00:15:42.360 --> 00:15:48.760
And the electron system, if it
carries some angular momentum,

00:15:48.760 --> 00:15:50.747
it also has a charge,
angular momentum charge.

00:15:50.747 --> 00:15:52.330
That means it's got
a magnetic moment.

00:15:52.330 --> 00:15:54.950
That's how much magnetic moment.

00:15:54.950 --> 00:15:58.180
So this is saying that the
magnetic moment of the electron

00:15:58.180 --> 00:16:01.270
bound to the proton
with angular momentum

00:16:01.270 --> 00:16:03.600
wants to anti-align
with the magnetic field.

00:16:03.600 --> 00:16:05.943
Or align in this case
because I put the wrong sign.

00:16:09.330 --> 00:16:14.123
But here's something really
upsetting about that.

00:16:14.123 --> 00:16:16.230
So what I just said
sounds crazy if you

00:16:16.230 --> 00:16:18.050
think about it in
the following way.

00:16:18.050 --> 00:16:21.180
Take an electron in
the coulomb system,

00:16:21.180 --> 00:16:22.530
just straight up coulomb.

00:16:22.530 --> 00:16:24.160
Take an electron in
the coulomb system,

00:16:24.160 --> 00:16:28.860
put it in, say, the ground
state, lowest energy state.

00:16:28.860 --> 00:16:30.350
Is it moving?

00:16:30.350 --> 00:16:32.500
This is problem four.

00:16:32.500 --> 00:16:34.850
So is it moving?

00:16:34.850 --> 00:16:36.490
The expectation
value of position

00:16:36.490 --> 00:16:38.360
doesn't change in time.

00:16:38.360 --> 00:16:40.180
It's a stationary state.

00:16:40.180 --> 00:16:44.677
Expectation value of
nothing changes in time.

00:16:44.677 --> 00:16:46.135
OK, fine, but it's
the ground state

00:16:46.135 --> 00:16:47.170
that carries your
angular momentum.

00:16:47.170 --> 00:16:48.170
That's not so upsetting.

00:16:48.170 --> 00:16:50.950
Go to the first excited
state with angular momentum.

00:16:50.950 --> 00:16:54.200
The n equals 2, l equals
1, m equals 1 state.

00:16:54.200 --> 00:16:58.390
So it's got as much angular
momentum in lz as possible.

00:16:58.390 --> 00:17:00.651
Is that thing moving?

00:17:00.651 --> 00:17:01.150
Yeah.

00:17:01.150 --> 00:17:02.870
It's not moving at all.

00:17:02.870 --> 00:17:05.670
And yet, we're saying
there's a current associated

00:17:05.670 --> 00:17:07.760
with it, an electric current.

00:17:07.760 --> 00:17:10.550
And that electric current is
inducing a magnetic moment,

00:17:10.550 --> 00:17:12.859
a la right-hand rule.

00:17:12.859 --> 00:17:15.780
Maybe via Biot-Savart,
if you want to be fancy.

00:17:15.780 --> 00:17:19.540
And that magnetic moment--
a little current loop--

00:17:19.540 --> 00:17:23.325
is leading to this interaction,
the Zeeman's interaction.

00:17:25.829 --> 00:17:28.750
We know that it's true because
experiments show the Zeeman

00:17:28.750 --> 00:17:30.070
splitting.

00:17:30.070 --> 00:17:32.300
So it is definitely
true that there

00:17:32.300 --> 00:17:34.050
is a magnetic moment
of this thing.

00:17:34.050 --> 00:17:37.805
But it's not moving,
how can that be?

00:17:37.805 --> 00:17:39.930
So how can there be a
current if nothing is moving?

00:17:44.394 --> 00:17:47.900
AUDIENCE: It is, in some
sense moving, [INAUDIBLE].

00:17:47.900 --> 00:17:51.539
Why is the expectation value of
the [INAUDIBLE] to stop moving?

00:17:53.806 --> 00:17:55.180
PROFESSOR: What's
the expectation

00:17:55.180 --> 00:17:55.900
value of the momentum?

00:17:55.900 --> 00:17:56.680
AUDIENCE: I think
it's also probability.

00:17:56.680 --> 00:17:57.680
PROFESSOR: Yeah, it's 0.

00:18:01.757 --> 00:18:05.813
AUDIENCE: I mean, first of
all, the world isn't classical.

00:18:05.813 --> 00:18:08.440
You can't use
classical intuition.

00:18:08.440 --> 00:18:09.590
PROFESSOR: OK, true.

00:18:09.590 --> 00:18:13.520
AUDIENCE: Second of
all, let's suppose

00:18:13.520 --> 00:18:18.753
you have a uniform ring
of classical charge

00:18:18.753 --> 00:18:19.785
and set it spinning.

00:18:19.785 --> 00:18:20.410
PROFESSOR: Yes.

00:18:20.410 --> 00:18:23.372
AUDIENCE: That ring is
as a ring, not moving,

00:18:23.372 --> 00:18:25.330
but there's still a
current associated with it.

00:18:25.330 --> 00:18:28.274
PROFESSOR: So you're saying
that the electron is a ring?

00:18:28.274 --> 00:18:31.060
AUDIENCE: It might make sense
if you give it that [INAUDIBLE].

00:18:31.060 --> 00:18:31.590
PROFESSOR: I like this.

00:18:31.590 --> 00:18:32.340
No, this is good.

00:18:32.340 --> 00:18:33.520
It's wrong, but it's good.

00:18:33.520 --> 00:18:35.610
And the reason it's good
is that you're really

00:18:35.610 --> 00:18:37.620
pushing your assumptions
to try to figure out

00:18:37.620 --> 00:18:39.205
how the experimental
data can possibly match.

00:18:39.205 --> 00:18:41.630
And you're saying, look, we have
to just reject our intuition.

00:18:41.630 --> 00:18:43.454
Our intuition is clearly
leading us astray.

00:18:43.454 --> 00:18:44.120
And I like that.

00:18:44.120 --> 00:18:45.000
That's correct.

00:18:45.000 --> 00:18:46.360
So what we're going to do
in the next few minutes

00:18:46.360 --> 00:18:48.950
is work through that and try
to find the best way to phrase

00:18:48.950 --> 00:18:49.680
that.

00:18:49.680 --> 00:18:52.030
Your strategy is
the correct one.

00:18:52.030 --> 00:18:54.420
So let me rephrase
that slightly.

00:18:54.420 --> 00:18:58.796
Look, if you have a classical
distribution of charge,

00:18:58.796 --> 00:19:00.170
that distribution
of charge could

00:19:00.170 --> 00:19:05.217
be a stationary
distribution-- a distribution,

00:19:05.217 --> 00:19:07.550
which as a distribution of
charge doesn't change a dime.

00:19:07.550 --> 00:19:09.508
But each individual charge
in that distribution

00:19:09.508 --> 00:19:10.559
is itself moving.

00:19:10.559 --> 00:19:12.850
The problem here is that we
just have the one electron.

00:19:12.850 --> 00:19:16.590
If you ever look, you will
find the electron at a spot.

00:19:16.590 --> 00:19:19.119
But what you're really
saying is, look,

00:19:19.119 --> 00:19:21.160
it's a mistake to think
about the electron having

00:19:21.160 --> 00:19:22.460
a definite position
in the first place.

00:19:22.460 --> 00:19:23.340
You just shouldn't
think about that.

00:19:23.340 --> 00:19:24.460
The best you can
say is that it has

00:19:24.460 --> 00:19:26.370
some probability of
being in any spot.

00:19:26.370 --> 00:19:27.746
So let's work with that.

00:19:27.746 --> 00:19:29.370
Let's take that idea
and let's push it.

00:19:29.370 --> 00:19:31.911
Let's see how far we can take
this idea that the electron has

00:19:31.911 --> 00:19:33.310
some probability [INAUDIBLE].

00:19:33.310 --> 00:19:36.280
So suppose we have an
electron in a stationary state

00:19:36.280 --> 00:19:37.476
of the coulomb potential.

00:19:37.476 --> 00:19:38.850
The stationary
states are labeled

00:19:38.850 --> 00:19:40.687
by three integers-- n, l, and m.

00:19:40.687 --> 00:19:42.270
And we'd written
their wave functions.

00:19:42.270 --> 00:19:44.270
There are, of course, an
infinite number of ways

00:19:44.270 --> 00:19:46.522
to write in different notation.

00:19:46.522 --> 00:19:48.230
There are an infinite
number of fonts one

00:19:48.230 --> 00:19:55.025
could use for the normalization
constant 1/r R nl of little

00:19:55.025 --> 00:19:59.500
r, Y lm of theta and phi.

00:20:03.070 --> 00:20:04.350
And I'm going to rewrite this.

00:20:04.350 --> 00:20:05.933
I'm going to expand
this out slightly.

00:20:05.933 --> 00:20:12.620
This is N 1/r R of R and l.

00:20:12.620 --> 00:20:20.310
And Ylm, remember, was of the
form Pl of cosine theta times

00:20:20.310 --> 00:20:23.442
e to the im phi.

00:20:23.442 --> 00:20:25.420
Everyone cool with that?

00:20:25.420 --> 00:20:27.790
So I just wrote the spherical
harmonic as a polynomial

00:20:27.790 --> 00:20:30.935
in cosine theta times
an exponential in phi.

00:20:30.935 --> 00:20:33.060
AUDIENCE: Where did the
1/r come from, on the left?

00:20:33.060 --> 00:20:36.010
PROFESSOR: The 1/r
was-- so most people

00:20:36.010 --> 00:20:38.430
call capital R the whole thing.

00:20:38.430 --> 00:20:40.060
But this is the
pulling out to 1/r

00:20:40.060 --> 00:20:41.604
to simplify the
radio wave equation.

00:20:41.604 --> 00:20:43.020
And the reason I
prefer writing it

00:20:43.020 --> 00:20:44.760
this way is just that
this guy satisfies

00:20:44.760 --> 00:20:47.200
a simple 1d
Schrodinger equation.

00:20:47.200 --> 00:20:48.450
AUDIENCE: That was the u of r?

00:20:48.450 --> 00:20:49.950
PROFESSOR: That was a thing
that we called u of r,

00:20:49.950 --> 00:20:51.190
and then I confused the
hell out of everyone

00:20:51.190 --> 00:20:53.106
by calling three different
things on the board

00:20:53.106 --> 00:20:55.160
u, which was sort
of unnecessary.

00:20:55.160 --> 00:20:58.540
So this is the artist
previously known as u.

00:21:03.320 --> 00:21:05.050
The dude can sing.

00:21:05.050 --> 00:21:09.980
OK, so the first
question I want to ask

00:21:09.980 --> 00:21:11.730
is, so this is a
stationary state.

00:21:11.730 --> 00:21:14.850
Is the electron moving?

00:21:14.850 --> 00:21:16.640
No, not in any
conventional sense.

00:21:16.640 --> 00:21:19.040
If you compute the expectation
value of the position

00:21:19.040 --> 00:21:21.700
and you take its time
derivative, this is zero.

00:21:21.700 --> 00:21:23.677
We could do this either
by calculating it

00:21:23.677 --> 00:21:25.760
or just by observing that
it's a stationary state.

00:21:25.760 --> 00:21:30.520
And on principle, it
can't change in time.

00:21:30.520 --> 00:21:34.740
So this guy is not moving
in any conventional sense.

00:21:39.191 --> 00:21:40.690
So why is there an
electric current?

00:21:40.690 --> 00:21:43.300
Why do we get the Zeeman
magnetic moment, the Bohr

00:21:43.300 --> 00:21:45.420
magneton?

00:21:45.420 --> 00:21:48.750
So as was pointed out, look,
this is quantum mechanics.

00:21:48.750 --> 00:21:50.480
It's not classical mechanics.

00:21:50.480 --> 00:21:54.720
And in quantum mechanics, the
electron isn't at any point.

00:21:58.020 --> 00:22:00.500
Rather, there's some
probability density.

00:22:00.500 --> 00:22:02.130
And the probability
density that we

00:22:02.130 --> 00:22:06.800
find the electron at
some point r is psi

00:22:06.800 --> 00:22:15.630
squared l m or r squared.

00:22:15.630 --> 00:22:18.620
A familiar beast.

00:22:18.620 --> 00:22:21.730
And meanwhile, a wonderful
thing about the probability

00:22:21.730 --> 00:22:23.270
distribution in
quantum mechanics

00:22:23.270 --> 00:22:26.890
that we've already discussed
here is that it's conserved.

00:22:26.890 --> 00:22:31.350
The time derivative of the
probability density or r--

00:22:31.350 --> 00:22:33.850
remember, this is a
density, not a probability.

00:22:33.850 --> 00:22:37.920
The time rate of change of
the probability distribution

00:22:37.920 --> 00:22:45.070
is minus the gradient-- the
divergence of the probability

00:22:45.070 --> 00:22:49.250
current, where the
probability current j

00:22:49.250 --> 00:22:55.520
is equal to h bar
over the mass, which

00:22:55.520 --> 00:22:56.770
I'm going to call the mass mu.

00:22:56.770 --> 00:22:58.144
Oh, gee, I don't
want-- I'm going

00:22:58.144 --> 00:23:00.900
to call the mass
capital M. That's

00:23:00.900 --> 00:23:03.364
the mass of the
electron, which is

00:23:03.364 --> 00:23:05.530
a little strange because
it's a very small quantity.

00:23:05.530 --> 00:23:10.020
But anyway, h over capital
M of the imaginary part

00:23:10.020 --> 00:23:16.370
of psi complex
conjugate gradient psi.

00:23:16.370 --> 00:23:21.600
And you showed on a problem
set long ago that this is true

00:23:21.600 --> 00:23:25.180
if j takes this form by virtue
of the Schrodinger equation.

00:23:25.180 --> 00:23:28.610
Now, we usually write this
imaginary 1 over 2i times psi

00:23:28.610 --> 00:23:32.280
gradient-- or psi star gradient
psi minus psi gradient psi

00:23:32.280 --> 00:23:34.980
star, but that's equal
to the imaginary part

00:23:34.980 --> 00:23:38.030
of the first term.

00:23:38.030 --> 00:23:40.280
The thing you want to emphasize
is the imaginary part.

00:23:44.690 --> 00:23:46.570
So we have this current.

00:23:46.570 --> 00:23:51.290
In our system, the
position expectation value

00:23:51.290 --> 00:23:52.660
is time independent.

00:23:52.660 --> 00:23:54.620
And indeed, it's a
stationary state.

00:23:54.620 --> 00:23:56.930
So beyond the position
expectation value being time

00:23:56.930 --> 00:23:58.720
independent, the
probability density

00:23:58.720 --> 00:24:01.020
itself is time independent
because the wave function

00:24:01.020 --> 00:24:03.977
evolves by an overall phase
and the probability density

00:24:03.977 --> 00:24:04.810
is the norm squared.

00:24:04.810 --> 00:24:06.850
So the phase goes away.

00:24:06.850 --> 00:24:10.630
So in our system, in an
electron in this state

00:24:10.630 --> 00:24:12.380
in the coulomb
potential, the time rate

00:24:12.380 --> 00:24:15.690
of change of the
probability density--

00:24:15.690 --> 00:24:17.265
let me actually
do this over here.

