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PROFESSOR: So today we're
going to do our last lecture
00:00:26.690 --> 00:00:30.089
on scattering in 1D
quantum mechanics,
00:00:30.089 --> 00:00:32.130
and we're going to introduce
some powerful ideas,
00:00:32.130 --> 00:00:34.129
in particular, the phase
shift and the S matrix,
00:00:34.129 --> 00:00:35.270
and they're cool.
00:00:35.270 --> 00:00:36.570
We'll use them for good.
00:00:36.570 --> 00:00:40.660
Before we get started,
questions from last time?
00:00:43.424 --> 00:00:45.340
AUDIENCE: What was the
music that was playing?
00:00:45.340 --> 00:00:46.180
PROFESSOR: Just now?
00:00:46.180 --> 00:00:48.820
It's a band called
Saint Germain.
00:00:48.820 --> 00:00:52.280
It's actually a guy,
but he refers to himself
00:00:52.280 --> 00:00:53.720
as a band called Saint Germain.
00:00:53.720 --> 00:00:59.200
Anyway, it's from an album
I think called "Traveler."
00:00:59.200 --> 00:01:02.480
Physics questions?
00:01:02.480 --> 00:01:04.530
Anyone?
00:01:04.530 --> 00:01:05.817
OK, good.
00:01:05.817 --> 00:01:06.650
Well, bad, actually.
00:01:06.650 --> 00:01:08.316
I'd much prefer it
if you had questions,
00:01:08.316 --> 00:01:11.296
but I'll take that as a sign
of knowledge, competence,
00:01:11.296 --> 00:01:11.795
and mastery.
00:01:16.680 --> 00:01:19.250
So last time, we
talked about scattering
00:01:19.250 --> 00:01:21.870
past a barrier
with some height L
00:01:21.870 --> 00:01:24.777
and some height V, which
I think we called V0,
00:01:24.777 --> 00:01:26.860
and we observed a bunch
of nice things about this.
00:01:26.860 --> 00:01:29.570
First off, we computed the
probability of transmission
00:01:29.570 --> 00:01:34.400
across this barrier as a
function of the energy,
00:01:34.400 --> 00:01:35.900
and it had a bunch
of nice features.
00:01:35.900 --> 00:01:39.160
First off, it asymptoted
to 1 at high energies,
00:01:39.160 --> 00:01:39.910
which makes sense.
00:01:39.910 --> 00:01:43.700
Things shouldn't really care if
you have a little tiny barrier.
00:01:43.700 --> 00:01:45.215
It went to 0 at 0 energy.
00:01:47.894 --> 00:01:49.560
I think we should all
find that obvious.
00:01:49.560 --> 00:01:50.430
If you have
extremely low energy,
00:01:50.430 --> 00:01:52.720
you're just going to bounce
off this very hard wall.
00:01:52.720 --> 00:01:55.200
In between, though,
there's some structure.
00:01:55.200 --> 00:01:57.950
In particular, at certain
values of the energy
00:01:57.950 --> 00:02:00.540
corresponding to certain
values of the wave
00:02:00.540 --> 00:02:04.280
number in the barrier region,
at certain values of the energy,
00:02:04.280 --> 00:02:06.841
we saw that the
transmission was perfect.
00:02:06.841 --> 00:02:07.340
Oops.
00:02:07.340 --> 00:02:08.464
This should have been here.
00:02:08.464 --> 00:02:09.740
Sorry.
00:02:09.740 --> 00:02:18.220
The transition was perfect at
special values of the energy,
00:02:18.220 --> 00:02:21.980
and that the reflection
at those points was 0.
00:02:21.980 --> 00:02:25.260
You transmitted perfectly.
00:02:25.260 --> 00:02:27.100
But meanwhile, the
reflection, which
00:02:27.100 --> 00:02:30.360
is 1 minus the transmission,
hit maxima at special points,
00:02:30.360 --> 00:02:31.740
and those special
points turn out
00:02:31.740 --> 00:02:37.440
to be half integer shifted
away from the special points
00:02:37.440 --> 00:02:38.875
for perfect transmission.
00:02:38.875 --> 00:02:41.000
So at certain points, we
have perfect transmission.
00:02:41.000 --> 00:02:45.049
At certain points, we have
extremely efficient reflection.
00:02:45.049 --> 00:02:46.590
One of our goals
today is going to be
00:02:46.590 --> 00:02:50.430
to understand this physics,
the physics of resonance
00:02:50.430 --> 00:02:53.667
in scattering off
of a potential.
00:02:53.667 --> 00:02:55.000
So the asymptotes we understand.
00:02:55.000 --> 00:02:56.010
Classically, that makes sense.
00:02:56.010 --> 00:02:57.259
Classically, that makes sense.
00:02:57.259 --> 00:03:00.732
But classically, this is a
really weird structure to see.
00:03:00.732 --> 00:03:02.690
And notice that it's also
not something that we
00:03:02.690 --> 00:03:05.970
saw when we looked at
scattering off of a simple step.
00:03:05.970 --> 00:03:08.130
When we looked at a
simple step, what we got
00:03:08.130 --> 00:03:13.582
was something that looked
like 0 and then this.
00:03:13.582 --> 00:03:14.540
There was no structure.
00:03:14.540 --> 00:03:16.740
It was just a
nice, simple curve.
00:03:16.740 --> 00:03:18.680
So something is
happening when we
00:03:18.680 --> 00:03:22.352
have a barrier as
opposed to a step.
00:03:22.352 --> 00:03:24.060
So our job is going
to be, in some sense,
00:03:24.060 --> 00:03:26.590
to answer why are
they so different.
00:03:26.590 --> 00:03:32.060
Before we get going, questions
about the step barrier.
00:03:32.060 --> 00:03:32.680
Yeah?
00:03:32.680 --> 00:03:36.408
AUDIENCE: That little line that
you drew over your other graph,
00:03:36.408 --> 00:03:37.810
is that to scale?
00:03:37.810 --> 00:03:39.310
PROFESSOR: Oh, sorry.
00:03:39.310 --> 00:03:40.212
This one?
00:03:40.212 --> 00:03:43.570
AUDIENCE: No, the little line
that you drew right there.
00:03:43.570 --> 00:03:44.530
PROFESSOR: This?
00:03:44.530 --> 00:03:46.030
Sorry, I should
drawn it separately.
00:03:46.030 --> 00:03:49.570
That was the transmission
as a function of energy
00:03:49.570 --> 00:03:54.310
across a single step, and
so that looked like this.
00:03:54.310 --> 00:03:55.937
AUDIENCE: So if we
were to overlap that
00:03:55.937 --> 00:03:57.260
onto our resonance one--
00:03:57.260 --> 00:03:58.051
PROFESSOR: Exactly.
00:03:58.051 --> 00:04:00.150
AUDIENCE: Would
it [INAUDIBLE] to?
00:04:00.150 --> 00:04:01.600
PROFESSOR: Oh, sorry.
00:04:01.600 --> 00:04:03.590
So I just meant to
compare the two.
00:04:03.590 --> 00:04:05.923
I just wanted to think of
them as two different systems.
00:04:05.923 --> 00:04:08.820
In this system, the transmission
curve is a nice, simple curve.
00:04:08.820 --> 00:04:09.920
It has no structure.
00:04:09.920 --> 00:04:13.120
In the case of a step barrier,
we get non-zero transmission
00:04:13.120 --> 00:04:15.370
at low energies
instead of going to 0
00:04:15.370 --> 00:04:17.431
and we get this
resonant structure.
00:04:17.431 --> 00:04:18.514
So it's just for contrast.
00:04:22.540 --> 00:04:24.460
Before going into
that in detail,
00:04:24.460 --> 00:04:27.820
I want to do one slight
variation of this problem.
00:04:27.820 --> 00:04:29.680
I'm not going to do any
of the computations.
00:04:29.680 --> 00:04:31.221
I'm just going to
tell you how to get
00:04:31.221 --> 00:04:33.600
the answer from the answers
we've already computed.
00:04:33.600 --> 00:04:36.160
So consider a square well.
00:04:36.160 --> 00:04:38.015
You guys have solved
the finite square well
00:04:38.015 --> 00:04:39.104
on your problem sets.
00:04:39.104 --> 00:04:41.020
Consider a finite square
well again of width L
00:04:41.020 --> 00:04:42.990
and of depth now minus V0.
00:04:42.990 --> 00:04:46.030
It's the same thing
with V goes to minus V.
00:04:46.030 --> 00:04:48.030
And again, I want to
consider scattering states.
00:04:48.030 --> 00:04:50.480
So I want to consider states
with positive energy, energy
00:04:50.480 --> 00:04:51.950
above the asymptotic potential.
00:04:54.530 --> 00:04:57.270
Now, at this point, by now we
all know how to solve this.
00:04:57.270 --> 00:04:59.755
We write plain waves
here, plain waves here,
00:04:59.755 --> 00:05:00.880
and plain waves in between.
00:05:00.880 --> 00:05:03.750
We solve the potential in each
region where it's constant,
00:05:03.750 --> 00:05:06.392
and then we impose continuity
of the wave function
00:05:06.392 --> 00:05:08.600
and continuity of the
derivative of the wave function
00:05:08.600 --> 00:05:10.080
at the matching points.
00:05:10.080 --> 00:05:11.580
So we know how to
solve this problem
00:05:11.580 --> 00:05:13.890
and deduce the transmission
reflection coefficients.
00:05:13.890 --> 00:05:15.890
We've done it for this
problem, and it's exactly
00:05:15.890 --> 00:05:16.760
the same algebra.
00:05:16.760 --> 00:05:18.712
And in fact, it's so
exactly the same algebra
00:05:18.712 --> 00:05:20.670
that we can just take
the results from this one
00:05:20.670 --> 00:05:23.840
and take V to minus V and
we'll get the right answer.
00:05:23.840 --> 00:05:24.970
You kind of have to.
00:05:24.970 --> 00:05:32.360
So if we do, what we get for
the transition probability--
00:05:32.360 --> 00:05:35.920
and now this is the transmission
probability for a square well--
00:05:35.920 --> 00:05:39.650
and again, I'm going to use the
same dimensionless constants.
00:05:39.650 --> 00:05:45.530
g0 squared is equal to 2mL
squared over h bar squared
00:05:45.530 --> 00:05:46.660
times V0.
00:05:46.660 --> 00:05:48.180
This is 1 over an energy.
00:05:48.180 --> 00:05:48.930
This is an energy.
00:05:48.930 --> 00:05:49.960
This is dimensionless.
00:05:49.960 --> 00:05:55.390
And the dimensionless energy
epsilon is equal to E over V.
00:05:55.390 --> 00:05:57.580
To express my
transmission probability,
00:05:57.580 --> 00:05:59.940
life is better when
it's dimensionless.
00:05:59.940 --> 00:06:02.140
So T, the transmission
probability,
00:06:02.140 --> 00:06:04.190
again, it's one of
these horrible 1 overs.
00:06:04.190 --> 00:06:13.780
1 over 1 plus 1 over 4
epsilon, epsilon plus 1,
00:06:13.780 --> 00:06:18.240
sine squared of g0, square
root of epsilon plus 1.
00:06:24.840 --> 00:06:27.580
This is again the same.
00:06:27.580 --> 00:06:30.081
You can see what we got last
time by now taking V to minus V
00:06:30.081 --> 00:06:30.580
again.
00:06:30.580 --> 00:06:32.160
That takes epsilon
to minus epsilon.
00:06:32.160 --> 00:06:36.360
So we get epsilon, 1 minus
epsilon, or epsilon minus 1
00:06:36.360 --> 00:06:38.894
picking up the minus
sign from this epsilon,
00:06:38.894 --> 00:06:40.810
and here we get a 1 minus
epsilon instead of 1
00:06:40.810 --> 00:06:41.450
plus epsilon.
00:06:41.450 --> 00:06:43.390
That's precisely what
we got last time.
00:06:43.390 --> 00:06:46.609
So if you're feeling punchy
at home tonight, check this.
00:06:46.609 --> 00:06:48.400
In some sense, you're
going to re-derive it
00:06:48.400 --> 00:06:49.486
on the problem set.
00:06:52.690 --> 00:06:55.290
So it looks basically
the same as before.
00:06:55.290 --> 00:06:57.480
We have a sine
function downstairs,
00:06:57.480 --> 00:07:00.011
which again will sometimes be 0.
00:07:00.011 --> 00:07:01.510
The sine will
occasionally be 0 when
00:07:01.510 --> 00:07:03.250
its argument is
a multiple of pi.