00:24:20.830 --> 00:24:30.980
So in the stationary
state, psi l, n, m,

00:24:30.980 --> 00:24:33.950
the time rate of change
of the density is 0.

00:24:33.950 --> 00:24:36.920
And this tells us by the
conservation equation

00:24:36.920 --> 00:24:41.530
that the divergence of
the current is also 0.

00:24:41.530 --> 00:24:43.609
Does that tell us
that the current is 0?

00:24:43.609 --> 00:24:44.150
AUDIENCE: No.

00:24:44.150 --> 00:24:45.174
PROFESSOR: Right.

00:24:45.174 --> 00:24:46.090
So what's the current?

00:24:51.410 --> 00:24:54.170
Well, we've written everything
here in spherical coordinates.

00:24:54.170 --> 00:25:00.080
And the current is given in
terms of the gradient operator.

00:25:00.080 --> 00:25:03.880
So let me remind you quickly of
what the gradient operator is

00:25:03.880 --> 00:25:05.080
in spherical coordinates.

00:25:05.080 --> 00:25:07.430
It has three components--
a radial component,

00:25:07.430 --> 00:25:11.680
a theta component, and
then a phi component.

00:25:11.680 --> 00:25:17.130
And the radial part is just D r.

00:25:17.130 --> 00:25:22.360
The theta component
is 1 over r D theta.

00:25:22.360 --> 00:25:26.310
And the phi around
the equator component

00:25:26.310 --> 00:25:33.460
is equal to 1 over
r sine theta D phi.

00:25:33.460 --> 00:25:36.360
Everyone cool with that?

00:25:36.360 --> 00:25:37.570
So what's j?

00:25:37.570 --> 00:25:39.450
Well, j is going to
have a component, j

00:25:39.450 --> 00:25:40.500
in the radial direction.

00:25:40.500 --> 00:25:42.680
The current in the
radial direction.

00:25:42.680 --> 00:25:45.660
And intuitively,
what should that be?

00:25:45.660 --> 00:25:50.560
Is there stuff going out or in?

00:25:50.560 --> 00:25:51.650
There shouldn't be.

00:25:51.650 --> 00:25:54.990
It's hydrogen.

00:25:54.990 --> 00:25:56.570
It's not doing this.

00:25:56.570 --> 00:25:57.729
So this should be 0.

00:25:57.729 --> 00:25:59.270
And we can just
quickly see that this

00:25:59.270 --> 00:26:01.187
is h bar upon M imaginary
part of-- well, this

00:26:01.187 --> 00:26:03.311
is the r component, it's
going to be the derivative

00:26:03.311 --> 00:26:04.610
in the radial direction.

00:26:04.610 --> 00:26:08.585
But the derivative in the radial
direction is going to be real.

00:26:11.132 --> 00:26:13.590
The derivative in the radial
direction is going to be real.

00:26:13.590 --> 00:26:14.810
So when we take
the norm squared,

00:26:14.810 --> 00:26:15.600
we don't pick up anything.

00:26:15.600 --> 00:26:16.740
We pick up an
overall coefficient.

00:26:16.740 --> 00:26:18.031
It's going to be strictly real.

00:26:18.031 --> 00:26:20.647
The phase, e to the
i m phi cancels out

00:26:20.647 --> 00:26:22.230
because this is the
complex conjugate.

00:26:22.230 --> 00:26:26.850
So the imaginary part of
this thing is going to be 0.

00:26:26.850 --> 00:26:28.020
OK?

00:26:28.020 --> 00:26:31.020
So J r is 0.

00:26:31.020 --> 00:26:36.935
And similarly, J theta
p is a real function.

00:26:36.935 --> 00:26:39.660
When we take a derivative,
we get a real function.

00:26:39.660 --> 00:26:41.850
And then when we multiply
by its complex conjugate,

00:26:41.850 --> 00:26:43.516
again, the phase
cancels out and we just

00:26:43.516 --> 00:26:47.650
get a whole bunch of real stuff
whose imaginary piece is 0.

00:26:47.650 --> 00:26:50.390
So J theta is 0.

00:26:50.390 --> 00:26:52.115
But J phi is a cool one.

00:26:55.680 --> 00:26:59.650
In my head, I was
just thinking J 5.

00:26:59.650 --> 00:27:01.115
OK, at least someone got that.

00:27:05.600 --> 00:27:09.655
So J phi, however,
is not going to be

00:27:09.655 --> 00:27:10.780
0 for the following reason.

00:27:10.780 --> 00:27:11.740
So what is it equal to?

00:27:11.740 --> 00:27:15.360
It's equal to h bar
upon M-- the mass,

00:27:15.360 --> 00:27:19.800
M-- times the imaginary
part of-- well,

00:27:19.800 --> 00:27:23.780
psi star, which is
psi complex conjugate.

00:27:23.780 --> 00:27:29.440
And then the gradient
with respect to phi.

00:27:29.440 --> 00:27:32.604
The phi component is
1 over r sine theta.

00:27:32.604 --> 00:27:34.270
And then derivative
with respect to phi.

00:27:34.270 --> 00:27:39.280
The derivative with respect
to phi pulls down a i m.

00:27:39.280 --> 00:27:45.640
Oh, look-- times psi, which
is equal to h bar upon m,

00:27:45.640 --> 00:27:48.000
times-- well, psi
is norm squared.

00:27:48.000 --> 00:27:49.820
That's real.

00:27:49.820 --> 00:27:51.040
Psi squared.

00:27:51.040 --> 00:27:53.020
1/r sine theta.

00:27:53.020 --> 00:27:54.560
That's real, so I
can pull that out.

00:27:54.560 --> 00:27:56.610
And we are left
with imaginary part

00:27:56.610 --> 00:28:02.510
of im, which is just
m, which is not 0.

00:28:02.510 --> 00:28:05.320
And in particular, it's
proportional to h bar m.

00:28:11.050 --> 00:28:13.075
Everyone see that?

00:28:13.075 --> 00:28:19.540
So what this is saying is that
the current, the probability

00:28:19.540 --> 00:28:24.130
current, for an electron
in the stationary state psi

00:28:24.130 --> 00:28:31.710
nlm of the Coulomb potential
is equal to norm psi squared

00:28:31.710 --> 00:28:41.390
over m r sine theta h bar
m in the phi direction.

00:28:41.390 --> 00:28:45.220
The phi uni-vector.

00:28:45.220 --> 00:28:47.230
Cool?

00:28:47.230 --> 00:28:49.660
So nothing is moving,
but there's a current.

00:28:52.350 --> 00:28:53.240
What is moving?

00:28:58.130 --> 00:29:00.260
What is the thing of
whose current-- of whom

00:29:00.260 --> 00:29:01.220
this is the current?

00:29:01.220 --> 00:29:02.250
AUDIENCE: Probability density.

00:29:02.250 --> 00:29:03.708
PROFESSOR: The
probability density.

00:29:03.708 --> 00:29:08.450
The probability
density is rotating.

00:29:08.450 --> 00:29:13.500
So what this is telling
us is that it's true

00:29:13.500 --> 00:29:15.680
the system is stationary.

00:29:15.680 --> 00:29:20.400
But I want to know-- I'm looking
down at the equatorial plane,

00:29:20.400 --> 00:29:20.900
OK.

00:29:20.900 --> 00:29:22.566
So this plot is going
to be looking down

00:29:22.566 --> 00:29:25.200
on the equatorial
plane of hydrogen.

00:29:25.200 --> 00:29:27.265
Here is the origin, the
center of the potential.

00:29:27.265 --> 00:29:28.640
And what this is
telling-- so I'm

00:29:28.640 --> 00:29:30.000
going to draw these vectors.

00:29:30.000 --> 00:29:31.740
They're in e phi direction.

00:29:41.770 --> 00:29:42.270
OK.

00:29:42.270 --> 00:29:44.650
But their magnitude
falls off with 1

00:29:44.650 --> 00:29:46.970
over r and the norm square
of the wave function, which

00:29:46.970 --> 00:29:48.700
also is falling
off exponentially.

00:29:48.700 --> 00:29:52.280
And it goes to 0 at the
origin as r to the l plus 1.

00:29:52.280 --> 00:29:56.530
And m must be no bigger than l.

00:29:56.530 --> 00:29:58.910
So in order for m to be
non-zero, l must be non-zero.

00:29:58.910 --> 00:30:02.550
Which means this must go like r
to the something greater than m

00:30:02.550 --> 00:30:03.450
plus 1.

00:30:03.450 --> 00:30:04.540
So the r's cancel out.

00:30:04.540 --> 00:30:06.230
And we have that it vanishes.

00:30:06.230 --> 00:30:09.290
So the contribution
vanishes at the origin,

00:30:09.290 --> 00:30:15.730
is small near the origin,
is largest at some radius,

00:30:15.730 --> 00:30:17.690
and then falls off again.

00:30:17.690 --> 00:30:19.106
So the magnitude
of the arrow here

00:30:19.106 --> 00:30:21.064
is meant to indicate the
magnitude of the flux.

00:30:25.190 --> 00:30:29.470
So what we see is that we have
a probability distribution where

00:30:29.470 --> 00:30:33.800
the probability,
as a distribution,

00:30:33.800 --> 00:30:34.680
is time independent.

00:30:37.110 --> 00:30:37.610
Right?

00:30:37.610 --> 00:30:40.090
The probability distribution
is, in fact, time independent.

00:30:40.090 --> 00:30:44.355
But there's a current which is a
persistent, stationary current.

00:30:47.370 --> 00:30:48.980
Everyone cool with that?

00:30:48.980 --> 00:30:51.520
So if I ask you,
where is the particle?

00:30:51.520 --> 00:30:54.320
Well, the probability
density tells you that.

00:30:54.320 --> 00:30:56.670
And if I ask you,
what's the current?

00:30:56.670 --> 00:31:03.780
The electric current,
minus e times j.

00:31:03.780 --> 00:31:04.360
At a point.

00:31:04.360 --> 00:31:06.850
And this is really the current
density and the probability

00:31:06.850 --> 00:31:08.740
current density.

00:31:08.740 --> 00:31:13.650
Now what is the consequence of
the statement, the probability

00:31:13.650 --> 00:31:15.300
density itself is
not changing in time.

00:31:15.300 --> 00:31:20.220
It's that there is zero
divergence of this current.

00:31:20.220 --> 00:31:22.814
And indeed, there
is zero divergence.

00:31:22.814 --> 00:31:24.230
Every little bit
going into a spot

00:31:24.230 --> 00:31:27.460
is matched by some
equivalent amount going out.

00:31:27.460 --> 00:31:29.000
There's zero divergence.

00:31:29.000 --> 00:31:29.500
Cool?

00:31:33.690 --> 00:31:40.110
And so we see that while it's
true that nothing is moving,

00:31:40.110 --> 00:31:44.070
it is also true that there is
a non-trivial electric current.

00:31:44.070 --> 00:31:46.020
And when you use
the Biot-Savart law

00:31:46.020 --> 00:31:49.090
to sum up the contributions
from each little bit

00:31:49.090 --> 00:31:53.790
of this current, what you
get is-- right hand rule--

00:31:53.790 --> 00:31:55.790
a magnetic field,
a magnetic moment.

00:31:55.790 --> 00:31:58.540
And following nothing
but the Biot-Savart law

00:31:58.540 --> 00:32:01.710
and using what you know
about the wave functions,

00:32:01.710 --> 00:32:06.250
this gives us that the magnetic
moment is equal to the Bohr

00:32:06.250 --> 00:32:17.290
magneton, b times m, z.

00:32:17.290 --> 00:32:18.990
OK?

00:32:18.990 --> 00:32:20.390
Everyone cool with that?

00:32:20.390 --> 00:32:21.580
Yeah?

00:32:21.580 --> 00:32:26.480
AUDIENCE: So when the perimeter
intuitively is [INAUDIBLE],

00:32:26.480 --> 00:32:29.920
still make all these
other expectation values?

00:32:29.920 --> 00:32:30.796
PROFESSOR: Yeah.

00:32:30.796 --> 00:32:32.504
AUDIENCE: Why is that
an intuitive force?

00:32:32.504 --> 00:32:33.410
Or maybe it's not.

00:32:33.410 --> 00:32:35.160
PROFESSOR: Well, here's
one way to say it.

00:32:35.160 --> 00:32:37.517
There are two-- let me
rephrase that question.

00:32:37.517 --> 00:32:39.350
Let me give you a
different question to ask.

00:32:39.350 --> 00:32:41.630
And let's think about the
answers to that question

00:32:41.630 --> 00:32:43.050
and I hope that will
answer your question,

00:32:43.050 --> 00:32:44.174
if it doesn't ask it again.

00:32:44.174 --> 00:32:47.780
So here's the question I
would suggest that you ask.

00:32:47.780 --> 00:32:49.060
Or that someone ask.

00:32:49.060 --> 00:32:49.850
I'll ask it.

00:32:55.357 --> 00:32:57.190
And I just totally lost
my train of thought.

00:32:57.190 --> 00:32:58.480
That's totally me.

00:32:58.480 --> 00:33:00.820
What was the question
I wanted you to ask?

00:33:00.820 --> 00:33:01.680
Wow, that's amazing.

00:33:01.680 --> 00:33:05.230
I completely just in the blink
of an eye totally lost my train

00:33:05.230 --> 00:33:07.170
of -- ah, yes.

00:33:07.170 --> 00:33:10.330
So here, it was crucial
in this calculation

00:33:10.330 --> 00:33:12.840
that the current was given
by the imaginary part

00:33:12.840 --> 00:33:13.730
of the gradient.

00:33:13.730 --> 00:33:16.720
And as you showed in a problem
set a while back, anytime

00:33:16.720 --> 00:33:21.130
you have a wave
function which is real,

00:33:21.130 --> 00:33:23.490
then the current will vanish.

00:33:23.490 --> 00:33:25.344
Here it was crucial--
in order to get

00:33:25.344 --> 00:33:27.010
a non-vanishing
current-- it was crucial

00:33:27.010 --> 00:33:28.510
that the wave
function was not real.

00:33:28.510 --> 00:33:31.160
It had a phase.

00:33:31.160 --> 00:33:32.920
But you also showed
on a problem set

00:33:32.920 --> 00:33:35.182
that you can always take
your energy eigenfunctions--

00:33:35.182 --> 00:33:37.640
for bounce-- you can always
take your energy eigenfunctions

00:33:37.640 --> 00:33:38.245
and make them real.

00:33:38.245 --> 00:33:40.790
You can always construct a
basis of energy eigenfunctions

00:33:40.790 --> 00:33:43.740
which are strictly real.

00:33:43.740 --> 00:33:44.534
Sorry?

00:33:44.534 --> 00:33:45.950
AUDIENCE: In one
dimension, right?

00:33:45.950 --> 00:33:46.872
PROFESSOR: Yeah well--

00:33:46.872 --> 00:33:48.330
AUDIENCE: Aren't
you in three also?

00:33:48.330 --> 00:33:50.431
PROFESSOR: What did
you use for that proof?

00:33:50.431 --> 00:33:52.540
AUDIENCE: Just used linear
combination of the two.

00:33:52.540 --> 00:33:52.606
PROFESSOR: Yeah.

00:33:52.606 --> 00:33:54.439
You just use the energy
eigenvalue equation,

00:33:54.439 --> 00:33:58.230
hermiticity, and the
linear, the energy operator.

00:33:58.230 --> 00:34:00.500
And that's perfectly true
in any number of dimensions.

00:34:00.500 --> 00:34:02.020
AUDIENCE: OK.

00:34:02.020 --> 00:34:04.290
PROFESSOR: So that sounds crazy.

00:34:04.290 --> 00:34:09.225
So first off, why do we have a
not real energy eigenfunction?

00:34:09.225 --> 00:34:11.350
Can we construct purely
real energy eigenfunctions?

00:34:11.350 --> 00:34:14.110
If we can, those
energy eigenfunctions

00:34:14.110 --> 00:34:17.340
would appear to have no current.

00:34:17.340 --> 00:34:19.620
So they would have
no magnetic moment.

00:34:19.620 --> 00:34:20.740
How do these fit together?

00:34:25.850 --> 00:34:27.522
Can you construct
energy eigenfunctions

00:34:27.522 --> 00:34:28.980
that are pure real
for this system,

00:34:28.980 --> 00:34:30.063
for the Coulomb potential?

00:34:36.451 --> 00:34:37.409
Let me give you a hint.

00:34:37.409 --> 00:34:41.530
In 1d for a free particle, what
are the energy eigenfunctions?

00:34:41.530 --> 00:34:42.565
1d,

00:34:42.565 --> 00:34:43.870
AUDIENCE: [INTERPOSING VOICES].

00:34:43.870 --> 00:34:45.780
PROFESSOR: E to the ikx, right?

00:34:45.780 --> 00:34:48.053
That's not real.

00:34:48.053 --> 00:34:49.469
But there's another
eigenfunction,

00:34:49.469 --> 00:34:50.750
you get the minus ikx.

00:34:50.750 --> 00:34:52.525
And if you take
the sum of those,

00:34:52.525 --> 00:34:54.250
you get e to the ikx
plus e to the minus

00:34:54.250 --> 00:34:57.990
ikx-- divide by 2 for fun-- and
that gives you cosine of kx.

00:34:57.990 --> 00:34:58.494
That's real.

00:34:58.494 --> 00:34:59.910
But there's a
second one, which is

00:34:59.910 --> 00:35:01.477
sine of kx, which is also real.

00:35:01.477 --> 00:35:03.310
Now you can take linear
combinations of them

00:35:03.310 --> 00:35:05.500
with i's and get the
exponentials back.

00:35:05.500 --> 00:35:08.290
But let's just take
the real part, right?

00:35:08.290 --> 00:35:10.360
Now, do those
carry any momentum?

00:35:10.360 --> 00:35:14.420
What's the momentum expectation
value for cosine of kx?

00:35:14.420 --> 00:35:15.120
AUDIENCE: 0.

00:35:15.120 --> 00:35:16.020
PROFESSOR: 0, because
you get confirmation

00:35:16.020 --> 00:35:17.870
from plus k and
one from minus k.

00:35:17.870 --> 00:35:19.830
They exactly cancel.

00:35:19.830 --> 00:35:21.970
Now look at this solution.

00:35:21.970 --> 00:35:25.030
Can you construct a real
energy eigenfunction

00:35:25.030 --> 00:35:26.359
of the Coulomb potential?

00:35:26.359 --> 00:35:26.984
AUDIENCE: Sure.

00:35:26.984 --> 00:35:27.878
PROFESSOR: How.

00:35:27.878 --> 00:35:30.011
AUDIENCE: [INAUDIBLE].

00:35:30.011 --> 00:35:30.760
PROFESSOR: Great..

00:35:30.760 --> 00:35:32.690
Let's take this and
its complex conjugate.

00:35:32.690 --> 00:35:33.930
So what that would be?

00:35:33.930 --> 00:35:36.710
Well, if we take this, psi nlm.

00:35:36.710 --> 00:35:39.390
And its complex conjugate, well,
what is its complex conjugate?

00:35:39.390 --> 00:35:40.230
These are real.

00:35:40.230 --> 00:35:41.520
That just gives me a minus.

00:35:41.520 --> 00:35:44.420
So that's the state
plus psi and l minus

00:35:44.420 --> 00:35:47.810
m, which is also an allowed
state with the same energy

00:35:47.810 --> 00:35:50.450
by rotational invariance.

00:35:50.450 --> 00:35:55.650
And this is thus
the state psi nl.

00:35:55.650 --> 00:36:00.890
And it doesn't have a definite
lz angular momentum, right?

00:36:00.890 --> 00:36:06.220
So I'll just call it
psi sub nl unhappy face.

00:36:06.220 --> 00:36:07.470
It is not an lz eigenfunction.

00:36:07.470 --> 00:36:09.170
But it is an energy
eigenfunction.

00:36:09.170 --> 00:36:12.940
And it could also have
built a minus with divide

00:36:12.940 --> 00:36:13.770
by 2i for fun.

00:36:13.770 --> 00:36:16.400
I could have built the
sine or the cosine of phi.

00:36:16.400 --> 00:36:20.460
One of which showed
up on your exam.

00:36:20.460 --> 00:36:23.480
So I can find a basis of
these states, sine and cosine,

00:36:23.480 --> 00:36:26.112
instead of, give me the im
phi, I need the minus im phi.

00:36:26.112 --> 00:36:27.820
And those would have
been perfectly real.

00:36:27.820 --> 00:36:30.445
And in those situations, what
would the current have been?

00:36:30.445 --> 00:36:31.600
AUDIENCE: [INAUDIBLE].

00:36:31.600 --> 00:36:33.966
PROFESSOR: If I put the system
in this stationary state,

00:36:33.966 --> 00:36:35.340
what would the
current have been?

00:36:35.340 --> 00:36:36.120
AUDIENCE: 0.

00:36:36.120 --> 00:36:38.680
PROFESSOR: Identically 0.

00:36:38.680 --> 00:36:43.160
However, that wouldn't be a
state with a definite angular

00:36:43.160 --> 00:36:43.660
momentum.

00:36:43.660 --> 00:36:46.436
So it wouldn't be surprising
that it has zero current.

00:36:46.436 --> 00:36:47.560
It's got some contribution.

00:36:47.560 --> 00:36:49.100
It's a super position of
having some angular momentum

00:36:49.100 --> 00:36:50.808
and having the opposite
angular momentum.

00:36:50.808 --> 00:36:54.840
And the expectation value,
the expected value, is zero.

00:36:54.840 --> 00:36:57.540
I can also study states with a
definite value of the angular

00:36:57.540 --> 00:36:58.040
momentum.

00:36:58.040 --> 00:36:59.520
Those are not real.

00:36:59.520 --> 00:37:01.190
There's nothing wrong with that.

00:37:01.190 --> 00:37:02.810
I could have
constructed states which

00:37:02.810 --> 00:37:05.126
don't have a definite angular
momentum but are real.

00:37:05.126 --> 00:37:06.750
But instead, I want
to work with states

00:37:06.750 --> 00:37:09.790
that have a definite
angular momentum.

00:37:09.790 --> 00:37:11.094
That cool?

00:37:11.094 --> 00:37:12.010
So there's no tension.

00:37:12.010 --> 00:37:13.051
There's no contradiction.

00:37:15.610 --> 00:37:18.260
It's just that, if we're
interested in finding states

00:37:18.260 --> 00:37:22.030
that have a definite energy and
a definite lz angular momentum,

00:37:22.030 --> 00:37:24.240
that is not going to
be a real function.

00:37:24.240 --> 00:37:26.640
And nothing tells us
that it has to be.

00:37:26.640 --> 00:37:28.170
And when you have
a non-trivial lz,

00:37:28.170 --> 00:37:30.419
when you have a non-trivial
angular momentum, just

00:37:30.419 --> 00:37:32.460
like a classical particle
with nontrivial angular

00:37:32.460 --> 00:37:35.940
momentum that has charge, we
find that there's a current.

00:37:35.940 --> 00:37:37.590
And thus a magnetic moment.

00:37:37.590 --> 00:37:40.820
So the important thing with
having a definite nonzero

00:37:40.820 --> 00:37:43.360
momentum, or in this
case angular momentum,

00:37:43.360 --> 00:37:45.467
to give us the current.

00:37:45.467 --> 00:37:46.960
Yeah?

00:37:46.960 --> 00:37:49.115
Questions about this?

00:37:49.115 --> 00:37:49.615
Yeah?

00:37:49.615 --> 00:37:51.842
AUDIENCE: Using the same
logic, it kind of looks

00:37:51.842 --> 00:37:54.317
like we're taking it
a charged particle

00:37:54.317 --> 00:37:55.750
and moving it around in circles.

00:37:55.750 --> 00:37:57.178
Like, where we
have a distribution

00:37:57.178 --> 00:37:57.654
and we're spinning.

00:37:57.654 --> 00:37:58.130
PROFESSOR: Yeah.

00:37:58.130 --> 00:37:59.082
AUDIENCE: With spin a
distribution of charge,

00:37:59.082 --> 00:38:01.325
we're accelerating the
charged particles in it.

00:38:01.325 --> 00:38:01.950
PROFESSOR: Yes.

00:38:01.950 --> 00:38:02.290
AUDIENCE: Individually.

00:38:02.290 --> 00:38:02.630
PROFESSOR: Yes.

00:38:02.630 --> 00:38:04.460
AUDIENCE: If we accelerate
them, they emit radiation.

00:38:04.460 --> 00:38:04.855
PROFESSOR: Yes.

00:38:04.855 --> 00:38:06.520
AUDIENCE: Hydrogen atoms
don't emit radiation.

00:38:06.520 --> 00:38:07.145
PROFESSOR: Yes.

00:38:07.145 --> 00:38:08.960
AUDIENCE: How does that go?

00:38:08.960 --> 00:38:11.300
PROFESSOR: That sounds
a lot like problem four

00:38:11.300 --> 00:38:13.494
on your problem set.

00:38:13.494 --> 00:38:14.630
AUDIENCE: [INAUDIBLE]

00:38:14.630 --> 00:38:15.380
PROFESSOR: Ah, OK.

00:38:15.380 --> 00:38:18.480
Well, that's a great question.

00:38:18.480 --> 00:38:21.960
And you should think about it.

00:38:21.960 --> 00:38:25.380
How could I say--
you read my mind.

00:38:25.380 --> 00:38:26.021
Yes.

00:38:26.021 --> 00:38:28.020
Struggling with this is
exactly the point of one

00:38:28.020 --> 00:38:29.850
of the problems on
your problem set.

00:38:29.850 --> 00:38:32.840
And ask me that again after the
problem set has been turned in,

00:38:32.840 --> 00:38:35.372
and I'll give you a
happy disquisition on it.

00:38:35.372 --> 00:38:36.830
But I want you to
struggle with it.

00:38:36.830 --> 00:38:39.411
Because it's hard and
interesting question.

00:38:39.411 --> 00:38:39.910
Yeah?

00:38:39.910 --> 00:38:43.330
AUDIENCE: So I don't think this
answers my question necessarily

00:38:43.330 --> 00:38:44.850
about the radiation.

00:38:44.850 --> 00:38:48.320
Because, analogously,
the classical enm,

00:38:48.320 --> 00:38:50.402
this would be the
magnetostatic case,

00:38:50.402 --> 00:38:50.890
where you just have a static--

00:38:50.890 --> 00:38:51.330
PROFESSOR: Precisely.

00:38:51.330 --> 00:38:51.755
Precisely.

00:38:51.755 --> 00:38:53.000
AUDIENCE: But with
radiation, you have, quickly

00:38:53.000 --> 00:38:54.791
oscillating fields and
the transition times

00:38:54.791 --> 00:38:57.477
are really small and the
frequencies of the wave

00:38:57.477 --> 00:38:58.560
functions are really fast.

00:38:58.560 --> 00:39:00.487
So I still don't know
if that's the same.

00:39:00.487 --> 00:39:02.320
PROFESSOR: So this
question is unfortunately

00:39:02.320 --> 00:39:05.260
a linear combination of a
really good independent question

00:39:05.260 --> 00:39:06.690
and that question.

00:39:06.690 --> 00:39:08.520
So ask me after the
lecture and I'll

00:39:08.520 --> 00:39:11.417
talk to you about the part
that's linearly independent.

00:39:11.417 --> 00:39:13.370
OK.

00:39:13.370 --> 00:39:14.700
I'm on thin ice here.

00:39:14.700 --> 00:39:15.230
OK.

00:39:15.230 --> 00:39:19.160
Anything else before we
dispense with hydrogen?

00:39:19.160 --> 00:39:19.730
OK.

00:39:19.730 --> 00:39:22.640
The question that
you all keep coming

00:39:22.640 --> 00:39:26.689
to about the-- well,
problem four on the-- I

00:39:26.689 --> 00:39:28.480
think it's problem four
on the problem set.

00:39:28.480 --> 00:39:31.190
Maybe it's problem two.

00:39:31.190 --> 00:39:32.890
It involves almost
no computation,

00:39:32.890 --> 00:39:35.056
but it's the most intellectually
challenging problem

00:39:35.056 --> 00:39:36.540
on the exam-- on
the problem set.

00:39:36.540 --> 00:39:38.873
So take it seriously, even
though it's calculation-free.

00:39:41.500 --> 00:39:43.830
OK.

00:39:43.830 --> 00:39:44.922
Yeah?

00:39:44.922 --> 00:39:49.104
AUDIENCE: Due to this
proton in the [INAUDIBLE].

00:39:49.104 --> 00:39:49.770
PROFESSOR: Yeah.

00:39:49.770 --> 00:39:50.269
Yeah.

00:39:50.269 --> 00:39:52.669
It's so that the--
it's not by symmetry,

00:39:52.669 --> 00:39:53.960
because there isn't a symmetry.

00:39:53.960 --> 00:39:56.930
The proton is 2,000 times
heavier than the electron.

00:39:56.930 --> 00:40:04.740
But-- let me make sure I'm
understanding your question.

00:40:04.740 --> 00:40:08.955
Your question is, we
found that the electron is

00:40:08.955 --> 00:40:11.500
in bound states in
the Coulomb potential.

00:40:11.500 --> 00:40:13.000
But in hydrogen,
you have two parts,

00:40:13.000 --> 00:40:14.416
you have an electron
and a proton.

00:40:14.416 --> 00:40:16.980
So is the electron in
that state and the proton

00:40:16.980 --> 00:40:19.270
is just free and cruising
on its own thing?

00:40:19.270 --> 00:40:22.330
That's exactly the topic
of the next 45 minutes.

00:40:22.330 --> 00:40:23.150
OK.

00:40:23.150 --> 00:40:23.760
Good question.

00:40:23.760 --> 00:40:27.900
So with that insight,
let me turn now

00:40:27.900 --> 00:40:30.630
to the question of
identical particles

00:40:30.630 --> 00:40:33.570
or multiple particles.

00:40:33.570 --> 00:40:34.070
OK.

00:40:34.070 --> 00:40:36.026
So we're done with
Coulomb for the moment.