00:07:03.250 --> 00:07:05.290
And when that sine
function is 0,
00:07:05.290 --> 00:07:07.940
then the transmission
probability is 1 over 1 plus 0.
00:07:07.940 --> 00:07:09.880
It's also known as 1.
00:07:09.880 --> 00:07:10.900
Transmission is perfect.
00:07:10.900 --> 00:07:12.480
So we again get a
resonant structure,
00:07:12.480 --> 00:07:14.900
and it's in fact exactly the
same plot, or almost exactly
00:07:14.900 --> 00:07:17.100
the same plot.
00:07:17.100 --> 00:07:19.885
I'm going to plot
transmission-- let's go ahead
00:07:19.885 --> 00:07:22.090
and do this--
transmission probability
00:07:22.090 --> 00:07:25.870
as a function, again, of
the dimensionless energy.
00:07:25.870 --> 00:07:37.450
And what we get is, again,
not very well drawn resonances
00:07:37.450 --> 00:07:39.720
where transmission
goes to 1 thanks
00:07:39.720 --> 00:07:43.030
to my beautiful artistic skills.
00:07:43.030 --> 00:07:46.690
Points where the reflection
hits a local maximum.
00:07:50.470 --> 00:07:52.540
But we know one more
thing about this system,
00:07:52.540 --> 00:07:55.930
which is that, in addition to
having scattering states whose
00:07:55.930 --> 00:07:58.160
transition probability are
indicated by this plot,
00:07:58.160 --> 00:08:02.480
we know we also have, for
negative energy, bound states.
00:08:02.480 --> 00:08:04.910
Unlike the step barrier,
for the step well,
00:08:04.910 --> 00:08:06.430
we also have bound states.
00:08:06.430 --> 00:08:07.960
So here's the
transmission curve,
00:08:07.960 --> 00:08:10.660
but I just want to remind
you that we have energies.
00:08:10.660 --> 00:08:14.650
At special values of energies,
we also have bound states.
00:08:14.650 --> 00:08:17.190
And precisely what
energies depends
00:08:17.190 --> 00:08:19.260
on the structure of
the well and the depth,
00:08:19.260 --> 00:08:24.240
but if the depth of the well
is, say, V0 so this is minus 1,
00:08:24.240 --> 00:08:27.230
we know that the lowest
bound state is always
00:08:27.230 --> 00:08:29.770
greater energy than
the bottom of the well.
00:08:29.770 --> 00:08:31.320
Cool?
00:08:31.320 --> 00:08:33.669
So epsilon, remember,
is E over V0,
00:08:33.669 --> 00:08:36.289
and in the square
well of depth V0,
00:08:36.289 --> 00:08:38.830
the lowest bound state cannot
possibly have energy lower than
00:08:38.830 --> 00:08:41.299
the bottom of the well, so
its epsilon must have epsilon
00:08:41.299 --> 00:08:44.864
greater than minus 1.
00:08:44.864 --> 00:08:46.280
Anyway, this is
just to remind you
00:08:46.280 --> 00:08:49.360
that there are bound states.
00:08:49.360 --> 00:08:51.030
Think of this like
a gun put down
00:08:51.030 --> 00:08:52.920
on a table in a play in Act One.
00:08:56.600 --> 00:08:58.744
It's that dramatic.
00:08:58.744 --> 00:08:59.660
It will show up again.
00:08:59.660 --> 00:09:03.677
It will come back and
play well with us.
00:09:03.677 --> 00:09:05.260
I want to talk about
these resonances.
00:09:05.260 --> 00:09:07.280
Let's understand
why they're there.
00:09:07.280 --> 00:09:09.030
There are a bunch of
ways of understanding
00:09:09.030 --> 00:09:11.770
why these resonances
are present, and let
00:09:11.770 --> 00:09:15.040
me just give you a couple.
00:09:15.040 --> 00:09:16.790
First, a heuristic
picture, and then I
00:09:16.790 --> 00:09:19.560
want to give you a very precise
computational picture of why
00:09:19.560 --> 00:09:21.300
these resonances are happening.
00:09:21.300 --> 00:09:23.600
So the first is imagine
we have a state where
00:09:23.600 --> 00:09:24.960
the transmission is perfect.
00:09:24.960 --> 00:09:26.980
What that tells you is
KL is a multiple of pi.
00:09:29.520 --> 00:09:32.060
If this is the
distance L of the well,
00:09:32.060 --> 00:09:35.240
in here the wave is
exactly one period
00:09:35.240 --> 00:09:38.960
if its KL is equal
to a multiple of pi.
00:09:47.150 --> 00:09:49.810
Let me simplify my life and
consider the case KL is 2 pi.
00:10:01.550 --> 00:10:02.710
Sorry.
00:10:02.710 --> 00:10:04.300
4 pi.
00:10:04.300 --> 00:10:05.950
I switched notations for you.
00:10:05.950 --> 00:10:10.120
I was using in my head, rather
in my notes, width of the well
00:10:10.120 --> 00:10:16.070
is 2L for reasons
you'll see on the notes
00:10:16.070 --> 00:10:18.380
if you look at the notes,
but let's ignore that.
00:10:18.380 --> 00:10:19.820
Let's focus on these guys.
00:10:19.820 --> 00:10:24.280
So consider the state,
configuration in energy,
00:10:24.280 --> 00:10:28.020
such that in the well, we have
exactly one period of the wave
00:10:28.020 --> 00:10:28.740
function.
00:10:28.740 --> 00:10:31.780
That means that the value of the
wave function at the two ends
00:10:31.780 --> 00:10:35.400
is the same and the
slope is the same.
00:10:35.400 --> 00:10:39.420
So whatever the
energy is out here,
00:10:39.420 --> 00:10:41.820
if it matches smoothly
and continuously
00:10:41.820 --> 00:10:44.485
and its derivative is continuous
here, it will match smoothly
00:10:44.485 --> 00:10:49.480
and continuously out here as
well with the same amplitude
00:10:49.480 --> 00:10:53.954
and the same period
inside and outside.
00:10:53.954 --> 00:10:56.120
The amplitude is the same
and the phase is the same.
00:10:56.120 --> 00:10:58.161
That means this wave must
have the same amplitude
00:10:58.161 --> 00:10:59.700
and the same period.
00:10:59.700 --> 00:11:01.210
It has to have the same period
because it's the same energy,
00:11:01.210 --> 00:11:03.460
but it must have the same
amplitude and the same slope
00:11:03.460 --> 00:11:05.100
at that point.
00:11:05.100 --> 00:11:07.450
And that means
that this wave has
00:11:07.450 --> 00:11:09.280
the same amplitude as this wave.
00:11:09.280 --> 00:11:11.449
The norm squared is the
transmission probability,
00:11:11.449 --> 00:11:13.990
the norm squared of this divided
by the norm squared of this.
00:11:13.990 --> 00:11:16.430
That means the transmission
probability has to be 1.
00:11:16.430 --> 00:11:18.730
What would have
happened had the system,
00:11:18.730 --> 00:11:21.899
instead of being perfectly
periodic inside the well--
00:11:21.899 --> 00:11:23.190
actually, let me leave that up.
00:11:28.207 --> 00:11:29.040
Let's do it as this.
00:11:36.501 --> 00:11:37.000
Shoot.
00:11:37.000 --> 00:11:39.208
I'm even getting my qualitative
wave functions wrong.
00:11:39.208 --> 00:11:40.970
Let's try this again.
00:11:40.970 --> 00:11:45.710
Start at the top,
go down, and then
00:11:45.710 --> 00:11:49.367
its deeper inside the well
so the amplitude out here,
00:11:49.367 --> 00:11:51.950
the difference in energy between
the energy and the potential,
00:11:51.950 --> 00:11:54.390
is less, which means
the period is longer
00:11:54.390 --> 00:11:56.176
and the amplitude is greater.
00:11:56.176 --> 00:11:58.300
The way I've drawn it, it's
got a particular value.
00:11:58.300 --> 00:12:07.510
It's got zero derivative
at this point,
00:12:07.510 --> 00:12:15.547
so it's got to-- so
there's our wave function.
00:12:15.547 --> 00:12:16.380
Same thing out here.
00:12:20.162 --> 00:12:21.870
The important thing
is the amplitude here
00:12:21.870 --> 00:12:23.328
has to be the same
as the amplitude
00:12:23.328 --> 00:12:26.080
here because the amplitude and
the amplitude and the phase
00:12:26.080 --> 00:12:29.760
were exactly as if there had
been no intervening region.
00:12:29.760 --> 00:12:31.670
Everyone agree with that?
00:12:31.670 --> 00:12:35.280
By contrast, if we had
looked at a situation where
00:12:35.280 --> 00:12:42.080
inside the well, it was not the
same amplitude, so for example,
00:12:42.080 --> 00:12:44.780
something like this, came
up with some slope which
00:12:44.780 --> 00:12:46.750
is different and a value
which is different,
00:12:46.750 --> 00:12:49.820
then this is going to match onto
something with the same period
00:12:49.820 --> 00:12:53.165
but with a different amplitude
than it would have over here.
00:12:56.015 --> 00:12:57.890
Same period because it's
got the same energy,
00:12:57.890 --> 00:12:59.680
but a different
amplitude because it
00:12:59.680 --> 00:13:03.810
has to match on with the
value and the derivative.
00:13:03.810 --> 00:13:06.850
So you can think through
this and pretty quickly
00:13:06.850 --> 00:13:10.790
convince yourself of the
necessity of the transmission
00:13:10.790 --> 00:13:14.050
aptitude being 1 if this
is exactly periodic.
00:13:14.050 --> 00:13:16.220
Again, if this is exactly
one period inside,
00:13:16.220 --> 00:13:17.770
you can just
imagine this is gone
00:13:17.770 --> 00:13:19.450
and you get a continuous
wave function,
00:13:19.450 --> 00:13:20.908
so the amplitude
and the derivative
00:13:20.908 --> 00:13:22.701
must be the same
on both sides as
00:13:22.701 --> 00:13:23.950
if there was no barrier there.
00:13:27.110 --> 00:13:29.660
When the wave function
doesn't have exactly one
00:13:29.660 --> 00:13:33.620
period inside the well,
you can't do that,
00:13:33.620 --> 00:13:37.120
so the amplitudes can't
be the same on both sides.
00:13:37.120 --> 00:13:38.996
But that's not a very
satisfying explanation.
00:13:38.996 --> 00:13:40.828
That's really an
explanation about solutions
00:13:40.828 --> 00:13:42.110
to the differential equation.
00:13:42.110 --> 00:13:44.609
I'm just telling you properties
of second order differential
00:13:44.609 --> 00:13:45.120
equations.
00:13:45.120 --> 00:13:46.440
Let's think of a
more physical, more
00:13:46.440 --> 00:13:47.731
quantum mechanical explanation.
00:13:47.731 --> 00:13:50.360
Why are we getting
these resonances?
00:13:50.360 --> 00:13:52.500
Well, I want to think
about this in the same way
00:13:52.500 --> 00:13:55.070
as we thought about the boxes
in the very first lecture.
00:13:57.740 --> 00:14:03.620
Suppose we have
this square well,
00:14:03.620 --> 00:14:06.889
and I know I have
some amplitude here.
00:14:06.889 --> 00:14:07.930
I've got a wave function.
00:14:07.930 --> 00:14:09.157
It's got some amplitude here.
00:14:09.157 --> 00:14:10.740
It's got some momentum
going this way,
00:14:10.740 --> 00:14:12.160
some positive momentum.
00:14:12.160 --> 00:14:15.770
And I want to ask, what's the
probability that I will scatter
00:14:15.770 --> 00:14:18.020
past the potential, let's
say to this point right here
00:14:18.020 --> 00:14:19.686
just on the other
side of the potential?
00:14:19.686 --> 00:14:22.849
What's the probability
that I will scatter across?
00:14:22.849 --> 00:14:24.640
Well, you say, we've
done this calculation.
00:14:24.640 --> 00:14:27.000
We know the probability to
transmit across this step
00:14:27.000 --> 00:14:27.870
potential.
00:14:27.870 --> 00:14:29.520
We did that last time.
00:14:29.520 --> 00:14:30.710
So that's T step.
00:14:35.410 --> 00:14:37.170
We know the probability
of scattering
00:14:37.170 --> 00:14:43.134
across this potential step,
and that's transmission up.
00:14:43.134 --> 00:14:45.550
And so the probability that
you transmit from here to here
00:14:45.550 --> 00:14:47.883
is the probability that you
transmit here first and then
00:14:47.883 --> 00:14:51.090
the probability that
you transmit here,
00:14:51.090 --> 00:14:52.757
the product of
the probabilities.