00:40:36.026 --> 00:40:37.650
Pretty much for the
rest of the course.

00:40:37.650 --> 00:40:40.270
So I want to move on to
the following question.

00:40:40.270 --> 00:40:42.769
Suppose I have a system-- we've
spent a lot of time thinking

00:40:42.769 --> 00:40:44.810
about a particle in a potential.

00:40:44.810 --> 00:40:46.860
I would like to think
about multiple particles

00:40:46.860 --> 00:40:47.560
for a minute.

00:40:51.320 --> 00:40:54.840
And neat things happen
for multiple particles

00:40:54.840 --> 00:41:00.450
that don't happen for
individual, isolated particles.

00:41:00.450 --> 00:41:02.104
So let's think
about what those--

00:41:02.104 --> 00:41:04.270
let's think about the physics
of multiple particles.

00:41:04.270 --> 00:41:08.570
So in particular, classically--
in classical mechanics--

00:41:08.570 --> 00:41:11.000
if I have two particles,
what is the information

00:41:11.000 --> 00:41:11.164
I have to specify?

00:41:11.164 --> 00:41:12.410
I have to specify the state.

00:41:12.410 --> 00:41:15.950
I have to specify the position
of the first particle.

00:41:15.950 --> 00:41:18.980
X1 and its momentum, p1.

00:41:18.980 --> 00:41:21.480
And then the position of the
second particle x2 and p2.

00:41:21.480 --> 00:41:23.670
I'm going to omit
vectors over everything,

00:41:23.670 --> 00:41:25.461
but everything is in
the appropriate number

00:41:25.461 --> 00:41:26.560
of dimensions.

00:41:26.560 --> 00:41:30.130
So in classical mechanics--
in quantum mechanics,

00:41:30.130 --> 00:41:32.130
the state of the
system is specified

00:41:32.130 --> 00:41:35.700
by a wave function, which is
a function of the positions--

00:41:35.700 --> 00:41:41.190
or the degrees of freedom,
let's say x1 and x2.

00:41:41.190 --> 00:41:47.840
And x1 and x2 and p1
and p2 are promoted

00:41:47.840 --> 00:41:51.739
to operators representing
these observables.

00:41:51.739 --> 00:41:53.280
And the wave function
is a function--

00:41:53.280 --> 00:41:55.760
now let me just quickly tell
you what this notation means.

00:41:55.760 --> 00:42:00.150
What this notation means is
x1 is some number, like 7.

00:42:00.150 --> 00:42:02.470
And the quantity
in the first spot,

00:42:02.470 --> 00:42:08.150
this means that the
first particle indicated

00:42:08.150 --> 00:42:12.450
by the first slot, is at x1.

00:42:12.450 --> 00:42:15.665
And the second
particle is at x2.

00:42:18.430 --> 00:42:20.430
So whenever I write a
wave function, what I mean

00:42:20.430 --> 00:42:23.010
is this is the probability
amplitude-- the thing

00:42:23.010 --> 00:42:24.967
whose norm squared is
probability-- to find

00:42:24.967 --> 00:42:27.300
the first particle at this
value and the second particle

00:42:27.300 --> 00:42:28.160
of that value.

00:42:28.160 --> 00:42:30.920
So the 1 and 2 label the
points, not the particles.

00:42:30.920 --> 00:42:33.270
The particle is labeled by
the position inside this wave

00:42:33.270 --> 00:42:33.770
function.

00:42:33.770 --> 00:42:35.438
Everyone cool with that?

00:42:35.438 --> 00:42:36.767
OK.

00:42:36.767 --> 00:42:37.850
So that's just a notation.

00:42:37.850 --> 00:42:40.290
So the quantum mechanical
description of two particles

00:42:40.290 --> 00:42:43.920
is a wave function of both
positions and operators

00:42:43.920 --> 00:42:48.510
representing the position
and the coordinates

00:42:48.510 --> 00:42:55.933
of the particles.

00:42:59.450 --> 00:43:04.010
And the probability--
actually, let's

00:43:04.010 --> 00:43:09.180
go ahead and finish up here--
and the probability density

00:43:09.180 --> 00:43:13.110
to find the first particle.

00:43:13.110 --> 00:43:22.500
First at x1, and
the second at x2.

00:43:22.500 --> 00:43:27.460
It's just the norm squared
of the amplitude, psi x1, x2,

00:43:27.460 --> 00:43:29.240
norm squared.

00:43:29.240 --> 00:43:30.480
Just as usual.

00:43:30.480 --> 00:43:31.790
That's the probability density.

00:43:34.718 --> 00:43:35.700
All right?

00:43:35.700 --> 00:43:37.770
Everyone cool with that?

00:43:37.770 --> 00:43:43.170
And what are x1, x2, commutator?

00:43:43.170 --> 00:43:44.170
What is this commutator?

00:43:47.937 --> 00:43:49.410
AUDIENCE: [INAUDIBLE].

00:43:49.410 --> 00:43:51.826
PROFESSOR: Can you know the
position of the first particle

00:43:51.826 --> 00:43:54.290
and the position of
the second particle?

00:43:54.290 --> 00:43:55.860
Simultaneously?

00:43:55.860 --> 00:43:56.450
Sure.

00:43:56.450 --> 00:43:56.990
I'm here.

00:43:56.990 --> 00:43:59.510
You're there.

00:43:59.510 --> 00:44:00.610
0.

00:44:00.610 --> 00:44:06.175
And x1 with p1-- just work in
one dimension for the moment--

00:44:06.175 --> 00:44:07.598
is equal to?

00:44:07.598 --> 00:44:09.510
AUDIENCE: [INAUDIBLE].

00:44:09.510 --> 00:44:10.840
PROFESSOR: Our h bar.

00:44:10.840 --> 00:44:13.630
And the same would have
been true of x2 and p2.

00:44:13.630 --> 00:44:18.621
And what about x1 with p2?

00:44:18.621 --> 00:44:19.120
0.

00:44:19.120 --> 00:44:21.420
These are independent
quantities.

00:44:21.420 --> 00:44:22.939
OK.

00:44:22.939 --> 00:44:24.480
So it's more or less
as you'd expect.

00:44:24.480 --> 00:44:28.900
So just to be explicit about
this, let's do an example.

00:44:28.900 --> 00:44:29.740
Two free particles.

00:44:37.640 --> 00:44:39.120
So what's the energy operator?

00:44:39.120 --> 00:44:42.620
Well, the energy operator is
equal to p squared upon 2m

00:44:42.620 --> 00:44:49.330
for the first particle, 1,
plus p squared 2 upon 2m 2.

00:44:49.330 --> 00:44:51.010
Second particle, its mass is m2.

00:44:51.010 --> 00:44:52.660
The first particle has mass m1.

00:44:52.660 --> 00:44:56.800
And the system is free,
so this is plus 0.

00:44:56.800 --> 00:44:57.687
Yeah?

00:44:57.687 --> 00:44:59.645
AUDIENCE: So for the
[INAUDIBLE] exponentially,

00:44:59.645 --> 00:45:05.124
is that true [INAUDIBLE]
particles [INAUDIBLE]?

00:45:05.124 --> 00:45:05.790
PROFESSOR: Yeah.

00:45:05.790 --> 00:45:06.290
OK.

00:45:06.290 --> 00:45:10.970
So the question is, x1,
x2, commutator equals 0?

00:45:10.970 --> 00:45:15.320
Is that true even if there are
forces between the particles?

00:45:15.320 --> 00:45:17.430
So what are the forces
between particles--

00:45:17.430 --> 00:45:19.466
what are they going
to contribute to?

00:45:19.466 --> 00:45:20.550
Yeah, the potential.

00:45:20.550 --> 00:45:22.280
And that's going to show
up in the energy operator.

00:45:22.280 --> 00:45:23.520
So that will certainly
matter and it

00:45:23.520 --> 00:45:25.353
will change what the
energy eigenvalues are.

00:45:25.353 --> 00:45:27.610
But it won't tell what
can or can't be measured.

00:45:27.610 --> 00:45:31.260
It won't tell you what
properties the system can have

00:45:31.260 --> 00:45:32.431
or not.

00:45:32.431 --> 00:45:32.930
Good.

00:45:32.930 --> 00:45:33.718
Yeah?

00:45:33.718 --> 00:45:35.176
AUDIENCE: What's
with the particles

00:45:35.176 --> 00:45:37.303
in the state in which
measuring the position of one

00:45:37.303 --> 00:45:39.460
makes the position of the
other one in certainty?

00:45:39.460 --> 00:45:41.110
PROFESSOR: We'll talk about
things like that in a minute.

00:45:41.110 --> 00:45:41.750
AUDIENCE: Does it screw this up?

00:45:41.750 --> 00:45:42.333
PROFESSOR: No.

00:45:42.333 --> 00:45:46.780
This is-- do the
commutation relations

00:45:46.780 --> 00:45:49.080
care what state you're in?

00:45:49.080 --> 00:45:50.990
They're relations
amongst operators.

00:45:50.990 --> 00:45:54.550
So they're independent
of the state always.

00:45:54.550 --> 00:45:55.990
OK.

00:45:55.990 --> 00:45:56.849
Yeah?

00:45:56.849 --> 00:45:58.390
AUDIENCE: For the
two free particles,

00:45:58.390 --> 00:46:00.310
are we assuming that they're
not charged particles?

00:46:00.310 --> 00:46:00.600
PROFESSOR: Yeah.

00:46:00.600 --> 00:46:02.540
I'm assuming that there are
no interactions whatsoever.

00:46:02.540 --> 00:46:04.000
Just totally free particles.

00:46:04.000 --> 00:46:05.490
Potential is zero.

00:46:05.490 --> 00:46:10.162
V of x1, x2 equals 0.

00:46:10.162 --> 00:46:11.120
So they're not charged.

00:46:11.120 --> 00:46:12.930
They're just totally
uninteresting.

00:46:12.930 --> 00:46:15.210
No interactions, no
forces, no potential.

00:46:15.210 --> 00:46:15.910
OK.

00:46:15.910 --> 00:46:18.310
So we know how to
solve this problem

00:46:18.310 --> 00:46:19.600
because we can use separation.

00:46:19.600 --> 00:46:25.550
Psi of x1, x2 can be
written as-- well,

00:46:25.550 --> 00:46:31.430
we know what the solutions
of p1 upon 2m is equal to er.

00:46:31.430 --> 00:46:35.200
So we can write
this as chi of x1.

00:46:35.200 --> 00:46:37.810
And actually, I'm going
to call the positions

00:46:37.810 --> 00:46:40.419
of these guys a and b.

00:46:40.419 --> 00:46:42.710
Because the subscriptions
are just terribly misleading.

00:46:42.710 --> 00:46:46.170
So the first point is called a,
the second point is called b.

00:46:46.170 --> 00:46:49.960
So a and b.

00:46:49.960 --> 00:46:56.430
And this is p1 and p2 the
momentum of the two guys.

00:46:56.430 --> 00:46:59.580
Chi of a and phi of b.

00:46:59.580 --> 00:47:01.605
So all I'm doing here
is I'm using separation.

00:47:05.310 --> 00:47:07.840
Pa, pb.

00:47:07.840 --> 00:47:10.091
So here's, instead of calling
the positions x1 and x2,

00:47:10.091 --> 00:47:12.090
I'm going to call them a
and b just because it's

00:47:12.090 --> 00:47:14.340
going to be easier to write
and make things clear.

00:47:14.340 --> 00:47:15.560
So I'm using separation.

00:47:15.560 --> 00:47:21.640
And then p1 squared, or
pa squared, upon 2m on cih

00:47:21.640 --> 00:47:26.870
of a is equal to ea, chi a.

00:47:26.870 --> 00:47:29.240
And ditto for b.

00:47:29.240 --> 00:47:33.850
This tells us that
chi of a is equal to e

00:47:33.850 --> 00:47:40.680
to the sub-coefficient
c, e to the i k a.

00:47:40.680 --> 00:47:42.900
Everyone cool with that?

00:47:42.900 --> 00:47:46.370
A is just replacing x1.

00:47:46.370 --> 00:47:47.504
So it's just the position.

00:47:47.504 --> 00:47:49.170
So this is just saying
that the solution

00:47:49.170 --> 00:47:52.370
to a single free
particle is a plane wave.

00:47:52.370 --> 00:47:55.200
And because it's just the
sum of the momentum squared,

00:47:55.200 --> 00:47:58.760
we can separate the equation and
the same thing obtains for psi.

00:47:58.760 --> 00:48:02.200
So similarly for phi.

00:48:02.200 --> 00:48:04.550
Ditto.

00:48:04.550 --> 00:48:09.820
Phi of b is equal to
the e to the i k sub b.

00:48:09.820 --> 00:48:13.030
I'll call this k sub
a to distinguish them.

00:48:13.030 --> 00:48:18.240
So the wave function's a
basis for the wave functions

00:48:18.240 --> 00:48:23.250
for two particles, psi
of a b with energy e

00:48:23.250 --> 00:48:29.590
is equal to e to
the i ka A plus kb

00:48:29.590 --> 00:48:39.390
B, Where E is equal to h
bar squared on 2m K sub

00:48:39.390 --> 00:48:43.820
a squared plus k sub b squared.

00:48:43.820 --> 00:48:45.820
Yeah?

00:48:45.820 --> 00:48:47.640
For a system of
two free particles,

00:48:47.640 --> 00:48:52.330
is every wave function of the
form chi of a times phi of b?

00:48:56.917 --> 00:48:58.500
If you have a system
that's separable,

00:48:58.500 --> 00:49:01.010
is every wave function, is every
solution itself, separated?

00:49:01.010 --> 00:49:01.750
AUDIENCE: No.

00:49:01.750 --> 00:49:02.333
PROFESSOR: No.

00:49:02.333 --> 00:49:04.180
Because we can have
arbitrary superpositions

00:49:04.180 --> 00:49:06.666
of forms of this type.

00:49:06.666 --> 00:49:08.290
So we get superpositions
of plane waves

00:49:08.290 --> 00:49:11.260
as long as the energies of each
plane wave in the superposition

00:49:11.260 --> 00:49:13.440
are equal to e-- the total
energy is equal to e-- I

00:49:13.440 --> 00:49:14.336
can just superpose
them and I still

00:49:14.336 --> 00:49:15.650
have an energy eigenfunction.

00:49:15.650 --> 00:49:17.900
Everyone cool with that?

00:49:17.900 --> 00:49:18.420
OK.

00:49:18.420 --> 00:49:20.040
So nothing shocking here.

00:49:20.040 --> 00:49:22.020
But I did this
example so that we'd

00:49:22.020 --> 00:49:27.540
have the notation of chi
and phi for the two states.

00:49:27.540 --> 00:49:30.170
OK.

00:49:30.170 --> 00:49:33.660
So now let me come to
this question of what

00:49:33.660 --> 00:49:34.370
about the proton?

00:49:34.370 --> 00:49:38.640
Well suppose I have a system now
which is not a free particle.

00:49:38.640 --> 00:49:40.330
It's two particles, a and b.

00:49:40.330 --> 00:49:46.530
And e is equal to pa squared
over 2a plus pb squared

00:49:46.530 --> 00:49:49.000
upon 2mb.