00:14:52.757 --> 00:14:53.465
Sound reasonable?
00:14:56.491 --> 00:14:56.990
Let's vote.
00:14:56.990 --> 00:14:59.700
How many people think that this
is equal to the transmission
00:14:59.700 --> 00:15:04.170
probability across
the potential well?
00:15:04.170 --> 00:15:05.047
Any votes?
00:15:05.047 --> 00:15:06.630
You have to vote one
way or the other.
00:15:06.630 --> 00:15:07.370
No, it is not.
00:15:07.370 --> 00:15:09.000
How many people vote?
00:15:09.000 --> 00:15:09.500
OK.
00:15:09.500 --> 00:15:11.500
Yes, it is.
00:15:11.500 --> 00:15:12.000
OK.
00:15:12.000 --> 00:15:13.950
The no's have it, and
that's not terribly
00:15:13.950 --> 00:15:16.660
surprising because, of course,
these have no resonance
00:15:16.660 --> 00:15:18.035
structure, so
where did that come
00:15:18.035 --> 00:15:20.080
from if it's just
that thing squared?
00:15:20.080 --> 00:15:23.440
So that probably can't be it,
but here's the bigger problem.
00:15:23.440 --> 00:15:24.910
Why is this the wrong argument?
00:15:27.379 --> 00:15:28.920
AUDIENCE: Because
there's reflection.
00:15:28.920 --> 00:15:29.640
PROFESSOR: Yeah, exactly.
00:15:29.640 --> 00:15:31.810
There's reflection, but that's
only one step in the answer
00:15:31.810 --> 00:15:33.018
or why it's the wrong answer.
00:15:33.018 --> 00:15:34.007
Why else?
00:15:34.007 --> 00:15:35.590
AUDIENCE: Your
transmission operations
00:15:35.590 --> 00:15:37.197
have to do it far away?
00:15:37.197 --> 00:15:39.530
PROFESSOR: That's, true but
I just want the probability,
00:15:39.530 --> 00:15:41.280
and if I got here with
positive momentum,
00:15:41.280 --> 00:15:43.113
I'm eventually going
to get out to infinity,
00:15:43.113 --> 00:15:45.080
so it's the same
probability because it's
00:15:45.080 --> 00:15:46.330
just an E to the ikx out here.
00:15:46.330 --> 00:15:48.300
The wave function
is just E to the ikx
00:15:48.300 --> 00:15:51.490
so the probability is
going to be the same.
00:15:51.490 --> 00:15:53.439
Other reasons?
00:15:53.439 --> 00:15:55.772
AUDIENCE: Well, the width of
the well is very important,
00:15:55.772 --> 00:15:57.272
but the first
argument ignores that.
00:15:57.272 --> 00:15:58.063
PROFESSOR: Exactly.
00:15:58.063 --> 00:15:59.090
That's also true.
00:15:59.090 --> 00:16:01.673
So far, the reasons we have are
we need the width of the well,
00:16:01.673 --> 00:16:03.120
doesn't appear.
00:16:03.120 --> 00:16:04.405
That seems probably wrong.
00:16:07.024 --> 00:16:09.440
The second is it's possible
that you could have reflected.
00:16:09.440 --> 00:16:10.898
We haven't really
incorporated that
00:16:10.898 --> 00:16:12.480
in any sort of elegant way.
00:16:12.480 --> 00:16:14.420
AUDIENCE: There are
other ways to transmit
00:16:14.420 --> 00:16:14.905
by reflecting twice.
00:16:14.905 --> 00:16:16.130
PROFESSOR: That's
absolutely true.
00:16:16.130 --> 00:16:17.160
There are other
ways to transmit,
00:16:17.160 --> 00:16:19.320
so you could transmit,
then you could reflect.
00:16:19.320 --> 00:16:22.581
So we could transmit then
reflect, and reflect again,
00:16:22.581 --> 00:16:23.330
and then transmit.
00:16:23.330 --> 00:16:26.190
We could transmit,
reflect, reflect, transmit.
00:16:26.190 --> 00:16:28.170
What else?
00:16:28.170 --> 00:16:30.590
Do probabilities add
in quantum mechanics?
00:16:33.929 --> 00:16:35.470
And when you have
products of events,
00:16:35.470 --> 00:16:37.470
do probabilities multiply?
00:16:37.470 --> 00:16:39.210
What adds in quantum mechanics?
00:16:39.210 --> 00:16:40.479
AUDIENCE: [INAUDIBLE].
00:16:40.479 --> 00:16:41.520
PROFESSOR: The amplitude.
00:16:41.520 --> 00:16:42.740
The wave function.
00:16:42.740 --> 00:16:45.520
We do not take the
product of probabilities.
00:16:45.520 --> 00:16:47.410
What we do is we ask,
what's the amplitude
00:16:47.410 --> 00:16:49.237
to get here from
there, and we take
00:16:49.237 --> 00:16:51.320
the amplitude norm squared
to get the probability.
00:16:54.290 --> 00:16:56.830
So the correct
question is what's
00:16:56.830 --> 00:16:59.099
the amplitude to get
from here to here?
00:16:59.099 --> 00:17:00.890
How does the wave
function of the amplitude
00:17:00.890 --> 00:17:03.610
change as you move
from here to here?
00:17:03.610 --> 00:17:06.534
And for that, think back
to the two slit experiment
00:17:06.534 --> 00:17:08.160
or think back to the boxes.
00:17:08.160 --> 00:17:11.134
We asked the following question.
00:17:11.134 --> 00:17:15.069
The amplitude that you should
transmit across this well
00:17:15.069 --> 00:17:18.740
has a bunch of components,
is a sum of a bunch of terms.
00:17:18.740 --> 00:17:22.564
You could transmit
down this well.
00:17:22.564 --> 00:17:24.230
Inside here, you know
your wave function
00:17:24.230 --> 00:17:30.910
is e to the i k prime L-- or
I think I'm calling this k2 x.
00:17:30.910 --> 00:17:34.220
And in moving across the
well, your wave function
00:17:34.220 --> 00:17:37.850
evolves by an e to the
i k2 L, and then you
00:17:37.850 --> 00:17:41.010
could transmit again with
some transmission amplitude.
00:17:41.010 --> 00:17:43.800
So this would be the
transmission down times e
00:17:43.800 --> 00:17:49.114
to the i k2 L times
the transmit up.
00:17:49.114 --> 00:17:50.780
As we saw last time,
these are the same,
00:17:50.780 --> 00:17:51.900
but I just want to
keep them separate
00:17:51.900 --> 00:17:53.520
so you know which
one talking about.
00:17:53.520 --> 00:17:54.330
This is a contribution.
00:17:54.330 --> 00:17:56.360
This is something that could
contribute to the amplitude.
00:17:56.360 --> 00:17:58.790
Is it the only thing that could
contribute to the amplitude?
00:17:58.790 --> 00:17:59.289
No.
00:17:59.289 --> 00:18:01.590
What else could contribute?
00:18:01.590 --> 00:18:02.450
Bounce, right?
00:18:02.450 --> 00:18:06.120
So to get from here, to here
I could transmit, evolve,
00:18:06.120 --> 00:18:07.030
transmit.
00:18:07.030 --> 00:18:11.280
I could also transmit,
evolve, reflect, evolve,
00:18:11.280 --> 00:18:13.840
reflect, evolve, transmit.
00:18:13.840 --> 00:18:18.490
So there's also a term
that's t, e to the ikL, r,
00:18:18.490 --> 00:18:24.030
e to the minus ikL,
r, e to the iKL.
00:18:26.697 --> 00:18:28.280
Sorry, e to the plus
ikL because we're
00:18:28.280 --> 00:18:30.509
increasing the
evolution of the phase.
00:18:30.509 --> 00:18:32.050
And then transmit
finally at the end.
00:18:36.060 --> 00:18:38.370
And these k's are
all k2, but I could
00:18:38.370 --> 00:18:41.242
have done that many times.
00:18:41.242 --> 00:18:43.450
But notice that each time
what I'm going to do is I'm
00:18:43.450 --> 00:18:47.110
going to transmit, reflect,
reflect, transmit or transmit,
00:18:47.110 --> 00:18:50.950
reflect, reflect, reflect,
reflect, transmit.
00:18:50.950 --> 00:18:58.890
So I'm always going to do
this some number of times.
00:18:58.890 --> 00:19:01.420
I do this once, I do this
twice, I do it thrice.
00:19:01.420 --> 00:19:03.080
This gives me a
geometric series.
00:19:03.080 --> 00:19:09.560
This is t e to
the i k2 L t times
00:19:09.560 --> 00:19:13.370
1 plus this quantity plus
this quantity squared.
00:19:13.370 --> 00:19:16.900
That's a geometric series,
1 over 1 plus this quantity
00:19:16.900 --> 00:19:19.610
squared.
00:19:19.610 --> 00:19:22.017
Sorry, 1 minus because
it's a geometric sum.
00:19:22.017 --> 00:19:23.100
And what is this quantity?
00:19:23.100 --> 00:19:24.900
Well, it's r squared,
and remember the r's
00:19:24.900 --> 00:19:26.733
in both directions are
the same, so I'm just
00:19:26.733 --> 00:19:30.452
going to write it as
r squared e to the 2i.
00:19:33.826 --> 00:19:34.686
r squared.
00:19:34.686 --> 00:19:36.060
It's a real number,
but I'm going
00:19:36.060 --> 00:19:38.259
to put the absolute
value on anyway.
00:19:38.259 --> 00:19:39.550
It's going to simplify my life.
00:19:39.550 --> 00:19:47.750
e to the 2i k2 L. So
this is our prediction
00:19:47.750 --> 00:19:50.002
from multiple bounces.
00:19:50.002 --> 00:19:51.960
What we're doing here is
we're taking seriously
00:19:51.960 --> 00:19:55.870
the superposition principle that
says given any process, any way
00:19:55.870 --> 00:19:58.440
that that process could happen,
you sum up the amplitudes
00:19:58.440 --> 00:20:00.106
and the probability
is the norm squared.
00:20:00.106 --> 00:20:03.100
If we have a source
and we have two slits
00:20:03.100 --> 00:20:06.226
and I ask you, what's the
probability that you land here,
00:20:06.226 --> 00:20:08.350
the probability is not the
sum of the probabilities
00:20:08.350 --> 00:20:09.690
for each individual transit.
00:20:09.690 --> 00:20:13.230
The probability is the
square of the amplitude where
00:20:13.230 --> 00:20:15.710
the amplitude is the
sum, amplitude top
00:20:15.710 --> 00:20:17.730
plus amplitude bottom.
00:20:17.730 --> 00:20:19.395
Here, there are
many, many slits.
00:20:19.395 --> 00:20:21.420
There are many different
ways this could happen.
00:20:21.420 --> 00:20:23.345
You could reflect
multiple times.
00:20:23.345 --> 00:20:25.809
Everyone cool with that?
00:20:25.809 --> 00:20:27.350
By the same token,
we could have done
00:20:27.350 --> 00:20:28.780
the same thing for
reflection, but let's stick
00:20:28.780 --> 00:20:30.154
with transmission
for the moment.
00:20:30.154 --> 00:20:31.930
This is what we get
for the transmission,
00:20:31.930 --> 00:20:34.800
and again, the transmission
amplitudes across the step,
00:20:34.800 --> 00:20:36.470
as we saw last
time, are the same.
00:20:36.470 --> 00:20:37.855
So this is, in fact, t squared.
00:20:42.720 --> 00:20:47.590
And this gives us a result for
transmission down the potential
00:20:47.590 --> 00:20:58.930
well, and if we use what the
reflection and transmission
00:20:58.930 --> 00:21:03.290
amplitudes were
for our step wells,
00:21:03.290 --> 00:21:10.040
the answer that this gives is
1 over e to the i k2 L minus 2i
00:21:10.040 --> 00:21:19.390
upon the transmission for
a step times sine of k2 L.
00:21:19.390 --> 00:21:24.799
Now, this isn't the
same as the probability
00:21:24.799 --> 00:21:26.340
that we derived over
here, but that's
00:21:26.340 --> 00:21:27.440
because this isn't
the probability.
00:21:27.440 --> 00:21:28.398
This was the amplitude.
00:21:28.398 --> 00:21:30.030
We just computed
the total amplitude.
00:21:30.030 --> 00:21:31.920
To get the probability,
we have to take
00:21:31.920 --> 00:21:34.300
the norm squared
of the amplitude.