00:49:49.000 --> 00:49:52.510
Plus a potential that
depends only on a minus b.

00:49:56.378 --> 00:49:57.280
OK?

00:49:57.280 --> 00:50:00.004
So this is, for example,
what happens in the Coulomb

00:50:00.004 --> 00:50:02.420
potential when you include the
proton having a finite mass

00:50:02.420 --> 00:50:04.586
instead of being infinitely
massive and stuck still.

00:50:07.250 --> 00:50:09.330
So now we can do
exactly the same thing

00:50:09.330 --> 00:50:12.531
we did in classical mechanics
when you have a potential that

00:50:12.531 --> 00:50:14.530
only depends on the
distance between two things.

00:50:14.530 --> 00:50:16.730
I can reorganize
degrees of freedom

00:50:16.730 --> 00:50:20.190
into the center
of mass position.

00:50:20.190 --> 00:50:32.360
R is equal to 1 over ma plus
mb of ma, a, plus mb, b.

00:50:32.360 --> 00:50:34.130
So that's the center
of mass motion.

00:50:34.130 --> 00:50:41.860
And the relative distance
is equal to 1 over-- whoops.

00:50:41.860 --> 00:50:43.130
I don't need that.

00:50:43.130 --> 00:50:43.860
A minus b.

00:50:47.370 --> 00:50:50.130
And then if you do this and you
write out the energy operator,

00:50:50.130 --> 00:50:52.250
e is equal to
minus h bar squared

00:50:52.250 --> 00:50:56.620
upon 2 capital M, total mass.

00:50:56.620 --> 00:51:07.310
Dr squared Plus 1 minus h bar
squared over 2 mu d little r

00:51:07.310 --> 00:51:11.675
squared plus v or r.

00:51:11.675 --> 00:51:13.149
OK?

00:51:13.149 --> 00:51:15.440
So this is exactly what
happens in classical mechanics.

00:51:15.440 --> 00:51:17.550
You work in terms of the
center of mass coordinate

00:51:17.550 --> 00:51:19.150
and the relative coordinate.

00:51:19.150 --> 00:51:21.820
The relative coordinate
becomes effectively

00:51:21.820 --> 00:51:24.159
an independent degree of
freedom with a potential,

00:51:24.159 --> 00:51:25.450
which is the central potential.

00:51:25.450 --> 00:51:28.766
And the center of mass
coordinate is a free particle.

00:51:28.766 --> 00:51:30.390
So if we have a proton
and our electron

00:51:30.390 --> 00:51:32.000
and they're attracted
to each other by Coulomb

00:51:32.000 --> 00:51:33.364
there's a center of mass motion.

00:51:33.364 --> 00:51:34.780
And then they do
together whatever

00:51:34.780 --> 00:51:37.115
they do together, a
la Coulomb potential.

00:51:37.115 --> 00:51:38.540
Everyone cool with that?

00:51:38.540 --> 00:51:43.050
The only difference is that the
mass in the Coulomb potential

00:51:43.050 --> 00:51:44.960
is not the mass of
the bare electron.

00:51:44.960 --> 00:51:46.720
But it's the
geometric mean mu is

00:51:46.720 --> 00:51:50.520
equal to ma and b
over ma plus mb.

00:51:54.130 --> 00:51:56.280
Now for a proton
and an electron,

00:51:56.280 --> 00:51:59.320
if ma is the proton
and mb is the electron,

00:51:59.320 --> 00:52:03.030
then this is proton electron
over proton plus electron.

00:52:03.030 --> 00:52:07.094
But proton is about 2,000
times the mass of the electron.

00:52:07.094 --> 00:52:09.760
So this is basically the mass of
the proton and they factor out.

00:52:09.760 --> 00:52:13.970
So the effective reduced mass
is roughly equal to the electron

00:52:13.970 --> 00:52:16.130
mass.

00:52:16.130 --> 00:52:20.242
Corrections of a part in 1,000.

00:52:20.242 --> 00:52:21.200
OK?

00:52:21.200 --> 00:52:23.840
So to answer your
question, is the proton

00:52:23.840 --> 00:52:26.900
also in some complicated
state described it?

00:52:26.900 --> 00:52:30.470
Well in fact, neither the
electron nor the proton

00:52:30.470 --> 00:52:32.790
are described by the
Coulomb potential.

00:52:32.790 --> 00:52:38.900
But the relative position,
the relative radial distance

00:52:38.900 --> 00:52:42.060
between them, is controlled
by the Coulomb potential

00:52:42.060 --> 00:52:46.371
and the center of mass degree
of freedom is a free particle.

00:52:46.371 --> 00:52:47.680
Does that make sense?

00:52:47.680 --> 00:52:48.180
OK.

00:52:48.180 --> 00:52:48.680
Good.

00:52:51.050 --> 00:52:51.680
OK.

00:52:51.680 --> 00:52:58.770
So, so much for that example.

00:52:58.770 --> 00:53:01.260
The free particle and
the central potential.

00:53:04.239 --> 00:53:05.780
Here's the much more
interesting that

00:53:05.780 --> 00:53:08.180
happens when we have
multiple particles.

00:53:08.180 --> 00:53:08.750
Yeah?

00:53:08.750 --> 00:53:11.426
AUDIENCE: Can you explain what
you mean by the [INAUDIBLE]?

00:53:11.426 --> 00:53:12.550
PROFESSOR: Oh, yeah, sorry.

00:53:12.550 --> 00:53:15.030
This is the-- I generated the
gradient with respect to r.

00:53:15.030 --> 00:53:16.970
So if r is a vector,
this is the gradient

00:53:16.970 --> 00:53:18.579
with respect to r, norm squared.

00:53:18.579 --> 00:53:20.120
And this is the
gradient with respect

00:53:20.120 --> 00:53:22.720
to the relative coordinate.

00:53:22.720 --> 00:53:27.540
So for example, this is--
if we're in one dimension,

00:53:27.540 --> 00:53:29.470
so this is strictly
one dimensional.

00:53:29.470 --> 00:53:34.610
Then this is just the derivative
with respect to r squared.

00:53:34.610 --> 00:53:40.260
And this is derivative with
respect to little r squared.

00:53:40.260 --> 00:53:42.320
Just the gradient squared.

00:53:42.320 --> 00:53:43.983
Did that answer your question?

00:53:43.983 --> 00:53:44.608
AUDIENCE: Yeah.

00:53:44.608 --> 00:53:46.520
What about in hydrogen?

00:53:46.520 --> 00:53:48.830
PROFESSOR: Well then
it's gradient operator.

00:53:48.830 --> 00:53:50.621
The thing that takes
the function gives you

00:53:50.621 --> 00:53:53.740
a vector which is the
directional derivative

00:53:53.740 --> 00:53:54.360
of that--

00:53:54.360 --> 00:53:55.280
AUDIENCE: [INAUDIBLE].

00:53:55.280 --> 00:53:55.740
PROFESSOR: Yeah.

00:53:55.740 --> 00:53:56.360
Exactly.

00:53:56.360 --> 00:53:59.395
In the r direction.

00:53:59.395 --> 00:54:01.260
AUDIENCE: Is that
subraction right there?

00:54:01.260 --> 00:54:02.230
PROFESSOR: Sorry?

00:54:02.230 --> 00:54:04.070
AUDIENCE: Is that a
subtraction between--

00:54:04.070 --> 00:54:05.211
PROFESSOR: A subtraction.

00:54:05.211 --> 00:54:05.710
Where?

00:54:05.710 --> 00:54:08.510
AUDIENCE: Between the first
and second [INAUDIBLE].

00:54:08.510 --> 00:54:10.730
PROFESSOR: Minus h bar
squared upon 2m, dr squared.

00:54:10.730 --> 00:54:13.091
Minus h bar squared
over 2 mu, dr-- this?

00:54:13.091 --> 00:54:15.215
AUDIENCE: OK, so that's--
those are separate terms,

00:54:15.215 --> 00:54:15.670
not multiplied?

00:54:15.670 --> 00:54:15.770
PROFESSOR: Yeah.

00:54:15.770 --> 00:54:17.140
They're not multiplied, right?

00:54:17.140 --> 00:54:17.959
It's just the sum.

00:54:17.959 --> 00:54:20.000
So the energy is kinetic
energy of the relative--

00:54:20.000 --> 00:54:21.440
of the center of mass.

00:54:21.440 --> 00:54:23.600
Kinetic energy of the
relative degree of freedom.

00:54:23.600 --> 00:54:26.250
And then potential energy.

00:54:26.250 --> 00:54:29.670
Which is exactly what happens
in classical mechanics.

00:54:29.670 --> 00:54:30.170
Yeah?

00:54:30.170 --> 00:54:33.747
AUDIENCE: What's that
circle under the first term?

00:54:33.747 --> 00:54:34.580
PROFESSOR: This one?

00:54:34.580 --> 00:54:35.357
M?

00:54:35.357 --> 00:54:36.190
It's the total mass.

00:54:36.190 --> 00:54:37.380
Ma plus mb.

00:54:37.380 --> 00:54:41.160
And this one is the reduced
mass ma mbu upon ma [INAUDIBLE].

00:54:44.760 --> 00:54:46.220
OK.

00:54:46.220 --> 00:54:47.760
So here's a cool
thing that happens

00:54:47.760 --> 00:54:51.080
with multiple particles that
didn't happen previously.

00:54:51.080 --> 00:54:52.615
Suppose we have
identical particles.

00:55:00.210 --> 00:55:04.070
So in particular, imagine
I have two billiard balls.

00:55:04.070 --> 00:55:06.580
So I have two billiard balls
and I shoot-- I send one

00:55:06.580 --> 00:55:07.732
in from one side.

00:55:07.732 --> 00:55:09.690
And I send in the other
in from the other side.

00:55:09.690 --> 00:55:11.126
And then they
collide and there's

00:55:11.126 --> 00:55:12.500
some horrible
chaos that happens.

00:55:12.500 --> 00:55:15.210
And one goes flying out
to this position, a.

00:55:15.210 --> 00:55:17.865
And the other goes flying
out to this position, b.

00:55:17.865 --> 00:55:18.365
OK?

00:55:20.772 --> 00:55:21.730
Now here's my question.

00:55:21.730 --> 00:55:24.609
Which ball went to a and
which ball went to b?

00:55:24.609 --> 00:55:27.150
Well if we did this experiment,
that would be easy to answer.

00:55:27.150 --> 00:55:30.490
Because we could paint a little
1 on this one and a little 2

00:55:30.490 --> 00:55:32.680
on this one and
they'd go flying out.

00:55:32.680 --> 00:55:35.726
And then at the end, when
you catch the ball at a

00:55:35.726 --> 00:55:38.350
and you catch the ball at b, you
can grab them and look at them

00:55:38.350 --> 00:55:40.800
and say aha, this one in my
left hand which I got from a

00:55:40.800 --> 00:55:44.380
has 1 on it, and this one
has 2 on it, and I'm done.

00:55:44.380 --> 00:55:47.570
The other way we could have
observed which ball went where

00:55:47.570 --> 00:55:49.670
is we could've taken a
high speed film of this

00:55:49.670 --> 00:55:52.042
and watched frame by frame
and said aha, particle 1,

00:55:52.042 --> 00:55:53.930
particle 1, particle
1, particle 1.

00:55:53.930 --> 00:55:54.920
Particle 2, 2, 2, 2.

00:55:54.920 --> 00:55:55.420
Right?

00:55:55.420 --> 00:55:57.322
We could have just
followed the paths.

00:55:57.322 --> 00:55:59.280
And we haven't done
anything to the experiment.

00:55:59.280 --> 00:56:00.390
We just took a film.

00:56:00.390 --> 00:56:02.660
We haven't messed with it.

00:56:02.660 --> 00:56:04.740
We don't change the
results of the experiment.

00:56:04.740 --> 00:56:06.410
We just watch.

00:56:06.410 --> 00:56:06.970
Right?

00:56:06.970 --> 00:56:08.444
Perfectly doable classically.

00:56:08.444 --> 00:56:09.985
Quantum mechanically,
is this doable?

00:56:09.985 --> 00:56:11.457
AUDIENCE: No.

00:56:11.457 --> 00:56:12.040
PROFESSOR: No.

00:56:12.040 --> 00:56:15.325
Because first off, if you watch
carefully along and figure out

00:56:15.325 --> 00:56:17.700
did it go through this slit,
did it go through that slit?

00:56:17.700 --> 00:56:20.270
You know you change the results.

00:56:20.270 --> 00:56:22.800
100% white versus 50-50.

00:56:22.800 --> 00:56:26.000
If you go back to the boxes.

00:56:26.000 --> 00:56:29.600
And meanwhile, if they're
truly identical particles

00:56:29.600 --> 00:56:32.520
like electrons, there's
no way to paint anything

00:56:32.520 --> 00:56:34.410
on the damn particle.

00:56:34.410 --> 00:56:35.601
They're just electrons.

00:56:35.601 --> 00:56:37.350
And they're completely,
as far as anyone's

00:56:37.350 --> 00:56:40.160
ever been able to tell,
completely and utterly

00:56:40.160 --> 00:56:40.700
identical.

00:56:40.700 --> 00:56:43.529
They cannot be distinguished
in any way whatsoever.

00:56:43.529 --> 00:56:45.820
So you can't do the thing
where you grab the one from a

00:56:45.820 --> 00:56:48.403
and grab the one from b and say
aha, this one had the 1 on it.

00:56:48.403 --> 00:56:49.920
They're indistinguishable.

00:56:49.920 --> 00:56:50.420
Yeah.

00:56:50.420 --> 00:56:51.295
AUDIENCE: [INAUDIBLE]

00:56:56.202 --> 00:56:58.410
PROFESSOR: We're going to
come to the Pauli exclusion

00:56:58.410 --> 00:56:59.430
principle.

00:56:59.430 --> 00:57:00.110
Hold on to that.

00:57:00.110 --> 00:57:00.860
Hold on to that question.

00:57:00.860 --> 00:57:02.398
We're going to come--
we're going to get there.

00:57:02.398 --> 00:57:02.898
OK.

00:57:07.080 --> 00:57:09.780
So we can't-- if we have
truly identical particles,

00:57:09.780 --> 00:57:13.840
what we mean by that is there's
no way to run this experiment

00:57:13.840 --> 00:57:16.170
and determine which
particle ended up--

00:57:16.170 --> 00:57:19.210
which of these two particles
went to a and which went to b.

00:57:19.210 --> 00:57:20.390
Everyone cool with that?

00:57:20.390 --> 00:57:21.500
They're identical.

00:57:21.500 --> 00:57:24.280
And what I want to understand
is, what are the consequences?

00:57:24.280 --> 00:57:26.340
So the first and basic
consequence of this

00:57:26.340 --> 00:57:30.660
is that the probability that
the first particle ends up at a

00:57:30.660 --> 00:57:33.250
and the second
particle ends up at b

00:57:33.250 --> 00:57:39.590
must be equal to the probability
that the first particle ends up

00:57:39.590 --> 00:57:42.020
at b and the second
particle ends up at a.

00:57:42.020 --> 00:57:45.331
Because you can't
tell which is which.

00:57:45.331 --> 00:57:47.580
If you can't tell, it must
be that those probabilities

00:57:47.580 --> 00:57:49.144
are equal.

00:57:49.144 --> 00:57:51.060
Because if they weren't
equal, you effectively

00:57:51.060 --> 00:57:55.110
have skewed the results and
they're distinguishable.

00:57:55.110 --> 00:57:56.610
These are totally
indistinguishable.

00:57:56.610 --> 00:57:58.170
So the probability that
the first particle ends up

00:57:58.170 --> 00:58:00.430
at a. second at b, must be equal
to the probability the first

00:58:00.430 --> 00:58:02.263
particle ends up at b
and the second ends up

00:58:02.263 --> 00:58:04.660
at a, because you cannot
tell the difference.

00:58:04.660 --> 00:58:07.862
This is what it means
to be identical.

00:58:07.862 --> 00:58:09.570
That equals sign is
what I mean by saying

00:58:09.570 --> 00:58:11.585
I have identical particles.

00:58:11.585 --> 00:58:12.085
Cool?

00:58:15.110 --> 00:58:18.300
So let's find out what the
consequences of this are.

00:58:18.300 --> 00:58:19.750
Define the following operator.

00:58:19.750 --> 00:58:22.590
And Dave Larson,
if you're watching,

00:58:22.590 --> 00:58:24.540
this is the [INAUDIBLE]
[INAUDIBLE] p.

00:58:24.540 --> 00:58:27.530
So I'm going to call this
operator script-y p sub 1,2.

00:58:27.530 --> 00:58:31.140
It's the operator-- so it's got
a little hat on it, one more

00:58:31.140 --> 00:58:36.960
offense-- which takes
the first particle

00:58:36.960 --> 00:58:40.110
and the second particle
and swaps them.

00:58:40.110 --> 00:58:41.460
OK?

00:58:41.460 --> 00:58:43.550
So-- and I guess I don't
even need the 1, 2.

00:58:43.550 --> 00:58:48.210
This p operator swaps
particle 1 and particle 2.

00:58:48.210 --> 00:58:52.170
So for example, it
takes probability

00:58:52.170 --> 00:58:58.820
of a to b to
probability of b to a.

00:59:06.090 --> 00:59:10.490
But more importantly,
this swapping operation

00:59:10.490 --> 00:59:13.490
takes the wave function,
the amplitude, of a and b,

00:59:13.490 --> 00:59:17.090
and it swaps a and b, psi of ba.

00:59:21.600 --> 00:59:23.330
Now it's clear that
the probability--

00:59:23.330 --> 00:59:27.099
that the swapping operation
does nothing to the probability,

00:59:27.099 --> 00:59:29.140
because the fact that
they're identical particles

00:59:29.140 --> 00:59:30.848
means that these are
equal to each other.

00:59:30.848 --> 00:59:33.610
So it hasn't changed the answer.

00:59:33.610 --> 00:59:35.040
But just because
the probabilities

00:59:35.040 --> 00:59:36.490
are equal to each
other, does that

00:59:36.490 --> 00:59:38.510
tell you that the
wave function is

00:59:38.510 --> 00:59:40.592
invariant under
swapping the particles?

00:59:40.592 --> 00:59:41.747
AUDIENCE: No.

00:59:41.747 --> 00:59:42.330
PROFESSOR: No.

00:59:42.330 --> 00:59:43.725
It doesn't have to be invariant.

00:59:43.725 --> 00:59:46.640
The important thing is that
the norm squared of psi.

00:59:46.640 --> 00:59:50.190
So in principle, this could
be equal to some phase,

00:59:50.190 --> 00:59:54.184
e to the i of phi sub ab.

00:59:54.184 --> 00:59:55.350
Let me call it theta sub ab.

00:59:58.430 --> 01:00:00.470
Psi of a,b.

01:00:00.470 --> 01:00:03.630
And if this was the case,
that there was a phase that we

01:00:03.630 --> 01:00:05.800
got here, when we
take the norm squared,

01:00:05.800 --> 01:00:08.670
the probability
remains the same.

01:00:08.670 --> 01:00:09.170
OK?

01:00:11.910 --> 01:00:13.660
On the other hand, we
know something else.

01:00:13.660 --> 01:00:18.540
We know that if we take
the wave function psi ab.

01:00:18.540 --> 01:00:23.650
And we swap it and then we
swap it again, what do we get?

01:00:23.650 --> 01:00:25.710
I have ab.

01:00:25.710 --> 01:00:31.312
But that means this is equal
to e to the 2i theta ab.

01:00:34.760 --> 01:00:37.400
So swapping twice
had better give me 1,

01:00:37.400 --> 01:00:39.675
so this had better
be equal to 1.

01:00:39.675 --> 01:00:41.300
Let me write this
slightly differently.

01:00:41.300 --> 01:00:46.446
This is e to the i theta
squared is equal to 1.

01:00:46.446 --> 01:00:48.695
So what must be true of e
to the i theta of our phase?

01:00:51.230 --> 01:00:53.120
It's a number that squares to 1.

01:00:53.120 --> 01:00:54.860
So it could be
one of two values.

01:00:54.860 --> 01:00:57.030
It could be 1 or it
could be minus 1.

01:00:57.030 --> 01:00:59.850
That's it.

01:00:59.850 --> 01:01:02.620
So what that tells
us is-- psi ab.

01:01:05.390 --> 01:01:08.470
So that tells us
that p-- whoops.

01:01:08.470 --> 01:01:19.661
P on psi ab is equal
plus or minus psi ba.

01:01:19.661 --> 01:01:20.160
Sorry, ab.

01:01:32.900 --> 01:01:35.540
Another way to say this
is just that-- another way

01:01:35.540 --> 01:01:39.850
to say this is that p
squared acting on psi

01:01:39.850 --> 01:01:41.360
is just psi again.

01:01:41.360 --> 01:01:44.210
So the eigenvalues of p
have to be plus or minus 1.

01:01:47.020 --> 01:01:47.860
Here they are.

01:01:51.780 --> 01:01:53.955
So in fact, this
is-- let me phrase

01:01:53.955 --> 01:01:56.760
this in a little
more correct way.

01:01:56.760 --> 01:02:04.805
This tells us that
the eigenvalues of p

01:02:04.805 --> 01:02:05.680
are plus and minus 1.

01:02:16.980 --> 01:02:18.390
Yeah?

01:02:18.390 --> 01:02:22.190
AUDIENCE: I'm trying to figure
out how it can be minus 1?

01:02:22.190 --> 01:02:25.515
If p squared is psi
then it [INAUDIBLE]

01:02:25.515 --> 01:02:26.950
has to be psi [INAUDIBLE]?

01:02:26.950 --> 01:02:27.965
PROFESSOR: Yes.

01:02:27.965 --> 01:02:30.074
AUDIENCE: How could we
add that other [? p? ?]

01:02:30.074 --> 01:02:30.740
PROFESSOR: Good.

01:02:30.740 --> 01:02:31.240
OK.

01:02:31.240 --> 01:02:33.250
So let's check this quickly.

01:02:39.230 --> 01:02:42.190
So the question is
how can it be minus 1.

01:02:42.190 --> 01:02:45.212
How can that-- that doesn't--
that would seem to violate

01:02:45.212 --> 01:02:45.920
our calculations.

01:02:45.920 --> 01:02:48.010
So what this is
saying is that, we

01:02:48.010 --> 01:02:52.120
know if we take p on, let's
say, p along psi of ab.

01:02:52.120 --> 01:02:56.700
Let's say-- let psi be
an eigenfunction of p.

01:02:56.700 --> 01:02:59.350
OK?

01:02:59.350 --> 01:03:04.010
So if psi if an eigenfunction
of p, with eigenvalue minus 1,

01:03:04.010 --> 01:03:07.040
then this p on psi is
equal to minus psi.

01:03:07.040 --> 01:03:09.760
Yeah?

01:03:09.760 --> 01:03:15.240
Now the probability is
equal to psi squared.

01:03:15.240 --> 01:03:24.100
And so p on psi--
sorry, p on psi squared

01:03:24.100 --> 01:03:31.240
is equal to minus psi squared.

01:03:31.240 --> 01:03:34.349
We take each side,
it goes to minus psi.

01:03:34.349 --> 01:03:35.890
So this is just
equal to psi squared.

01:03:40.111 --> 01:03:41.050
Yeah?

01:03:41.050 --> 01:03:42.170
Who asked the question?

01:03:42.170 --> 01:03:42.800
Sorry.

01:03:42.800 --> 01:03:43.130
Right.

01:03:43.130 --> 01:03:43.629
OK.

01:03:43.629 --> 01:03:48.410
So it leaves the norm
squared invariant.

01:03:48.410 --> 01:03:51.750
So it's OK to have a minus
1 eigenvalue under p,

01:03:51.750 --> 01:03:54.310
because that doesn't change
the probability distribution.

01:03:54.310 --> 01:03:56.760
The probability distribution
is left invariant.

01:03:56.760 --> 01:04:01.280
However, if we take p
squared on psi, that's

01:04:01.280 --> 01:04:08.230
equal to p on p on psi, is equal
to the p on minus psi, which

01:04:08.230 --> 01:04:13.584
is equal to minus minus
psi, which is equal to psi.

01:04:13.584 --> 01:04:14.540
OK?

01:04:14.540 --> 01:04:17.000
So this is the statement
that the square

01:04:17.000 --> 01:04:17.875
acts as the identity.

01:04:21.060 --> 01:04:23.004
Did that answer your question?

01:04:23.004 --> 01:04:23.920
AUDIENCE: [INAUDIBLE].

01:04:23.920 --> 01:04:24.880
PROFESSOR: OK.

01:04:24.880 --> 01:04:26.320
OK.

01:04:26.320 --> 01:04:30.538
AUDIENCE: Along the [INAUDIBLE],
we need p squared on side B

01:04:30.538 --> 01:04:33.140
to go back to side
B. It would preserve

01:04:33.140 --> 01:04:36.244
the probability regardless if
we had had the two [INAUDIBLE].

01:04:36.244 --> 01:04:36.910
PROFESSOR: Good.

01:04:36.910 --> 01:04:39.960
Why did p squared have
to be the identity?

01:04:39.960 --> 01:04:41.390
Because what is p doing?

01:04:41.390 --> 01:04:43.240
p takes two particles
and it swaps them.

01:04:43.240 --> 01:04:44.340
AUDIENCE: Oh.

01:04:44.340 --> 01:04:47.180
PROFESSOR: And if it swaps
them again, what do you get?

01:04:47.180 --> 01:04:48.992
The original configuration.

01:04:48.992 --> 01:04:50.470
Right?

01:04:50.470 --> 01:04:52.840
If by swapping,
if by p, you mean

01:04:52.840 --> 01:04:55.510
the thing that swaps those
particles, then doing it twice

01:04:55.510 --> 01:04:56.980
is like not doing
anything at all.

01:04:56.980 --> 01:04:58.250
You can define a
different quantity

01:04:58.250 --> 01:05:00.510
which isn't this swapping
operation which does this

01:05:00.510 --> 01:05:02.170
twice and gives
you something else.

01:05:02.170 --> 01:05:03.410
That's perfectly reasonable.

01:05:03.410 --> 01:05:04.770
But I'm going to be interested
in the operator, which

01:05:04.770 --> 01:05:05.420
is just swap.

01:05:05.420 --> 01:05:07.410
And if you do it twice, you
get back to the identity.

01:05:07.410 --> 01:05:07.634
AUDIENCE: Oh.

01:05:07.634 --> 01:05:08.420
OK.

01:05:08.420 --> 01:05:09.346
PROFESSOR: Cool?

01:05:09.346 --> 01:05:09.846
OK.

01:05:09.846 --> 01:05:10.346
Yeah?

01:05:10.346 --> 01:05:13.790
AUDIENCE: So regarding
the defaults,

01:05:13.790 --> 01:05:17.241
is what the operation is doing
is changed particle number

01:05:17.241 --> 01:05:21.190
1, particle number
2, [INAUDIBLE]?

01:05:21.190 --> 01:05:22.210
PROFESSOR: Well--

01:05:22.210 --> 01:05:24.162
AUDIENCE: What does it
do with [INAUDIBLE]?

01:05:24.162 --> 01:05:24.870
PROFESSOR: Right.

01:05:24.870 --> 01:05:28.230
So at any-- given any
configuration, at any moment

01:05:28.230 --> 01:05:30.877
time, right, pick your wave
function, pick your state.

01:05:30.877 --> 01:05:32.210
For example, two particles here.

01:05:32.210 --> 01:05:34.765
What the script p operator
with swapping operator does,

01:05:34.765 --> 01:05:35.916
it just swaps them.

01:05:35.916 --> 01:05:39.350
So it swaps the position of one
and the position of the other.

01:05:39.350 --> 01:05:40.330
AUDIENCE: [INAUDIBLE].

01:05:40.330 --> 01:05:40.650
PROFESSOR: Right.

01:05:40.650 --> 01:05:41.110
Exactly.

01:05:41.110 --> 01:05:42.760
And that-- you do that,
at some moment in time.

01:05:42.760 --> 01:05:43.920
You do that to a state.

01:05:43.920 --> 01:05:46.330
So in that experiment,
what-- I mean,

01:05:46.330 --> 01:05:48.390
there's no answer
the question, you

01:05:48.390 --> 01:05:51.025
know, does p swap them before
or does it swap them after.

01:05:51.025 --> 01:05:51.650
It's up to you.

01:05:51.650 --> 01:05:54.740
You can apply the p
operator anytime you want.

01:05:54.740 --> 01:05:55.522
Cool?

01:05:55.522 --> 01:05:56.506
OK.

01:05:56.506 --> 01:05:57.490
Yeah?

01:05:57.490 --> 01:06:01.426
AUDIENCE: So when you
extracted the probabilities,

01:06:01.426 --> 01:06:04.870
obviously if you have a case
where it's really far apart

01:06:04.870 --> 01:06:08.314
and the two particles
end up [INAUDIBLE],

01:06:08.314 --> 01:06:10.774
aren't they more likely
that they have not

01:06:10.774 --> 01:06:13.240
moved the entire
distance in between?

01:06:13.240 --> 01:06:15.960
PROFESSOR: Yeah, that
sounds reasonable.

01:06:15.960 --> 01:06:17.904
AUDIENCE: It sounds like--

01:06:17.904 --> 01:06:18.570
PROFESSOR: Yeah.

01:06:18.570 --> 01:06:19.682
This is disconcerting.

01:06:19.682 --> 01:06:22.015
AUDIENCE: --kind of like the
wave function. [INAUDIBLE].

01:06:24.727 --> 01:06:26.060
PROFESSOR: That's exactly right.

01:06:26.060 --> 01:06:27.640
So that's an
excellent observation.

01:06:27.640 --> 01:06:28.700
Let me rephrase that slightly.

01:06:28.700 --> 01:06:30.210
So here's the observation
that she's making.

01:06:30.210 --> 01:06:31.668
It's exactly correct
and it's where

01:06:31.668 --> 01:06:33.520
we're going to get
in a few minutes.

01:06:33.520 --> 01:06:35.020
So the observation is this.