00:21:34.300 --> 00:21:36.175
And when we take
the norm squared,
00:21:36.175 --> 00:21:43.940
what we get is 1 upon 1 plus 1
over 4 epsilon, epsilon plus 1,
00:21:43.940 --> 00:21:48.170
sine squared of g0
root epsilon plus 1.
00:21:52.892 --> 00:21:53.850
We get the same answer.
00:21:59.317 --> 00:22:02.310
AUDIENCE: Is that an equality?
00:22:02.310 --> 00:22:04.480
PROFESSOR: This is an equality.
00:22:04.480 --> 00:22:05.010
Oh, sorry.
00:22:08.530 --> 00:22:10.297
Thank you, Barton.
00:22:10.297 --> 00:22:11.880
We get to the
transmission probability
00:22:11.880 --> 00:22:13.230
when we take the norm
squared of the amplitude.
00:22:13.230 --> 00:22:13.860
Thank you.
00:22:13.860 --> 00:22:17.050
Is equal to this, which is
the same as we got before.
00:22:17.050 --> 00:22:17.550
Thanks.
00:22:21.280 --> 00:22:22.445
Yeah, please?
00:22:22.445 --> 00:22:25.770
AUDIENCE: Tell me why kL
equals pi doesn't work.
00:22:28.269 --> 00:22:29.810
PROFESSOR: kL equals
pi doesn't work.
00:22:29.810 --> 00:22:30.350
It does.
00:22:30.350 --> 00:22:31.891
You just have to be
careful what kL--
00:22:35.554 --> 00:22:37.012
AUDIENCE: On the
first drawing, you
00:22:37.012 --> 00:22:38.678
changed your kL [INAUDIBLE].
00:22:43.247 --> 00:22:44.580
PROFESSOR: Because I'm an idiot.
00:22:44.580 --> 00:22:45.810
Because I got a
factor of 2 wrong.
00:22:45.810 --> 00:22:46.309
Thank you.
00:22:52.280 --> 00:22:55.200
Thank you, Matt.
00:22:55.200 --> 00:22:57.750
Thank you.
00:22:57.750 --> 00:22:58.422
Answer analysis.
00:22:58.422 --> 00:22:59.380
It's a wonderful thing.
00:23:06.800 --> 00:23:10.140
This does something
really nice for us.
00:23:10.140 --> 00:23:15.230
Why are we getting a
resonance at special values
00:23:15.230 --> 00:23:16.630
at the energy?
00:23:16.630 --> 00:23:17.890
What's happening?
00:23:17.890 --> 00:23:22.090
Well, in this quantum
mechanical process
00:23:22.090 --> 00:23:25.250
of multiple interactions,
multiple scatterings,
00:23:25.250 --> 00:23:29.659
there are many terms in the
amplitude for transmitting.
00:23:29.659 --> 00:23:31.450
There are terms that
involve no reflection,
00:23:31.450 --> 00:23:33.425
there are terms that
involve two reflections,
00:23:33.425 --> 00:23:36.570
there are terms that
involve four reflections,
00:23:36.570 --> 00:23:41.170
and they all come with an
actual magnitude and a phase.
00:23:41.170 --> 00:23:47.270
And when the phase is the
same, they add constructively,
00:23:47.270 --> 00:23:51.200
and when the phases are not
the same, they interfere.
00:23:51.200 --> 00:23:53.850
And when the phases
are exactly off,
00:23:53.850 --> 00:23:57.500
they interfere
destructively, and that
00:23:57.500 --> 00:23:59.980
is why you're
getting a resonance.
00:23:59.980 --> 00:24:02.760
Multiple terms in
your superposition
00:24:02.760 --> 00:24:04.620
interfere with each
other, something
00:24:04.620 --> 00:24:07.060
that does not
happen classically.
00:24:07.060 --> 00:24:09.120
Classically, the
probabilities are products.
00:24:09.120 --> 00:24:11.440
Quantum mechanically,
we have superposition
00:24:11.440 --> 00:24:14.110
and probabilities are the
squares of the amplitude,
00:24:14.110 --> 00:24:16.630
and we get interference
effects in the probabilities.
00:24:16.630 --> 00:24:18.590
Cool?
00:24:18.590 --> 00:24:22.530
To me, this is a nice, glorious
version of the two slit
00:24:22.530 --> 00:24:25.260
experiment, and we're going
to bump up again into it later
00:24:25.260 --> 00:24:31.610
when we talk about the physics
of solids in the real world.
00:24:31.610 --> 00:24:33.870
Questions at this point?
00:24:33.870 --> 00:24:34.689
Yeah?
00:24:34.689 --> 00:24:36.730
AUDIENCE: Question on
something you said earlier.
00:24:36.730 --> 00:24:38.399
What is the k2 L
equals [INAUDIBLE]?
00:24:38.399 --> 00:24:40.190
PROFESSOR: Yeah, what's
special about that?
00:24:40.190 --> 00:24:46.660
What's special about that is at
this point, where kL is n pi,
00:24:46.660 --> 00:24:48.670
we get perfect transmission.
00:24:48.670 --> 00:24:52.200
When kL is equal
to n plus 1/2 pi,
00:24:52.200 --> 00:24:55.054
the reflection is
as good as it gets.
00:24:55.054 --> 00:24:56.720
What that's really
doing is it's saying,
00:24:56.720 --> 00:25:00.550
when is this locally largest?
00:25:00.550 --> 00:25:03.324
So that's the special
points when the transmission
00:25:03.324 --> 00:25:05.740
is as small as possible, which
means the reflection, which
00:25:05.740 --> 00:25:07.406
is 1 minus the
transmission probability,
00:25:07.406 --> 00:25:09.620
is as large as possible.
00:25:09.620 --> 00:25:13.659
And you can get that, again,
from these expressions.
00:25:13.659 --> 00:25:14.325
Other questions?
00:25:28.790 --> 00:25:31.780
So I want to think a little more
about these square barriers.
00:25:31.780 --> 00:25:34.080
And in particular, in
thinking about the square well
00:25:34.080 --> 00:25:36.360
barrier, what we've
been talking about
00:25:36.360 --> 00:25:39.650
all along are
monochromatic wave packets.
00:25:39.650 --> 00:25:42.180
We've been talking about
plain waves, just simply
00:25:42.180 --> 00:25:48.990
e to the ikx, but you
can't put a single particle
00:25:48.990 --> 00:25:52.630
in a state which
is a plain wave.
00:25:52.630 --> 00:25:53.710
It's not normalizable.
00:25:53.710 --> 00:25:55.490
What we really mean
at the end of the day
00:25:55.490 --> 00:25:57.198
when we talk about
single particles is we
00:25:57.198 --> 00:26:00.940
put them in some well localized
wave packet, which at time 0,
00:26:00.940 --> 00:26:05.200
let's say, is at position x0,
which in this case is negative,
00:26:05.200 --> 00:26:09.840
and which has some well
defined average momentum, k0.
00:26:09.840 --> 00:26:12.340
I'll say it's the expectation
value of momentum in this wave
00:26:12.340 --> 00:26:14.077
packet.
00:26:14.077 --> 00:26:15.660
And the question we
really want to ask
00:26:15.660 --> 00:26:18.970
when we talk about scattering
is, what happens to this beast
00:26:18.970 --> 00:26:23.310
as it hits the
barrier, which I'm
00:26:23.310 --> 00:26:24.810
going to put the
left hand side at 0
00:26:24.810 --> 00:26:27.170
and the right hand
side at L and let
00:26:27.170 --> 00:26:29.810
the depth be minus v0 again.
00:26:29.810 --> 00:26:32.740
What happens as this incident
wave packet hits the barrier
00:26:32.740 --> 00:26:33.859
and then scatters off?
00:26:33.859 --> 00:26:36.150
Well, we know what would
happen if it was a plain wave,
00:26:36.150 --> 00:26:37.520
but a plain wave
wouldn't be localized.
00:26:37.520 --> 00:26:39.160
So this is the
question I want to ask,
00:26:39.160 --> 00:26:42.530
and I want to use the
results that we already have.
00:26:42.530 --> 00:26:43.750
Now, here's the key thing.
00:26:46.490 --> 00:26:49.100
Consider to begin
with just a wave
00:26:49.100 --> 00:26:51.890
packet for a free
particle centered
00:26:51.890 --> 00:26:58.520
at x0 and with momentum k0,
just look for a free particle.
00:26:58.520 --> 00:27:01.350
We know how to write this.
00:27:01.350 --> 00:27:02.980
We can put the
system, for example,
00:27:02.980 --> 00:27:04.650
we can take our wave
function at time 0
00:27:04.650 --> 00:27:07.560
to be a Gaussian, some
normalization times
00:27:07.560 --> 00:27:13.270
e to the minus x minus x0
squared over 2a squared.
00:27:13.270 --> 00:27:15.309
And we want to give
it some momentum k0,
00:27:15.309 --> 00:27:17.600
and you know how to do that
after the last problem set,
00:27:17.600 --> 00:27:20.066
e to the i k0 x.
00:27:20.066 --> 00:27:22.830
Everyone cool with that?
00:27:22.830 --> 00:27:26.995
So there's our
initial wave function,
00:27:26.995 --> 00:27:28.870
and we want to know how
it evolves with time,
00:27:28.870 --> 00:27:30.130
and we know how to do that.
00:27:30.130 --> 00:27:31.930
To evolve it in time,
we first expand it
00:27:31.930 --> 00:27:33.820
in energy eigenstates.
00:27:33.820 --> 00:27:39.260
So psi of x0 is equal to--
well, the energy eigenstates
00:27:39.260 --> 00:27:46.150
in this case are plain waves,
dk e to the ikx over root 2 pi
00:27:46.150 --> 00:27:49.990
times some coefficients, f of
k, the expansion coefficients.
00:27:49.990 --> 00:27:52.400
But these are just
the Fourier transform
00:27:52.400 --> 00:27:55.550
of our initial
Gaussian wave packet,
00:27:55.550 --> 00:28:02.740
and we know what the form
of f of k is equal to.
00:28:02.740 --> 00:28:05.560
Well, it's a Gaussian
of width 1 upon alpha
00:28:05.560 --> 00:28:10.130
e to the minus k minus k0
because it has momentum k0,
00:28:10.130 --> 00:28:13.460
so it's centered around k0,
squared over 2 times a squared,
00:28:13.460 --> 00:28:15.310
and the a goes upstairs.
00:28:15.310 --> 00:28:19.700
And the position of
the initial wave packet
00:28:19.700 --> 00:28:21.300
is encoded in the
Fourier transform,
00:28:21.300 --> 00:28:23.050
and I'm going to put
a normalization here,
00:28:23.050 --> 00:28:26.560
which I'm not going to worry
about, with an overall phase
00:28:26.560 --> 00:28:31.900
e to the minus ik x0.
00:28:31.900 --> 00:28:34.360
So in just the same say
that adding on a phase,
00:28:34.360 --> 00:28:36.667
e to the i k0 x, in
the position space
00:28:36.667 --> 00:28:39.250
wave function tells you what the
expectation value of momentum
00:28:39.250 --> 00:28:41.285
is, tacking on the phase--
and we can get this
00:28:41.285 --> 00:28:43.368
from just Fourier transform--
tacking on the phase
00:28:43.368 --> 00:28:45.470
e to the ik x0 in
the Fourier transform
00:28:45.470 --> 00:28:48.210
tells you the center point,
the central position,
00:28:48.210 --> 00:28:49.292
of the wave packet.
00:28:49.292 --> 00:28:50.750
So this you did on
the problem set,
00:28:50.750 --> 00:28:52.650
and this is a trivial
momentum space
00:28:52.650 --> 00:28:55.170
version of the same thing.
00:28:55.170 --> 00:28:56.890
So here's our wave
packet expanded
00:28:56.890 --> 00:28:58.890
in plain waves, which
are energy eigenstates.
00:28:58.890 --> 00:29:01.015
And the statement that
these are energy eigenstates
00:29:01.015 --> 00:29:03.650
is equivalent to the statement
that under time evolution,
00:29:03.650 --> 00:29:06.207
they do nothing but
rotate by a phase.
00:29:06.207 --> 00:29:08.165
So if we want to know
what the wave function is
00:29:08.165 --> 00:29:12.540
as a function of time, psi of x
and t is equal to the integral,
00:29:12.540 --> 00:29:18.470
dk, the Fourier mode f of k.
00:29:18.470 --> 00:29:20.052
I'll write it out.
00:29:20.052 --> 00:29:22.610
Actually, I'm going to
write this out explicitly.
00:29:22.610 --> 00:29:28.540
f of k 1 over root
2 pi e to the ikx
00:29:28.540 --> 00:29:33.410
minus omega t where omega, of
course, is a function of t.
00:29:33.410 --> 00:29:39.750
It's a free particle, so
for our free particle,
00:29:39.750 --> 00:29:45.462
h bar squared k squared upon
2m is equal to h bar omega.
00:29:48.410 --> 00:29:50.300
We've done this before.
00:29:50.300 --> 00:29:52.680
But now what I want to do
is I want to take exactly
00:29:52.680 --> 00:29:54.910
the same system and I
want to add, at position
00:29:54.910 --> 00:30:00.120
zero, a well of
depth v0 and width L.
00:30:00.120 --> 00:30:03.553
How is our story
going to change?
00:30:03.553 --> 00:30:05.886
Well, we want our initial
wave packet, which is up to us
00:30:05.886 --> 00:30:08.330
to choose, we want our initial
wave packet to be the same.
00:30:08.330 --> 00:30:09.871
We want to start
with a Gaussian far,
00:30:09.871 --> 00:30:12.359
far, far away from the barrier.
00:30:12.359 --> 00:30:14.150
We want it to be well
localized in position
00:30:14.150 --> 00:30:17.090
and well localized in momentum
space, not perfectly localized,
00:30:17.090 --> 00:30:17.659
of course.
00:30:17.659 --> 00:30:19.700
It's a finite Gaussian to
satisfy the uncertainty
00:30:19.700 --> 00:30:22.520
principle, but it's
well localized.
00:30:22.520 --> 00:30:24.096
So how does this story change?
00:30:24.096 --> 00:30:25.470
Well, this doesn't
change at all.
00:30:25.470 --> 00:30:27.080
It's the same wave function.
00:30:27.080 --> 00:30:29.980
However, when we expand
in energy eigenstates,
00:30:29.980 --> 00:30:37.350
the energy eigenstates are
no longer simple plain waves.
00:30:37.350 --> 00:30:40.600
The energy eigenstates,
as we know for the system,
00:30:40.600 --> 00:30:43.570
take a different form.
00:30:43.570 --> 00:30:48.990
For the square well and positive
energy scattering states,
00:30:48.990 --> 00:30:51.170
the plain wave, or the
energy eigenstates,
00:30:51.170 --> 00:30:55.470
which I will label by k, just
because I'm going to call e
00:30:55.470 --> 00:31:01.121
is equal to h bar squared
k squared upon 2m--
00:31:01.121 --> 00:31:03.620
the energy is a constant-- which
is the k asymptotically far
00:31:03.620 --> 00:31:04.620
away from the potential.
00:31:06.850 --> 00:31:12.050
The wave function I can write
as 1 over root 2 pi times
00:31:12.050 --> 00:31:16.300
e to the ikx when we're
on the left hand side,
00:31:16.300 --> 00:31:18.920
but being on the left hand side
is equivalent to multiplying
00:31:18.920 --> 00:31:21.870
by a theta function of minus x.
00:31:21.870 --> 00:31:24.750
This is the function which is
0 when its argument is negative
00:31:24.750 --> 00:31:30.350
and 1 when its
argument is positive.
00:31:33.530 --> 00:31:36.930
Plus we have the
reflected term, which
00:31:36.930 --> 00:31:39.680
has an amplitude, r, which
is again a function of k.
00:31:39.680 --> 00:31:41.250
This is the
reflection amplitude,
00:31:41.250 --> 00:31:43.850
also known as c upon a.
00:31:43.850 --> 00:31:48.020
e to the minus ikx-- again,
on the left hand side--
00:31:48.020 --> 00:31:53.980
theta of minus x plus a
transition amplitude whose
00:31:53.980 --> 00:31:57.070
norm squared is transmission
probability e to the ikx when
00:31:57.070 --> 00:32:01.470
we're on the right, theta of x.
00:32:06.280 --> 00:32:08.930
So this is just a slightly
different notation
00:32:08.930 --> 00:32:11.810
than what we usually write
with left and right separated.
00:32:17.950 --> 00:32:20.565
So what we want to do now is
we want to decompose our wave
00:32:20.565 --> 00:32:22.690
function in terms of the
actual energy eigenstates.
00:32:29.440 --> 00:32:33.860
The way we're going to do that,
and having done that, having
00:32:33.860 --> 00:32:38.275
expanded our wave
function in this basis,
00:32:38.275 --> 00:32:40.650
we can determine the time
evolution in the following way.
00:32:40.650 --> 00:32:43.420
First, we expand
the wave function
00:32:43.420 --> 00:32:48.230
at time 0 as an
integral, dk, and I'm
00:32:48.230 --> 00:32:50.176
going to pull the root 2 pi out.
00:32:53.880 --> 00:32:55.340
And we have some
Fourier transform,
00:32:55.340 --> 00:32:56.890
which I'm now going
to call f tilde
00:32:56.890 --> 00:32:59.223
because is slightly different
than the f we used before,
00:32:59.223 --> 00:33:01.500
but it's what the expansion
coefficients have to be,
00:33:01.500 --> 00:33:11.260
f tilde of k times
this beast, phi of k.
00:33:24.745 --> 00:33:26.120
Let me actually
take this product
00:33:26.120 --> 00:33:27.995
and write it out in
terms of the three terms.
00:33:27.995 --> 00:33:33.680
So those three terms are going
to be f of k, again tilde,
00:33:33.680 --> 00:33:43.340
e to the ikx, theta of minus x,
plus f tilde r e to the minus
00:33:43.340 --> 00:33:53.272
ikx, theta of minus
x plus f tilde t
00:33:53.272 --> 00:33:57.100
e to the ikx, theta of x.
00:34:02.490 --> 00:34:03.590
Let's look at these terms.
00:34:06.510 --> 00:34:08.639
Finally, we want to look
at the time evolution,
00:34:08.639 --> 00:34:11.300
but we started out as a
superposition of states
00:34:11.300 --> 00:34:13.159
with definite
energy labeled by k,
00:34:13.159 --> 00:34:17.346
so we know the time evolution
is e to the ikx minus omega t, e
00:34:17.346 --> 00:34:22.000
to the i minus ikx plus
omega t, so minus omega t,
00:34:22.000 --> 00:34:24.130
and kx minus omega t.
00:34:24.130 --> 00:34:27.310
So we can immediately, from this
time evolving wave function,
00:34:27.310 --> 00:34:29.699
identify these
two terms as terms
00:34:29.699 --> 00:34:33.350
with a central peak
moving to the right,
00:34:33.350 --> 00:34:36.588
and this, central peak moving
to the left, kx plus omega t.
00:34:40.040 --> 00:34:41.040
Everyone cool with that?
00:34:44.530 --> 00:34:45.030
Questions?
00:34:49.326 --> 00:34:50.826
AUDIENCE: Can you
explain real quick
00:34:50.826 --> 00:34:53.666
one more time how
that's the [INAUDIBLE]?
00:34:53.666 --> 00:34:55.290
PROFESSOR: How that--
sorry, say again.
00:34:55.290 --> 00:34:57.166
AUDIENCE: How the top
equation [INAUDIBLE].
00:34:57.166 --> 00:35:00.450
PROFESSOR: How the top
equation led to this,
00:35:00.450 --> 00:35:01.697
or just where this came from?
00:35:01.697 --> 00:35:02.950
AUDIENCE: [INAUDIBLE].
00:35:02.950 --> 00:35:04.800
PROFESSOR: Good.
00:35:04.800 --> 00:35:06.710
This is just a notational thing.
00:35:06.710 --> 00:35:11.900
Usually when I say phi sub
k is equal to, on the left,
00:35:11.900 --> 00:35:16.560
e to the ikx with some
overall amplitude.
00:35:16.560 --> 00:35:21.870
Let's say 1 over
2 pi e to the ikx
00:35:21.870 --> 00:35:28.040
plus c over a e to the
minus ikx on the left,
00:35:28.040 --> 00:35:30.770
and because this is
the reflected wave,
00:35:30.770 --> 00:35:35.460
I'm just going to call this
R. And then on the right,
00:35:35.460 --> 00:35:38.099
we have e to the ikx.
00:35:38.099 --> 00:35:39.890
There's only a wave
travelling to the right
00:35:39.890 --> 00:35:42.162
and the coefficient is the
transmission amplitude.
00:35:42.162 --> 00:35:44.120
So this is what we normally
write, but then I'm
00:35:44.120 --> 00:35:46.760
using the so-called
theta function.
00:35:46.760 --> 00:35:51.500
Theta of x is defined as 0
when x is less than 0 and 1
00:35:51.500 --> 00:35:53.680
when x is greater than 0.
00:35:53.680 --> 00:35:55.520
This is a function 1.
00:35:55.520 --> 00:35:57.310
Using theta function
allows me to write
00:35:57.310 --> 00:35:59.510
this thing as a single
function without having
00:35:59.510 --> 00:36:02.490
to goose around
with lots of terms.
00:36:02.490 --> 00:36:03.440
Is that cool?
00:36:03.440 --> 00:36:04.035
Great.
00:36:04.035 --> 00:36:06.460
AUDIENCE: Are we just
thinking about one
00:36:06.460 --> 00:36:09.612
step here or the whole
well, because always, we
00:36:09.612 --> 00:36:11.320
should have something
[INAUDIBLE].
00:36:11.320 --> 00:36:11.900
PROFESSOR: Fantastic.
00:36:11.900 --> 00:36:12.640
Excellent, excellent.
00:36:12.640 --> 00:36:13.390
Thank you so much.
00:36:16.180 --> 00:36:19.130
What I want to do is I want
to think of the wave function.
00:36:19.130 --> 00:36:22.539
This is a good description when
the particle is far, far away,
00:36:22.539 --> 00:36:24.830
and this is a good description
when the particle is not
00:36:24.830 --> 00:36:27.240
in the potential.
00:36:27.240 --> 00:36:27.740
Sorry.
00:36:27.740 --> 00:36:28.050
Thank you.
00:36:28.050 --> 00:36:29.520
I totally glossed
over this step.
00:36:29.520 --> 00:36:31.860
I want to imagine
this as a potential
00:36:31.860 --> 00:36:40.422
where all of the matching is
implemented at x equals 0,
00:36:40.422 --> 00:36:41.880
so when I write it
in this fashion.
00:36:44.816 --> 00:36:46.565
Another equivalent way
to think about this
00:36:46.565 --> 00:36:48.648
is this is a good description
of the wave function
00:36:48.648 --> 00:36:49.940
when we're not inside the well.
00:36:49.940 --> 00:36:52.273
So for the purposes of the
rest of the analysis that I'm
00:36:52.273 --> 00:36:53.970
going to do, this
is exactly what
00:36:53.970 --> 00:36:59.010
the form of the wave function is
when we're not inside the well,
00:36:59.010 --> 00:37:01.060
and then let's just
not use this to ask
00:37:01.060 --> 00:37:03.380
about questions inside the well.
00:37:03.380 --> 00:37:04.849
When I write it in
this theta form,
00:37:04.849 --> 00:37:06.890
or for that matter, when
I write it in this form,
00:37:06.890 --> 00:37:08.130
this is not the form.
00:37:08.130 --> 00:37:13.780
What I mean is left of the
well and right of the well,
00:37:13.780 --> 00:37:20.180
and inside-- t e to
the ikx-- inside,
00:37:20.180 --> 00:37:23.789
it's doing something
else, but we
00:37:23.789 --> 00:37:25.330
don't want to ask
questions about it.
00:37:28.080 --> 00:37:31.140
That's just going
to simplify my life.
00:37:31.140 --> 00:37:31.810
Yeah?
00:37:31.810 --> 00:37:34.250
AUDIENCE: And is
the lowercase t then
00:37:34.250 --> 00:37:35.977
just the square
root of capital T?
00:37:35.977 --> 00:37:37.810
PROFESSOR: It's the
square root of capital T
00:37:37.810 --> 00:37:40.060
but you've got to be careful
because there can be a phase.
00:37:40.060 --> 00:37:42.185
Remember, this is the
amplitude, and what it really
00:37:42.185 --> 00:37:45.430
is is this reflection
is B over A,
00:37:45.430 --> 00:37:47.240
and these guys are
complex numbers.
00:37:47.240 --> 00:37:48.990
And it's true that B
over A norm squared
00:37:48.990 --> 00:37:52.660
is the transmission
probability, but B over A
00:37:52.660 --> 00:37:55.090
has a phase, and that
quantity I'm going to call t,
00:37:55.090 --> 00:37:57.500
and we'll interpret that in
more detail in a few minutes.
00:38:01.610 --> 00:38:03.520
So it's of this form.