01:06:35.020 --> 01:06:38.161
Look, imagine I take two
electrons, which for the moment

01:06:38.161 --> 01:06:39.660
we'll just call
identical particles,

01:06:39.660 --> 01:06:41.000
so we take two
identical particles.

01:06:41.000 --> 01:06:42.916
Put one in my right hand,
one in my left hand.

01:06:42.916 --> 01:06:45.940
And I just hold them there
and I wait for a while.

01:06:45.940 --> 01:06:50.570
A while later, is it the same
electron in my right hand?

01:06:50.570 --> 01:06:53.070
AUDIENCE: Probably.

01:06:53.070 --> 01:06:54.160
PROFESSOR: I don't know.

01:06:54.160 --> 01:06:55.930
They're identical.

01:06:55.930 --> 01:06:57.460
I can't tell.

01:06:57.460 --> 01:06:59.000
They're completely identical.

01:06:59.000 --> 01:07:00.625
And so if you think
about this like you

01:07:00.625 --> 01:07:03.250
do some like weak scattering--
if you did some weak scattering

01:07:03.250 --> 01:07:06.050
process between these where you
pick them very far apart, very

01:07:06.050 --> 01:07:07.730
slowly moving
along-- but they're

01:07:07.730 --> 01:07:09.771
very far away so the
electrostatic interaction is

01:07:09.771 --> 01:07:13.440
very small and so they repel
each other just a little bit.

01:07:13.440 --> 01:07:15.264
If this system is,
in fact, identical

01:07:15.264 --> 01:07:17.555
and if the wave function is,
let's say for the moment--

01:07:17.555 --> 01:07:18.920
and we'll talk about
whether this is correct

01:07:18.920 --> 01:07:21.680
or not-- if the wave function
is invariant under swapping

01:07:21.680 --> 01:07:24.402
the particles-- let's just
imagine that it's invariant

01:07:24.402 --> 01:07:26.110
and they're swapping
the particles-- then

01:07:26.110 --> 01:07:27.530
there are two things
that could have happened.

01:07:27.530 --> 01:07:29.340
The particles could
have done this.

01:07:29.340 --> 01:07:32.590
Or there's also a contribution
where they do this.

01:07:32.590 --> 01:07:34.700
Which kind of hurts.

01:07:34.700 --> 01:07:37.110
And in order for the
system to be symmetric,

01:07:37.110 --> 01:07:38.610
you have to have
both contributions.

01:07:38.610 --> 01:07:39.600
So let's come to that.

01:07:39.600 --> 01:07:43.970
But indeed, it's as if there's
some additional interactions,

01:07:43.970 --> 01:07:45.370
or some additional correlations.

01:07:45.370 --> 01:07:47.078
And that's exactly
what we want to study.

01:07:47.078 --> 01:07:48.480
So let's get to that.

01:07:48.480 --> 01:07:49.890
Very good observation.

01:07:49.890 --> 01:07:52.570
So what I'd like to do
is make that precise.

01:07:52.570 --> 01:07:56.120
So there are two
kinds of particles--

01:07:56.120 --> 01:07:59.020
or three kinds of particles,
I should say-- in the world

01:07:59.020 --> 01:08:00.880
from this point of view.

01:08:00.880 --> 01:08:04.210
The first kind of particle
are distinguishable particles.

01:08:04.210 --> 01:08:06.570
Suppose I have two
particles, one with a mass m

01:08:06.570 --> 01:08:09.930
and one with a mass 2000m.

01:08:09.930 --> 01:08:12.151
Say just to pick
randomly a number.

01:08:12.151 --> 01:08:12.650
Right?

01:08:12.650 --> 01:08:15.920
Those are distinguishable
because you can weigh them.

01:08:15.920 --> 01:08:18.170
So you can tell which one
is the heavy one, which

01:08:18.170 --> 01:08:19.080
one is the light one.

01:08:19.080 --> 01:08:20.700
And you can tell.

01:08:20.700 --> 01:08:21.500
Cool?

01:08:21.500 --> 01:08:23.170
So there are
distinguishable particles.

01:08:23.170 --> 01:08:24.919
And if we have
distinguishable particles--

01:08:24.919 --> 01:08:27.439
I'll call psi sub d
for distinguishable--

01:08:27.439 --> 01:08:30.670
then it's OK to have
the following thing.

01:08:30.670 --> 01:08:32.250
Psi distinguishable
first particle

01:08:32.250 --> 01:08:34.338
is-- the amplitude for
the first particle would

01:08:34.338 --> 01:08:36.629
be an a and the amplitude
for the second particle would

01:08:36.629 --> 01:08:41.880
be a b, could be chi of a, some
function of a, and phi of b.

01:08:41.880 --> 01:08:45.579
This is not invariant
under a goes to b.

01:08:45.579 --> 01:08:47.120
Because under a goes
to b, it becomes

01:08:47.120 --> 01:08:49.139
chi of b, some
function of b, phi

01:08:49.139 --> 01:08:51.330
of a, some different
function of a.

01:08:51.330 --> 01:08:52.494
That's distinct.

01:08:52.494 --> 01:08:53.410
But it doesn't matter.

01:08:53.410 --> 01:08:54.810
They're distinguishable.

01:08:54.810 --> 01:08:56.939
So that's perfectly fine.

01:08:56.939 --> 01:09:05.040
It's not true p of ab
is not equal to p of ba.

01:09:05.040 --> 01:09:08.979
But that's OK, because
they're distinguishable.

01:09:08.979 --> 01:09:11.609
Everyone cool with that?

01:09:11.609 --> 01:09:14.144
AUDIENCE: So p of ba is i of p?

01:09:14.144 --> 01:09:14.810
PROFESSOR: Yeah.

01:09:14.810 --> 01:09:15.309
Exactly.

01:09:15.309 --> 01:09:17.090
So p of ab, this
is by definition

01:09:17.090 --> 01:09:20.544
equal to norm squared of
psi d-- and I should say d.

01:09:20.544 --> 01:09:26.450
D. Norm squared of psi
d of ab squared, which

01:09:26.450 --> 01:09:32.330
is equal to norm squared of
chi of a, phi of b squared.

01:09:32.330 --> 01:09:36.080
Whereas this guy would have
been chi of b, phi of a, norm

01:09:36.080 --> 01:09:36.600
squared.

01:09:36.600 --> 01:09:38.600
And since those are just
some stupid functions--

01:09:38.600 --> 01:09:39.529
I haven't told
you what they are,

01:09:39.529 --> 01:09:40.903
just some random
functions-- then

01:09:40.903 --> 01:09:43.096
they're just different
probability distributions.

01:09:46.439 --> 01:09:49.490
So on the other hand, if we
have indistinguishable wave

01:09:49.490 --> 01:09:56.040
functions, then psi
indistinguishable,

01:09:56.040 --> 01:10:04.895
we know that psi squared of a,b
squared is equal to psi of ba

01:10:04.895 --> 01:10:05.710
norm squared.

01:10:08.690 --> 01:10:12.690
And this is not of that form.

01:10:12.690 --> 01:10:14.990
So we have two possibilities.

01:10:14.990 --> 01:10:16.617
I'll write this
as psi plus minus.

01:10:16.617 --> 01:10:18.950
If I know one of the particles
is in the state described

01:10:18.950 --> 01:10:22.110
by chi, and the other particle
is in this state described

01:10:22.110 --> 01:10:26.500
by phi, this would be
an example of a wave

01:10:26.500 --> 01:10:28.170
function with that property.

01:10:28.170 --> 01:10:30.960
However, it's not invariant
under swapping a and b.

01:10:30.960 --> 01:10:34.141
So how could I make it invariant
under swapping a and b?

01:10:34.141 --> 01:10:35.390
Well I could do the following.

01:10:35.390 --> 01:10:39.660
1 over root 2, chi
of a, phi of b.

01:10:39.660 --> 01:10:41.980
And if I want to
make it invariant,

01:10:41.980 --> 01:10:46.050
I could add plus
chi of b, phi of a.

01:10:46.050 --> 01:10:49.412
And now if I swap a and b, here
this becomes chi of b, phi a,

01:10:49.412 --> 01:10:50.620
but that's exactly this term.

01:10:50.620 --> 01:10:52.000
And this becomes
chi of a, phi b.

01:10:52.000 --> 01:10:52.708
That's this term.

01:10:52.708 --> 01:10:54.136
So just swap them.

01:10:54.136 --> 01:10:55.642
Yeah?

01:10:55.642 --> 01:10:57.100
But we don't need
the wave function

01:10:57.100 --> 01:10:59.850
to be invariant under swapping.

01:10:59.850 --> 01:11:03.780
We just need it to be
invariant under-- up to a sine.

01:11:03.780 --> 01:11:05.870
So the other option is to
have a minus sign here.

01:11:09.700 --> 01:11:12.720
And this gives us that
the swapping operation, p,

01:11:12.720 --> 01:11:21.540
acting on psi plus minus of
a,b is equal to plus minus psi

01:11:21.540 --> 01:11:22.040
of a,b.

01:11:27.210 --> 01:11:31.190
And the plus is generally called
the symmetric, and the minus,

01:11:31.190 --> 01:11:33.270
the anti-symmetric combination.

01:11:37.540 --> 01:11:38.040
OK.

01:11:38.040 --> 01:11:40.790
So distinguishable particles can
just be in some random state,

01:11:40.790 --> 01:11:42.420
but there are
constraints on what

01:11:42.420 --> 01:11:44.620
states, what combinations
of states are allowed,

01:11:44.620 --> 01:11:46.110
for indistinguishable particles.

01:11:46.110 --> 01:11:48.710
If you can't tell the difference
between two indistinguishable

01:11:48.710 --> 01:11:51.220
particles and you know one is
in the state chi and the other

01:11:51.220 --> 01:11:54.580
in the state phi, this
cannot be the wave function.

01:11:54.580 --> 01:11:58.270
It must be either chi phi
plus chi phi in this fashion,

01:11:58.270 --> 01:11:59.146
or minus.

01:11:59.146 --> 01:12:01.740
Everyone cool with that?

01:12:01.740 --> 01:12:02.684
Yeah?

01:12:02.684 --> 01:12:07.624
AUDIENCE: I have a question
about what we mean by p of ab.

01:12:07.624 --> 01:12:10.588
So normally, when we
talk about probabilities

01:12:10.588 --> 01:12:15.231
we say that, yes, if you
measure a system what's

01:12:15.231 --> 01:12:17.480
the probability that the
metric value will equal that?

01:12:17.480 --> 01:12:18.170
PROFESSOR: Yes.

01:12:18.170 --> 01:12:20.211
AUDIENCE: But if we can't
even determine anything

01:12:20.211 --> 01:12:23.654
from that type of system, what
do we have a probability of?

01:12:23.654 --> 01:12:24.320
PROFESSOR: Good.

01:12:24.320 --> 01:12:26.110
So what this
probability means is,

01:12:26.110 --> 01:12:29.900
what's the probability that
if-- that upon observation

01:12:29.900 --> 01:12:32.760
in the system, I find
the first particle

01:12:32.760 --> 01:12:35.720
to be at a and the
second part to be at b.

01:12:35.720 --> 01:12:36.220
Right?

01:12:36.220 --> 01:12:38.026
And I can check that
by saying like, look,

01:12:38.026 --> 01:12:39.150
I catch the first particle.

01:12:39.150 --> 01:12:40.640
I catch the second particle.

01:12:40.640 --> 01:12:41.260
Is this a?

01:12:41.260 --> 01:12:41.760
No.

01:12:41.760 --> 01:12:42.260
OK.

01:12:42.260 --> 01:12:45.099
Then that gives zero to the
probability distribution.

01:12:45.099 --> 01:12:47.265
I do that a billion times
and I build up statistics.

01:12:47.265 --> 01:12:50.262
And if I'm a fourth--
one out of four times,

01:12:50.262 --> 01:12:52.345
I'll find a particle, the
first particle-- or I'll

01:12:52.345 --> 01:12:53.550
find a particle at a.

01:12:53.550 --> 01:12:55.589
One out of four times,
I'll find a partial at b.

01:12:55.589 --> 01:12:57.880
And the probability that I
find the first particle at a

01:12:57.880 --> 01:13:00.658
and the second particle
at b is one tenth, say.

01:13:00.658 --> 01:13:01.158
OK.

01:13:01.158 --> 01:13:04.010
AUDIENCE: So we can never make
that if they're identical,

01:13:04.010 --> 01:13:04.510
right?

01:13:04.510 --> 01:13:06.551
PROFESSOR: What you can't
do if they're identical

01:13:06.551 --> 01:13:08.580
is you can't say which
particle you caught at a.

01:13:08.580 --> 01:13:12.850
This is saying a particle at
a or a particle at b, right?

01:13:12.850 --> 01:13:16.380
But it's-- but whether this is
the same or not of probability

01:13:16.380 --> 01:13:19.473
that I find a particle at
b-- the first particle a b

01:13:19.473 --> 01:13:21.499
and the second particle at a.

01:13:21.499 --> 01:13:23.040
If you can't tell
the difference then

01:13:23.040 --> 01:13:25.200
they're just a particle
at a and a particle at b.

01:13:29.682 --> 01:13:32.172
OK.

01:13:32.172 --> 01:13:33.670
OK.

01:13:33.670 --> 01:13:37.010
So what does this give us?

01:13:37.010 --> 01:13:39.360
So this gives us a
couple of nice facts.

01:13:39.360 --> 01:13:48.010
So imagine-- that's
an exciting sound.

01:13:48.010 --> 01:13:50.650
So this gives us a
couple of nice facts

01:13:50.650 --> 01:13:54.840
with which we can find
awesomeness in the world.

01:13:54.840 --> 01:13:57.440
The first is the following.

01:14:00.320 --> 01:14:03.710
If you have identical
particles, then the energy

01:14:03.710 --> 01:14:05.150
can't depend on the order.

01:14:05.150 --> 01:14:07.650
If you have identical particles
and they're truly identical,

01:14:07.650 --> 01:14:09.690
you swap them, then the
energy will be the same.

01:14:09.690 --> 01:14:11.148
If it wasn't the
same, then they're

01:14:11.148 --> 01:14:13.470
distinguishable by figuring
out what the energy is.

01:14:13.470 --> 01:14:15.220
So in order that
they're identical,

01:14:15.220 --> 01:14:19.730
it must be true that if
you swap the particles,

01:14:19.730 --> 01:14:21.710
and then compute
the energy, this

01:14:21.710 --> 01:14:23.757
should be the same as
what you get if you first

01:14:23.757 --> 01:14:25.340
compute the energy
and then swap them.

01:14:28.180 --> 01:14:32.020
Which is to say that
the commutator of e

01:14:32.020 --> 01:14:36.695
with the swapping
operator, p, is 0.

01:14:36.695 --> 01:14:37.660
OK?

01:14:37.660 --> 01:14:43.967
But what that tells you is
that the expectation value of p

01:14:43.967 --> 01:14:44.925
doesn't change in time.

01:14:52.620 --> 01:14:56.110
In particular, if it's
some initial state--

01:14:56.110 --> 01:15:10.310
if you were initially p on psi
is equal to plus psi at time 0,

01:15:10.310 --> 01:15:15.200
then psi-- then p
at psi, p psi, is

01:15:15.200 --> 01:15:18.875
equal to plus psi
for all future times.