00:38:03.520 --> 00:38:06.420
I just want to look at
each of these three terms.
00:38:06.420 --> 00:38:09.040
In particular, I want to
focus on them at time 0.
00:38:09.040 --> 00:38:11.380
So at t equals 0, what
does this look like?
00:38:11.380 --> 00:38:17.740
Well, that first term
is integral dk of f of k
00:38:17.740 --> 00:38:25.630
e to the ikx minus omega
t, theta of minus x.
00:38:25.630 --> 00:38:28.065
Notice that theta of minus
x is independent of k.
00:38:28.065 --> 00:38:29.440
It's independent
of the integral,
00:38:29.440 --> 00:38:33.930
so this is just a function
times theta of minus x.
00:38:33.930 --> 00:38:35.917
And this function
was constructed
00:38:35.917 --> 00:38:37.250
to give us the initial Gaussian.
00:38:42.210 --> 00:38:43.650
And this was at t equals 0.
00:38:43.650 --> 00:38:49.090
So this is just our Gaussian
at position x0 centered
00:38:49.090 --> 00:38:55.292
around value k0 at time
equals 0, theta of minus x.
00:39:01.555 --> 00:39:07.830
Sorry, at position x0.
00:39:07.830 --> 00:39:10.880
All this function is, this
is the Fourier transform
00:39:10.880 --> 00:39:13.270
of our Gaussian and we're
undoing the Fourier transform.
00:39:13.270 --> 00:39:16.090
This is just giving
us our Gaussian back.
00:39:16.090 --> 00:39:18.830
And as long as the particle
is far away from the well--
00:39:18.830 --> 00:39:21.992
so here's the well and
here's our wave packet--
00:39:21.992 --> 00:39:23.700
that theta function
is totally irrelevant
00:39:23.700 --> 00:39:26.272
because the Gaussian makes
it 0 away from the center
00:39:26.272 --> 00:39:27.230
of the Gaussian anyway.
00:39:34.800 --> 00:39:36.520
And so just as a
quick question, if we
00:39:36.520 --> 00:39:39.260
look at this is a function of
t so we put back in the minus
00:39:39.260 --> 00:39:49.760
omega t, so if we put back
in the time dependence,
00:39:49.760 --> 00:39:52.870
this is a wave that's
moving to the right,
00:39:52.870 --> 00:39:54.705
and to be more
precise about that,
00:39:54.705 --> 00:39:57.080
what does it mean to say it's
a wave moving to the right?
00:39:57.080 --> 00:40:02.700
Well, this is an envelope
on this set of plain waves,
00:40:02.700 --> 00:40:04.890
and the envelope,
by construction,
00:40:04.890 --> 00:40:06.770
was well localized
around position x
00:40:06.770 --> 00:40:09.994
but it was also well
localized in momentum,
00:40:09.994 --> 00:40:11.660
and in particular,
the Fourier transform
00:40:11.660 --> 00:40:14.130
is well localized
around the momentum k0.
00:40:17.930 --> 00:40:20.370
And so using the method
of stationary phase,
00:40:20.370 --> 00:40:27.660
or just asking, where is the
phase constant stationary,
00:40:27.660 --> 00:40:33.440
we get that the central
peak of the wave function
00:40:33.440 --> 00:40:36.890
satisfies the equation d d k0.
00:40:36.890 --> 00:40:38.940
If you're not familiar
with stationary phase,
00:40:38.940 --> 00:40:42.240
let the recitation
instructors know
00:40:42.240 --> 00:40:45.110
and they will
discuss it for you.
00:40:45.110 --> 00:40:49.890
The points of stationary
phase of this superposition
00:40:49.890 --> 00:40:58.690
of this wave packet lie
at the position d dk of kx
00:40:58.690 --> 00:41:04.930
minus omega of kt
evaluated at the peak, k0,
00:41:04.930 --> 00:41:06.530
of the distribution.
00:41:06.530 --> 00:41:10.860
But d dk of kx minus omega
t is, for the first term x,
00:41:10.860 --> 00:41:13.350
and for the second
term, d dk of omega.
00:41:13.350 --> 00:41:16.650
Well, we know that omega, we're
in the free particle regime.
00:41:16.650 --> 00:41:19.490
Omega, we all know what it is.
00:41:19.490 --> 00:41:21.952
It's h bar k squared upon 2m.
00:41:21.952 --> 00:41:23.660
We take the derivative
with respect to k.
00:41:23.660 --> 00:41:25.300
We get h bar k over m.
00:41:25.300 --> 00:41:32.860
The 2's cancel, so x
minus h bar k over m t.
00:41:32.860 --> 00:41:37.140
And a place where
this phase is 0,
00:41:37.140 --> 00:41:40.920
where the phase is
stationary, moves over time
00:41:40.920 --> 00:41:46.190
as x is equal to
h bar k over m t.
00:41:46.190 --> 00:41:49.880
But h bar k over m,
that's the momentum.
00:41:49.880 --> 00:41:52.930
p over m, that's the
classical velocity.
00:41:52.930 --> 00:41:58.140
So this is v0 t, the velocity
associated with that momentum.
00:41:58.140 --> 00:41:58.919
So this is good.
00:41:58.919 --> 00:42:01.210
We've done the right job of
setting up our wave packet.
00:42:01.210 --> 00:42:03.120
We built a Gaussian
that was far away that
00:42:03.120 --> 00:42:05.794
was moving in with a fixed
velocity towards the barrier,
00:42:05.794 --> 00:42:08.210
and now we want to know what
happens after it collides off
00:42:08.210 --> 00:42:09.300
the barrier.
00:42:09.300 --> 00:42:10.012
Cool?
00:42:10.012 --> 00:42:11.970
So what we really want
to ask is at late times,
00:42:11.970 --> 00:42:13.553
what does the wave
function look like?
00:42:15.794 --> 00:42:17.460
Again, we're focusing
on the first term.
00:42:17.460 --> 00:42:20.450
The position of this wave packet
at late times, a positive t,
00:42:20.450 --> 00:42:22.932
is positive, and
when it's positive,
00:42:22.932 --> 00:42:24.890
then this theta function
kills its contribution
00:42:24.890 --> 00:42:27.860
to the overall wave function.
00:42:27.860 --> 00:42:30.410
This theta is now theta
of minus a positive number
00:42:30.410 --> 00:42:34.060
and this Gaussian is gone.
00:42:34.060 --> 00:42:35.510
What is it replaced by?
00:42:35.510 --> 00:42:38.060
Well, these two terms
aren't necessarily 0.
00:42:38.060 --> 00:42:40.350
In particular, this one
is moving to the left.
00:42:40.350 --> 00:42:43.120
So as time goes forward, x
is moving further and further
00:42:43.120 --> 00:42:47.190
to the left, and so this theta
function starts turning on.
00:42:47.190 --> 00:42:49.490
And similarly, as
x goes positive,
00:42:49.490 --> 00:42:53.290
this theta function
starts turning on as well.
00:42:53.290 --> 00:42:59.959
Let's focus on this third
term, the transmitted term.
00:42:59.959 --> 00:43:01.250
Let's focus on this third term.
00:43:08.120 --> 00:43:12.400
In particular, that term looks
like integral dk over root 2
00:43:12.400 --> 00:43:18.130
pi, f times the transmission
amplitude times e
00:43:18.130 --> 00:43:21.590
to the ikx minus omega t.
00:43:24.540 --> 00:43:26.680
And there's an overall
theta of x outside,
00:43:26.680 --> 00:43:28.810
but for late times where
the center of the wave
00:43:28.810 --> 00:43:31.510
packet where the transmitted
wave packet should be positive,
00:43:31.510 --> 00:43:34.255
this theta is just going to be
1 so we can safely ignore it.
00:43:34.255 --> 00:43:37.740
It's just saying we're
far off to the right.
00:43:37.740 --> 00:43:41.700
And now I want to
do one last thing.
00:43:41.700 --> 00:43:44.310
This was an overall envelope.
00:43:44.310 --> 00:43:46.025
This t was our
scattering amplitude,
00:43:46.025 --> 00:43:47.900
and I want to write it
in the following form.
00:43:47.900 --> 00:43:53.410
I want to write it as root
T e to the minus i phi
00:43:53.410 --> 00:43:54.287
where phi is k.
00:43:54.287 --> 00:43:55.870
So what this is
saying is that indeed,
00:43:55.870 --> 00:43:57.703
as was pointed out
earlier, the norm squared
00:43:57.703 --> 00:44:01.010
of this coefficient T is the
transmission probability,
00:44:01.010 --> 00:44:02.540
but it has a phase.
00:44:02.540 --> 00:44:05.520
And what I want to know is
what does this phase mean?
00:44:05.520 --> 00:44:08.360
What information is
contained in this phase?
00:44:08.360 --> 00:44:10.110
And that's what
we're about to find.
00:44:10.110 --> 00:44:11.470
So let's put that in.
00:44:11.470 --> 00:44:18.030
Here we have root
T and minus phi
00:44:18.030 --> 00:44:20.952
where omega and phi
are both functions of k
00:44:20.952 --> 00:44:22.410
because the
transmission amplitudes
00:44:22.410 --> 00:44:27.260
depend on the momentum,
or the energy.
00:44:27.260 --> 00:44:30.840
And now I again want to know,
how does this wave packet move?
00:44:30.840 --> 00:44:33.930
If I look at this wave packet,
how does it move en masse?
00:44:33.930 --> 00:44:37.510
As a group of waves, how
does this wave packet move?
00:44:37.510 --> 00:44:40.130
In particular,
with what velocity?
00:44:40.130 --> 00:44:42.560
I'll again make an argument
by stationary phase.
00:44:42.560 --> 00:44:44.960
I'll look at a point
of phase equals 0
00:44:44.960 --> 00:44:46.565
and ask how it moves over time.
00:44:46.565 --> 00:44:47.940
And the point of
stationary phase
00:44:47.940 --> 00:44:54.860
is again given by d dk of
the phase kx minus omega
00:44:54.860 --> 00:45:01.220
of kt minus phi of k is equal
to 0 evaluated at k0, which
00:45:01.220 --> 00:45:03.200
is where our envelope
is sharply peaked.
00:45:07.470 --> 00:45:09.420
This expression
is equal to-- this
00:45:09.420 --> 00:45:12.730
is, again, x minus d omega dk.
00:45:12.730 --> 00:45:22.250
That's the classical velocity,
v0 t-- minus d phi dk.
00:45:22.250 --> 00:45:28.495
But just as a note, d phi
dk is equal to d omega dk,
00:45:28.495 --> 00:45:34.290
d phi d omega, but
this is equal to-- this
00:45:34.290 --> 00:45:36.110
is just the chain rule.
d omega dk, that's
00:45:36.110 --> 00:45:38.310
the classical velocity.
00:45:38.310 --> 00:45:45.230
d phi d omega, that's
d phi dE times h bar.
00:45:45.230 --> 00:45:48.510
I just multiplied by h
bar on the top and bottom.
00:45:48.510 --> 00:45:57.280
So this is equal to x minus
v0 t plus h bar d phi dE.
00:46:00.480 --> 00:46:03.100
So first off, let's just make
sure that the units make sense.
00:46:03.100 --> 00:46:04.997
That's a length,
that's a velocity,
00:46:04.997 --> 00:46:06.580
so this had better
have units of time.
00:46:06.580 --> 00:46:07.950
Time, that's good.
00:46:07.950 --> 00:46:10.890
So h bar times d phase over dE.
00:46:10.890 --> 00:46:13.970
Well, phase is dimensionless,
energy is units of energy,
00:46:13.970 --> 00:46:17.390
h is energy times time, so
this dimensionally works out.
00:46:17.390 --> 00:46:18.500
So this is 0.
00:46:18.500 --> 00:46:20.630
So the claim is the
point of stationary phase
00:46:20.630 --> 00:46:24.260
has this derivative equal to
0, so setting this equal to 0
00:46:24.260 --> 00:46:26.580
tells us that the peak
of the wave function
00:46:26.580 --> 00:46:29.650
moves according
to this equation.
00:46:33.420 --> 00:46:34.915
So this is really satisfying.
00:46:34.915 --> 00:46:37.290
This should be really satisfying
for a couple of reasons.
00:46:37.290 --> 00:46:41.440
First off, it tells you that
the peak of the transmitted wave
00:46:41.440 --> 00:46:44.460
packet, not just a plain wave,
the peak of the actual wave
00:46:44.460 --> 00:46:48.100
packet, well localized,
moves with overall velocity
00:46:48.100 --> 00:46:50.860
v0, the constant velocity
that we started with,
00:46:50.860 --> 00:46:51.530
and that's good.