01:15:18.875 --> 01:15:23.251
P psi of p is equal
to plus psi of t.

01:15:23.251 --> 01:15:23.750
OK.

01:15:23.750 --> 01:15:25.810
So if you have two
identical particles

01:15:25.810 --> 01:15:28.460
and the wave function is
invariant under swapping them

01:15:28.460 --> 01:15:30.410
at some moment in time,
then it will always

01:15:30.410 --> 01:15:32.610
be invariant under
swapping them.

01:15:32.610 --> 01:15:35.230
It's a persistent
property of particles

01:15:35.230 --> 01:15:38.730
that the wave function is
invariant under swapping them.

01:15:38.730 --> 01:15:40.480
Yeah?

01:15:40.480 --> 01:15:41.563
Yeah.

01:15:41.563 --> 01:15:44.229
AUDIENCE: What about things that
like, become indistinguishable.

01:15:44.229 --> 01:15:47.638
For example, you have atomic
nuclei like, for uranium.

01:15:47.638 --> 01:15:50.560
And one of them is in
like a heavier isotope.

01:15:50.560 --> 01:15:52.749
And during the time
that you're like,

01:15:52.749 --> 01:15:54.332
holding them in your
hands one of them

01:15:54.332 --> 01:15:55.874
decays, now they're
the same isotope.

01:15:55.874 --> 01:15:56.540
PROFESSOR: Yeah.

01:15:56.540 --> 01:15:57.090
That's-- OK.

01:15:57.090 --> 01:15:58.200
AUDIENCE: [INAUDIBLE].

01:15:58.200 --> 01:15:59.700
PROFESSOR: This
tells you something.

01:15:59.700 --> 01:16:00.908
This is a very good question.

01:16:00.908 --> 01:16:02.830
So the question is,
suppose I have an excited

01:16:02.830 --> 01:16:04.620
isotope of uranium
that' s distinguishable

01:16:04.620 --> 01:16:07.450
from some other
isotope of uranium.

01:16:07.450 --> 01:16:10.230
I wait for a while
and then this thing

01:16:10.230 --> 01:16:12.050
decays down to the
state-- it won't

01:16:12.050 --> 01:16:13.290
decay to the stabilized
isotope of uranium,

01:16:13.290 --> 01:16:15.081
but whatever-- it decays
down to-- we could

01:16:15.081 --> 01:16:16.990
imagine a universe
in which it did.

01:16:16.990 --> 01:16:19.400
It decays down to
a stable state.

01:16:19.400 --> 01:16:21.390
And-- I mean,
uranium's never stable.

01:16:21.390 --> 01:16:23.880
But anyway, you get the idea.

01:16:23.880 --> 01:16:28.059
At which point they're
indistinguishable.

01:16:28.059 --> 01:16:28.850
This sounds better.

01:16:28.850 --> 01:16:32.370
Because now the wave
function should be invariant.

01:16:32.370 --> 01:16:34.350
But it started out
not being invariant.

01:16:34.350 --> 01:16:37.950
What's the problem
in this argument?

01:16:37.950 --> 01:16:39.000
The system has changed.

01:16:39.000 --> 01:16:40.957
In particular, something
went flying out.

01:16:40.957 --> 01:16:42.540
So this is actually
kind of a nice way

01:16:42.540 --> 01:16:45.010
to argue that there must
have been something else.

01:16:45.010 --> 01:16:47.180
The wave function
describes a full system.

01:16:47.180 --> 01:16:51.421
But if something leaves,
then it's not the same system

01:16:51.421 --> 01:16:52.920
anymore and the
wave functions isn't

01:16:52.920 --> 01:16:55.690
describing the same
degrees of freedom.

01:16:55.690 --> 01:16:56.315
Something left.

01:16:56.315 --> 01:16:58.315
AUDIENCE: But suppose we
keep track of that one,

01:16:58.315 --> 01:17:00.139
we still can't swap
the two uraniums.

01:17:00.139 --> 01:17:00.930
PROFESSOR: Exactly.

01:17:00.930 --> 01:17:02.760
Then-- well, then there's some
additional constraint, right?

01:17:02.760 --> 01:17:04.320
It must be invariant
under swapping that.

01:17:04.320 --> 01:17:05.986
But it must-- but the
wave function also

01:17:05.986 --> 01:17:08.039
knows about that extra
bit that went flying off.

01:17:08.039 --> 01:17:09.455
And so the whole
wave function has

01:17:09.455 --> 01:17:11.750
to be invariant under
swapping the identical parts,

01:17:11.750 --> 01:17:15.950
but not invariant under
swap-- the invariance is not

01:17:15.950 --> 01:17:17.450
just those two things.

01:17:17.450 --> 01:17:20.434
They're correlated with that
thing that went flying away.

01:17:20.434 --> 01:17:21.850
So another way to
think about this

01:17:21.850 --> 01:17:24.540
is imagine two different
hydrogen atoms.

01:17:24.540 --> 01:17:25.850
Here I've got a hydrogen atom.

01:17:25.850 --> 01:17:27.130
It's an electron and a proton.

01:17:27.130 --> 01:17:29.110
Here's a deuterium atom.

01:17:29.110 --> 01:17:32.550
It's an electron bound to a
proton and a neutron glued

01:17:32.550 --> 01:17:34.400
together, deuteron.

01:17:34.400 --> 01:17:40.270
So are the electrons identical?

01:17:40.270 --> 01:17:42.410
Yeah, they're totally identical.

01:17:42.410 --> 01:17:43.910
So is the wave
function invariant--

01:17:43.910 --> 01:17:45.680
does the wave function-- or
the probability distribution

01:17:45.680 --> 01:17:47.832
have to be invariant under
swapping the electrons?

01:17:47.832 --> 01:17:48.790
Yes, they're identical.

01:17:48.790 --> 01:17:50.000
So the probability
distribution must

01:17:50.000 --> 01:17:51.610
be invariant under
swapping the electrons.

01:17:51.610 --> 01:17:53.870
However, is an identical
under swapping the hydrogen

01:17:53.870 --> 01:17:55.129
with the deuterium?

01:17:55.129 --> 01:17:55.670
AUDIENCE: No.

01:17:55.670 --> 01:17:56.600
PROFESSOR: No.

01:17:56.600 --> 01:17:59.990
So it's invariant under swapping
the identical parts and not

01:17:59.990 --> 01:18:01.650
the non-identical parts.

01:18:01.650 --> 01:18:02.160
Cool?

01:18:02.160 --> 01:18:02.785
AUDIENCE: Yeah.

01:18:02.785 --> 01:18:03.970
PROFESSOR: OK.

01:18:03.970 --> 01:18:09.870
So this tells us that there
are two kinds of particles.

01:18:09.870 --> 01:18:12.060
There are persistent--
or sorry, there

01:18:12.060 --> 01:18:13.450
are three kinds of particles.

01:18:13.450 --> 01:18:15.640
And these properties
are persistent.

01:18:15.640 --> 01:18:18.171
The first kind of particle--
sets of particles--

01:18:18.171 --> 01:18:19.420
are distinguishable particles.

01:18:23.977 --> 01:18:26.310
If you have two particles
which are distinguishable then

01:18:26.310 --> 01:18:26.810
you're done.

01:18:26.810 --> 01:18:27.809
They're distinguishable.

01:18:27.809 --> 01:18:28.650
Nothing else to say.

01:18:28.650 --> 01:18:34.140
Two, you can have identical
particles with the property

01:18:34.140 --> 01:18:39.750
that if you take p on psi
a,b-- sorry, p on psi.

01:18:39.750 --> 01:18:43.450
If you swap the particles,
this is equal to plus psi.

01:18:43.450 --> 01:18:47.840
And then you have-- so
identical with plus.

01:18:47.840 --> 01:18:51.230
And you have three
identical particles

01:18:51.230 --> 01:18:55.280
where if you swap the particles,
you get a minus sign on psi.

01:18:58.250 --> 01:18:59.820
These particles are
called-- we have

01:18:59.820 --> 01:19:01.236
a name for particles
of this kind.

01:19:01.236 --> 01:19:02.907
Bosons.

01:19:02.907 --> 01:19:03.990
These are called fermions.

01:19:11.760 --> 01:19:14.990
My TA did a bad thing to me when
I was taking quantum mechanics.

01:19:14.990 --> 01:19:17.110
And said, just imagine
them as little tiny Fermis.

01:19:17.110 --> 01:19:18.943
So just take a picture
of Fermi in your head

01:19:18.943 --> 01:19:21.200
and imagine little
tiny-- and this is cruel,

01:19:21.200 --> 01:19:22.200
because I can't help it.

01:19:22.200 --> 01:19:24.090
Every time someone in a seminar
is like, blah, blah, blah,

01:19:24.090 --> 01:19:24.360
Fermi.

01:19:24.360 --> 01:19:24.970
And I'm like, damn it.

01:19:24.970 --> 01:19:25.511
Little Fermi.

01:19:28.890 --> 01:19:30.880
It's really quite annoying.

01:19:30.880 --> 01:19:32.090
So now you have it too.

01:19:32.090 --> 01:19:34.390
Great.

01:19:34.390 --> 01:19:37.735
So what are the consequences
of the fact that there

01:19:37.735 --> 01:19:40.160
are two kinds of
particles in the universe?

01:19:40.160 --> 01:19:44.390
These fermions and bosons?

01:19:44.390 --> 01:19:46.000
This has a really
lovely consequence.

01:19:46.000 --> 01:19:49.710
The first is, suppose
we have two fermions.

01:19:49.710 --> 01:19:50.280
OK?

01:19:50.280 --> 01:19:51.800
Examples of fermions
are electrons.

01:19:51.800 --> 01:19:55.330
Suppose we have a wave
function for two fermions.

01:19:55.330 --> 01:19:58.000
The first might-- what's
that probability amplitude

01:19:58.000 --> 01:20:00.180
that the first is at a
and the second is t b?

01:20:00.180 --> 01:20:03.789
Well it's this psi of a,b.

01:20:03.789 --> 01:20:05.330
And the statement
that it's a fermion

01:20:05.330 --> 01:20:09.310
is the statement that this is
equal to minus psi of b, a.

01:20:09.310 --> 01:20:12.220
If we swap the positions
of the two particles,

01:20:12.220 --> 01:20:14.300
we must pick up a minus sign.

01:20:14.300 --> 01:20:17.250
This tells us in particular
that the probability amplitude

01:20:17.250 --> 01:20:19.625
for the first particle to be
at a and the second particle

01:20:19.625 --> 01:20:22.700
to be at a is equal
to minus itself.

01:20:22.700 --> 01:20:24.200
Because upon swapping
the particles,

01:20:24.200 --> 01:20:27.190
we get minus psi of a,a.

01:20:27.190 --> 01:20:29.820
So the probability amplitude
to find two fermions

01:20:29.820 --> 01:20:33.390
at the same place is equal to 0.

01:20:33.390 --> 01:20:38.046
Two fermions cannot
occupy the same state.

01:20:38.046 --> 01:20:39.670
This was the Pauli
exclusion principle,

01:20:39.670 --> 01:20:41.420
which we needed to get
the periodic table.

01:20:44.800 --> 01:20:47.620
Pauli.

01:20:47.620 --> 01:20:49.500
Two.

01:20:49.500 --> 01:20:53.400
If we have-- so this is
fermions-- if we have bosons,

01:20:53.400 --> 01:20:57.590
psi of a,b-- let
me write this out.

01:20:57.590 --> 01:21:02.180
So psi fermion or
boson-- so fermion

01:21:02.180 --> 01:21:03.862
is going to be with
a minus and boson

01:21:03.862 --> 01:21:05.570
is going to be with
a plus-- is equal to,

01:21:05.570 --> 01:21:08.260
suppose I have two
particles 1 over root 2.

01:21:08.260 --> 01:21:10.550
And one particle
is in the state chi

01:21:10.550 --> 01:21:12.654
and the other particle
is in the state phi.

01:21:12.654 --> 01:21:15.070
But in order to be fermionic
or bosonic, in order for this

01:21:15.070 --> 01:21:16.695
to be invariant under
swapping a and b,

01:21:16.695 --> 01:21:24.150
we have to have a plus or
minus chi of b, phi of a.

01:21:24.150 --> 01:21:26.594
And here we immediately see
this Pauli principle at work.

01:21:26.594 --> 01:21:28.760
If I could take the fermionic
example with the minus

01:21:28.760 --> 01:21:34.370
sign, then psi evaluated at a,
a must be chi at a, phi at b,

01:21:34.370 --> 01:21:36.770
minus chi at b, phi at a.

01:21:36.770 --> 01:21:40.600
But if a is b, this is chi
a, phi a, minus chi a, phi a.

01:21:40.600 --> 01:21:42.950
That's zero.

01:21:42.950 --> 01:21:44.624
But let's think about
the bosonic case.

01:21:44.624 --> 01:21:46.790
If we have a bosonic field--
or, if we have-- sorry.

01:21:46.790 --> 01:21:50.260
If we have bosonic
identical particles, then

01:21:50.260 --> 01:21:55.540
psi b at a with a-- for
our fermion, it was zero.

01:21:55.540 --> 01:22:01.410
But psi b of the amplitude to
be at two at the same place

01:22:01.410 --> 01:22:04.500
is equal to-- well, if b
is a, then these two terms

01:22:04.500 --> 01:22:05.930
are identical and
we have a plus.

01:22:05.930 --> 01:22:10.380
This is root 2 chi
at a, phi at a.

01:22:10.380 --> 01:22:11.880
Which is greater
than what you might

01:22:11.880 --> 01:22:13.255
have naively
guessed, which would

01:22:13.255 --> 01:22:15.410
have been just chi a, phi a.

01:22:15.410 --> 01:22:18.660
For bosons, they really like
being next to each other.

01:22:18.660 --> 01:22:20.800
They really like being
in the same place.

01:22:20.800 --> 01:22:23.990
And this will eventually
lead to lasers.

01:22:23.990 --> 01:22:26.540
So from this simple
statistical property

01:22:26.540 --> 01:22:28.440
under swapping, picking
up a minus sign,

01:22:28.440 --> 01:22:29.814
we get the Pauli
principle, which

01:22:29.814 --> 01:22:30.970
gave us the periodic table.

01:22:30.970 --> 01:22:33.670
And is going to give us
in the next lecture bands

01:22:33.670 --> 01:22:35.910
and solids in conductivity.

01:22:35.910 --> 01:22:39.670
From the same principle but
with a plus, and the persistence

01:22:39.670 --> 01:22:43.300
of this sine, from the
persistence of the statistics,

01:22:43.300 --> 01:22:46.040
from the fact that we
have two identical bosons,

01:22:46.040 --> 01:22:47.930
we get that they like
to be in the same spot.

01:22:47.930 --> 01:22:50.361
And we'll get lasers and
boson [INAUDIBLE] condensates.

01:22:50.361 --> 01:22:52.110
And next time, we'll
pick up with fermions

01:22:52.110 --> 01:22:53.750
in a periodic
potential and we'll

01:22:53.750 --> 01:22:56.900
study solids and get to diamond.