00:46:51.530 --> 00:46:53.280
If it was moving with
some other velocity,
00:46:53.280 --> 00:46:54.696
we would have lost
energy somehow.
00:46:54.696 --> 00:46:57.600
That would be not so sensible.
00:46:57.600 --> 00:47:00.240
So this wave packet is moving
with an overall velocity v0.
00:47:00.240 --> 00:47:06.650
However, it doesn't
move along just
00:47:06.650 --> 00:47:09.300
as the original wave
packet's peak had.
00:47:09.300 --> 00:47:11.550
It moves as if shifted in time.
00:47:14.560 --> 00:47:16.920
So the phase, and more to
the point, the gradient
00:47:16.920 --> 00:47:18.420
of the phase with
respect to energy,
00:47:18.420 --> 00:47:22.150
the rate of change with energy
of the phase times h bar
00:47:22.150 --> 00:47:28.690
is giving us a shift in the time
of where the wave packet is.
00:47:28.690 --> 00:47:29.530
What does that mean?
00:47:32.730 --> 00:47:33.910
Let's be precise about this.
00:47:43.120 --> 00:47:45.545
Let's interpret this.
00:47:45.545 --> 00:47:47.420
Here's the interpretation
I want to give you.
00:47:51.160 --> 00:47:53.070
Consider classically the system.
00:47:53.070 --> 00:47:57.180
Classically, we have an
object with some energy,
00:47:57.180 --> 00:48:01.390
and it rolls along and it finds
a potential well, and what
00:48:01.390 --> 00:48:04.200
happens when it gets
into the potential well?
00:48:04.200 --> 00:48:05.494
It speeds up.
00:48:05.494 --> 00:48:06.910
because it's got
a lot more energy
00:48:06.910 --> 00:48:07.790
relative to the potential.
00:48:07.790 --> 00:48:10.290
So it goes much faster in here
and it gets to the other side
00:48:10.290 --> 00:48:11.460
and it slows down again.
00:48:11.460 --> 00:48:13.571
So if I had taken a
particle with velocity--
00:48:13.571 --> 00:48:15.070
let's call this
position 0 and let's
00:48:15.070 --> 00:48:17.680
say it gets to this
wall at time 0,
00:48:17.680 --> 00:48:21.390
so it's moving with
x is equal to v0 t,
00:48:21.390 --> 00:48:23.060
if there had been
no barrier there,
00:48:23.060 --> 00:48:25.018
then at subsequent times,
it would get out here
00:48:25.018 --> 00:48:29.160
in a time that distance
over v0, right?
00:48:29.160 --> 00:48:32.570
However, imagine this well
was extraordinarily deep.
00:48:32.570 --> 00:48:34.330
If this well were
extraordinarily deep,
00:48:34.330 --> 00:48:35.530
what would happen?
00:48:35.530 --> 00:48:38.840
Well basically, in here, its
velocity is arbitrarily large,
00:48:38.840 --> 00:48:42.145
and it would just immediately
jump across this well.
00:48:45.180 --> 00:48:48.460
This would be a perfectly
good description of the motion
00:48:48.460 --> 00:48:51.020
before it gets to the well,
but after it leaves the well,
00:48:51.020 --> 00:48:57.290
the position is going
to be v0 t plus-- well,
00:48:57.290 --> 00:48:58.520
what's the time shift?
00:48:58.520 --> 00:49:00.430
The time shift is
the time that we
00:49:00.430 --> 00:49:04.080
didn't need to cross this gap.
00:49:04.080 --> 00:49:06.540
And how much would
that time have been?
00:49:06.540 --> 00:49:10.220
Well, it's the distance
divided by the velocity.
00:49:10.220 --> 00:49:13.300
That's the time we didn't need.
00:49:13.300 --> 00:49:16.740
So t plus-- it moves as
if it's at a later time--
00:49:16.740 --> 00:49:20.430
t plus L over v0.
00:49:20.430 --> 00:49:22.944
Cool?
00:49:22.944 --> 00:49:25.360
So if we had a really deep
well and we watched a particle,
00:49:25.360 --> 00:49:29.120
we would watch it move x0,
x v0 t, v0 t, v0 t, v0 t
00:49:29.120 --> 00:49:33.550
plus L over v, v0
t plus L over v.
00:49:33.550 --> 00:49:36.560
So it's the time that we made
up by being deep in the well.
00:49:36.560 --> 00:49:40.450
So there's a classical picture
because it goes faster inside.
00:49:40.450 --> 00:49:44.520
So comparing these, here
we've done a calculation
00:49:44.520 --> 00:49:48.760
of the time shift due
to the quantum particle
00:49:48.760 --> 00:49:54.390
quantum mechanically
transiting the potential well.
00:49:54.390 --> 00:49:55.430
So let's compare these.
00:49:58.420 --> 00:50:02.530
This says the classical
prediction is the time it took,
00:50:02.530 --> 00:50:08.130
delta t, classical is
equal to L over v0,
00:50:08.130 --> 00:50:10.940
and the question is,
is this the same?
00:50:10.940 --> 00:50:13.880
One way to phrase this question
is, is it the same as h bar d
00:50:13.880 --> 00:50:21.350
phi dE evaluated at k0?
00:50:28.100 --> 00:50:35.290
From our results last time,
for the amplitude c over a,
00:50:35.290 --> 00:50:39.860
we get that phi is equal
to-- which is just minus
00:50:39.860 --> 00:50:46.430
the argument, or the phase, of
c over a, or of the transmission
00:50:46.430 --> 00:50:48.254
amplitude little
t-- phi turns out
00:50:48.254 --> 00:50:58.670
to be equal to k2 L minus arctan
of k1 squared plus k2 squared
00:50:58.670 --> 00:51:06.290
over 2 k1 k2, tan of
k2 L. I look at this
00:51:06.290 --> 00:51:07.870
and it doesn't tell
me all that much.
00:51:07.870 --> 00:51:10.444
It's a little bit bewildering,
so let's unpack this.
00:51:10.444 --> 00:51:11.860
What we really
want to know is, is
00:51:11.860 --> 00:51:13.318
this close to the
classical result?
00:51:20.050 --> 00:51:22.120
Here's a quick way to check.
00:51:22.120 --> 00:51:24.630
We know this expression is
going to simplify near resonance
00:51:24.630 --> 00:51:27.100
where the sine vanishes,
so let's look, just
00:51:27.100 --> 00:51:31.130
for simplicity,
near the resonance.
00:51:31.130 --> 00:51:34.380
And in particular, let's
look near the resonance k2 L
00:51:34.380 --> 00:51:36.974
is equal to n pi.
00:51:36.974 --> 00:51:39.140
Then it turns out that a
quick calculation gives you
00:51:39.140 --> 00:51:43.840
that h bar d phi dE
at this value of k,
00:51:43.840 --> 00:51:50.540
at that value of the energy,
goes as L over 2 v0 times
00:51:50.540 --> 00:51:56.840
1 plus the energy over
the depth of the well, v0.
00:51:56.840 --> 00:52:00.070
Now remember, the classical
approximation was L over v0.
00:52:00.070 --> 00:52:01.330
We just did this very quickly.
00:52:01.330 --> 00:52:03.310
We did it assuming an
arbitrarily deep well.
00:52:03.310 --> 00:52:05.820
So v0 is arbitrarily
larger in magnitude than E,
00:52:05.820 --> 00:52:07.222
so this term is negligible.
00:52:07.222 --> 00:52:09.180
Where we should compare
that very simple, naive
00:52:09.180 --> 00:52:12.510
classical result is
here, L over 2 v0.
00:52:12.510 --> 00:52:15.180
And what we see is that the
quantum mechanical result
00:52:15.180 --> 00:52:18.030
gives a time shift which
is down by a factor of 2.
00:52:21.920 --> 00:52:22.880
So what's going on?
00:52:22.880 --> 00:52:26.950
Well, apparently, the
things slow down inside.
00:52:26.950 --> 00:52:29.630
The time that it
took us to cross
00:52:29.630 --> 00:52:31.810
was greater than you
would have naively guessed
00:52:31.810 --> 00:52:34.530
by making it arbitrarily deep.
00:52:34.530 --> 00:52:37.590
And we can make that
a little more sharp
00:52:37.590 --> 00:52:46.380
by plotting, as a function of
E over v0, the actual phase
00:52:46.380 --> 00:52:47.984
shift.
00:52:47.984 --> 00:52:50.150
If do a better job than
saying it's infinitely deep,
00:52:50.150 --> 00:52:55.720
the classical prediction
looks something like this,
00:52:55.720 --> 00:53:01.109
and this is for delta t,
the time shift, classical.
00:53:01.109 --> 00:53:03.400
When you look at the correct
quantum mechanical result,
00:53:03.400 --> 00:53:19.920
here's what you find,
where the difference is
00:53:19.920 --> 00:53:24.570
a factor of 2, 1/2 the
height down, and again,
00:53:24.570 --> 00:53:26.450
1/2 the height down.
00:53:26.450 --> 00:53:28.810
So this is that factor
of 2 downstairs.
00:53:28.810 --> 00:53:32.442
So the wave packet goes
actually a little bit faster
00:53:32.442 --> 00:53:33.900
than the classical
prediction would
00:53:33.900 --> 00:53:36.440
guess except near
resonance, and these
00:53:36.440 --> 00:53:38.815
are at the resonant
values of the momentum.
00:53:38.815 --> 00:53:40.440
At the resonant values
of the momentum,
00:53:40.440 --> 00:53:41.971
it takes much longer
to get across.
00:53:41.971 --> 00:53:43.470
Instead of going a
little bit faster
00:53:43.470 --> 00:53:45.430
than the classical
result, it goes a factor
00:53:45.430 --> 00:53:48.570
of 2 slower than the
classical result.
00:53:48.570 --> 00:53:50.050
And so now I ask
you the question,
00:53:50.050 --> 00:53:51.450
why is it going more slowly?
00:53:51.450 --> 00:53:55.190
Why does it spend so much more
time inside the well quantum
00:53:55.190 --> 00:53:57.810
mechanically than it
would have classically?
00:53:57.810 --> 00:54:00.830
Why is the particle effectively
taking so much longer
00:54:00.830 --> 00:54:03.990
to transit the well
near resonance?
00:54:03.990 --> 00:54:06.449
AUDIENCE: Because it can
reflect and it can keep going
00:54:06.449 --> 00:54:08.490
and a classical particle
is not going to do that.
00:54:08.490 --> 00:54:09.280
PROFESSOR: Yeah, exactly.
00:54:09.280 --> 00:54:10.950
So the classical particle
just goes across.
00:54:10.950 --> 00:54:13.160
The quantum mechanical
particle has a superposition
00:54:13.160 --> 00:54:14.534
of contributions
to its amplitude
00:54:14.534 --> 00:54:17.792
where it transits-- transit,
bounce, bounce, transit,
00:54:17.792 --> 00:54:20.560
transit, bounce, bounce,
bounce, bounce, transit.
00:54:20.560 --> 00:54:22.250
And now you can
ask, how much time
00:54:22.250 --> 00:54:25.540
was spent by each of those
imaginary particles imaginarily
00:54:25.540 --> 00:54:27.005
moving across?
00:54:27.005 --> 00:54:29.380
And if you're careful about
how you set up that question,
00:54:29.380 --> 00:54:33.740
you can recover this factor of
2, which is kind of beautiful.
00:54:33.740 --> 00:54:36.640
But the important thing here is
when you're hitting resonance,
00:54:36.640 --> 00:54:39.780
the multiple scattering
processes are important.
00:54:39.780 --> 00:54:41.040
They're not canceling out.
00:54:41.040 --> 00:54:42.270
They're not at random phase.
00:54:42.270 --> 00:54:44.030
They're not interfering
destructively with each other.
00:54:44.030 --> 00:54:45.488
They're interfering
constructively,
00:54:45.488 --> 00:54:47.700
and you get perfect
transmission precisely
00:54:47.700 --> 00:54:51.160
because of the
constructive interference
00:54:51.160 --> 00:54:53.050
of an infinite number
of contributions
00:54:53.050 --> 00:54:54.882
to the quantum
mechanical amplitude.
00:54:54.882 --> 00:54:56.840
And this is, again, we're
seeing the same thing
00:54:56.840 --> 00:54:59.550
in this annoying slow down.
00:54:59.550 --> 00:55:01.150
This tells us another
thing, though,
00:55:01.150 --> 00:55:04.110
which is that the
scattering phase, the phase
00:55:04.110 --> 00:55:06.650
in the transmission
amplitude, contains
00:55:06.650 --> 00:55:08.400
an awful lot of the
physics of the system.
00:55:08.400 --> 00:55:12.350
It's telling us about how long
it takes for the wave packet
00:55:12.350 --> 00:55:14.680
to transit across the
potential, effectively.
00:55:14.680 --> 00:55:15.347
Yeah?
00:55:15.347 --> 00:55:16.600
AUDIENCE: What's the
vertical axis being used for?
00:55:16.600 --> 00:55:17.308
PROFESSOR: Sorry.
00:55:17.308 --> 00:55:21.620
The vertical axis
here is the shift
00:55:21.620 --> 00:55:24.300
in the time due to the fact
that it went across this well
00:55:24.300 --> 00:55:26.700
and went a little
bit faster inside.
00:55:26.700 --> 00:55:30.580
So empirically, what it means is
when you get out very far away
00:55:30.580 --> 00:55:33.070
and you watch the motion
of the wave packet,
00:55:33.070 --> 00:55:35.794
and you ask, how long
has it been since it got
00:55:35.794 --> 00:55:37.210
to the barrier in
the first place,
00:55:37.210 --> 00:55:38.640
it took less time than
you would have guessed
00:55:38.640 --> 00:55:41.400
by knowing that its velocity is
v0, and the amount of time less
00:55:41.400 --> 00:55:42.757
is this much time.
00:55:42.757 --> 00:55:43.840
That answer your question?
00:55:43.840 --> 00:55:44.720
Good.
00:55:44.720 --> 00:55:45.972
Yeah?
00:55:45.972 --> 00:55:48.710
AUDIENCE: Why is each
of the amplitudes 1/2?
00:55:48.710 --> 00:55:50.880
Why is the first one, and
it bounces around twice.
00:55:54.847 --> 00:55:56.930
PROFESSOR: When we compare
the classical amplitude
00:55:56.930 --> 00:56:00.140
and the limit that
v0 goes to infinity?
00:56:00.140 --> 00:56:03.660
The comparison is
just a factor of 2.
00:56:03.660 --> 00:56:07.150
It's more complicated out here.
00:56:07.150 --> 00:56:10.428
In the limit that v0 is
large, this is exactly 1/2.
00:56:10.428 --> 00:56:11.344
AUDIENCE: [INAUDIBLE].
00:56:13.779 --> 00:56:15.320
PROFESSOR: The
resonances are leading
00:56:15.320 --> 00:56:16.486
to this extra factor of 1/2.
00:56:18.907 --> 00:56:20.490
I have to say I don't
remember exactly
00:56:20.490 --> 00:56:24.020
whether, as you include
the sub-leading terms of 1
00:56:24.020 --> 00:56:26.590
over v0, whether it stays
1/2 or whether it doesn't,
00:56:26.590 --> 00:56:28.480
but in the limit
that v0 is large,
00:56:28.480 --> 00:56:30.500
it remains either close
to 1/2 or exactly 1/2,
00:56:30.500 --> 00:56:31.750
I just don't remember exactly.
00:56:31.750 --> 00:56:34.000
The important thing is
that there's a sharp dip.
00:56:34.000 --> 00:56:38.680
It takes much longer to transit,
and so you get less bonus time,
00:56:38.680 --> 00:56:40.240
as it were.
00:56:40.240 --> 00:56:43.370
You've gained less time in
the quantum mechanical model
00:56:43.370 --> 00:56:44.822
than the classical model.
00:56:44.822 --> 00:56:47.280
And when you do the experiment,
you get the quantum result.
00:56:50.940 --> 00:56:52.580
That's the crucial point.
00:56:52.580 --> 00:56:53.260
Other questions?
00:56:56.680 --> 00:56:59.290
So the phase contains an
awful lot of the physics.
00:57:04.240 --> 00:57:08.440
So I want to generalize
this whole story
00:57:08.440 --> 00:57:11.920
in a very particular way,
this way of reorganizing
00:57:11.920 --> 00:57:13.140
the scattering in 1D.
00:57:13.140 --> 00:57:14.060
What we're doing
right now is we're
00:57:14.060 --> 00:57:16.060
studying scattering
problems in one dimension,
00:57:16.060 --> 00:57:17.900
but we live in three dimensions.
00:57:17.900 --> 00:57:19.983
The story is going to be
more complicated in three
00:57:19.983 --> 00:57:20.483
dimensions.
00:57:20.483 --> 00:57:22.649
It's going to be more
complicated in two dimensions,
00:57:22.649 --> 00:57:24.220
but the basic ideas
are all the same.
00:57:24.220 --> 00:57:26.219
It's just the details are
going to be different.
00:57:26.219 --> 00:57:29.420
And one thing that turns out
to be very useful in organizing
00:57:29.420 --> 00:57:31.449
scattering, both
in one dimension
00:57:31.449 --> 00:57:32.990
and in three
dimensions, is something
00:57:32.990 --> 00:57:34.198
called the scattering matrix.
00:57:34.198 --> 00:57:35.870
I'm going to talk
about that now.
00:57:35.870 --> 00:57:39.380
In three dimensions,
it's essential,
00:57:39.380 --> 00:57:41.720
but even in one dimension,
where it's usually not used,
00:57:41.720 --> 00:57:43.360
it's a very powerful
way to organize
00:57:43.360 --> 00:57:49.350
our knowledge of the system as
encoded by the scattering data.
00:57:49.350 --> 00:57:51.000
So here's the basic idea.
00:57:51.000 --> 00:57:52.800
As we discussed
before, what we really
00:57:52.800 --> 00:57:55.420
want to do in the ideal world
is take some unknown potential
00:57:55.420 --> 00:57:58.510
in some bounded
region, some region,
00:57:58.510 --> 00:58:01.660
and outside, we have the
potential is constant.
00:58:01.660 --> 00:58:03.300
[INAUDIBLE] my bad
artistic skills.
00:58:03.300 --> 00:58:06.740
So potential is
constant out here.
00:58:06.740 --> 00:58:08.990
And the potential could be
some horrible thing in here
00:58:08.990 --> 00:58:10.490
that we don't happen
to know, and we
00:58:10.490 --> 00:58:12.530
want to read off of
the scattering process,
00:58:12.530 --> 00:58:15.500
we want to be able to deduce
something about the potential.
00:58:15.500 --> 00:58:26.190
So for example, we
can deduce the energy
00:58:26.190 --> 00:58:30.920
by looking at the
position of the barriers,
00:58:30.920 --> 00:58:33.380
and we can disentangle
the position of the energy
00:58:33.380 --> 00:58:37.350
and the depth and the width
by looking at the phase shift,
00:58:37.350 --> 00:58:38.680
by looking at the time delay.
00:58:38.680 --> 00:58:40.940
So we can deduce all the
parameters of our potential
00:58:40.940 --> 00:58:43.830
by looking at the resonance
points and the phase shift.
00:58:43.830 --> 00:58:46.665
I want to do this more
generally for general potential.
00:58:46.665 --> 00:58:48.040
And to set that
up, we need to be
00:58:48.040 --> 00:58:50.150
a bit more general
than we've been.
00:58:54.060 --> 00:58:56.730
So in general, if we
solve this potential,
00:58:56.730 --> 00:59:02.270
as we talked about before, we
have A and B, e to the ikx,
00:59:02.270 --> 00:59:05.850
e to the minus ikx out
here, and out here, we
00:59:05.850 --> 00:59:11.278
have C e to the ikx and
D e to the minus ikx.
00:59:14.235 --> 00:59:16.110
And I'm not going to
ask what happens inside.
00:59:19.377 --> 00:59:20.960
Now we can do, as
discussed, two kinds
00:59:20.960 --> 00:59:22.043
of scattering experiments.
00:59:22.043 --> 00:59:24.220
We can send things in from
the left, in which case
00:59:24.220 --> 00:59:27.100
A is nonzero, and then things
can either transmit or reflect,
00:59:27.100 --> 00:59:29.755
but nothing's going to come
in from infinity, so D is 0.
00:59:29.755 --> 00:59:31.130
Or we can do the
same in reverse.
00:59:31.130 --> 00:59:33.254
Send things in from here,
and that corresponds to A
00:59:33.254 --> 00:59:36.630
is 0, nothing coming in
this way but d0 and 0.
00:59:36.630 --> 00:59:39.900
And more generally, we can
ask the following question.
00:59:39.900 --> 00:59:42.950
Suppose I send some amount
of stuff in from the left
00:59:42.950 --> 00:59:46.010
and I send some amount of stuff
in from the right, D. Then
00:59:46.010 --> 00:59:48.650
that will tell me how much stuff
will be going out to the right
00:59:48.650 --> 00:59:52.150
and how much stuff will
be going out to the left.
00:59:52.150 --> 00:59:54.040
If you tell me how
much is coming in,
00:59:54.040 --> 00:59:59.440
I will tell you how much
is coming out, B, C.
00:59:59.440 --> 01:00:02.630
So if you could
solve this problem,
01:00:02.630 --> 01:00:05.480
the answer is just some
paralinear relations
01:00:05.480 --> 01:00:09.080
between these, and we can write
this as a matrix, which I will
01:00:09.080 --> 01:00:15.030
S11, S12, S21, S22.
01:00:15.030 --> 01:00:17.650
What this matrix is doing is
it takes the amplitude you're
01:00:17.650 --> 01:00:19.120
sending in from
the left to right
01:00:19.120 --> 01:00:21.344
and tells you the amplitude
coming out to the left
01:00:21.344 --> 01:00:22.010
or to the right.
01:00:22.010 --> 01:00:22.833
Yes?
01:00:22.833 --> 01:00:25.870
AUDIENCE: How do we know
this relation is linear?
01:00:25.870 --> 01:00:28.850
PROFESSOR: If you double the
amount of stuff coming in,
01:00:28.850 --> 01:00:31.030
then you must double the
amount of stuff going out
01:00:31.030 --> 01:00:32.420
or probability is not conserved.
01:00:35.070 --> 01:00:37.120
Also, we've derived
these relations.
01:00:37.120 --> 01:00:38.730
You know how the relations work.
01:00:38.730 --> 01:00:42.769
The relations work by satisfying
a series of linear equations
01:00:42.769 --> 01:00:44.310
between the various
coefficients such
01:00:44.310 --> 01:00:46.855
that you have continuity
and differentiability at all
01:00:46.855 --> 01:00:48.634
the matching points.
01:00:48.634 --> 01:00:50.050
But the crucial
thing of linearity
01:00:50.050 --> 01:00:54.034
is probability is conserved
and time evolution is linear.
01:00:54.034 --> 01:00:54.700
Other questions?
01:00:58.540 --> 01:01:01.100
So this is just some stupid
matrix, and we call it,
01:01:01.100 --> 01:01:05.505
not surprisingly, the S
matrix in all of its majesty.
01:01:09.510 --> 01:01:10.810
The basic idea is this.
01:01:10.810 --> 01:01:13.685
For scattering problems, if
someone tells you the S matrix,
01:01:13.685 --> 01:01:15.310
and in particular,
if someone tells you
01:01:15.310 --> 01:01:18.892
how all the coefficients of
the S matrix vary with energy,
01:01:18.892 --> 01:01:21.100
then you've completely solved
any scattering problem.
01:01:21.100 --> 01:01:22.950
You tell me what A and D are.
01:01:22.950 --> 01:01:23.450
Great.
01:01:23.450 --> 01:01:26.071
I'll tell you exactly what
B and C are mode by mode,
01:01:26.071 --> 01:01:27.570
and I can do this
for superposition.
01:01:27.570 --> 01:01:30.377
So this allows you to completely
solve any scattering problem
01:01:30.377 --> 01:01:31.960
in quantum mechanics
once you know it.
01:01:31.960 --> 01:01:35.176
So it suffices to know S to
solve all scattering problems.
01:01:38.270 --> 01:01:40.520
I want to now spend just a
little bit of time thinking
01:01:40.520 --> 01:01:43.310
about what properties the
S matrix and its components
01:01:43.310 --> 01:01:45.580
must satisfy.
01:01:45.580 --> 01:01:47.210
What properties
must the S matrix
01:01:47.210 --> 01:01:51.400
satisfy in order to
jibe well with the rest
01:01:51.400 --> 01:01:54.610
of the rules of
quantum mechanics?
01:01:54.610 --> 01:01:56.760
I'm not going to study
any particular system.
01:01:56.760 --> 01:01:59.480
I just want to ask
general questions.
01:01:59.480 --> 01:02:01.510
So the first thing
that must be true
01:02:01.510 --> 01:02:04.285
is that stuff doesn't disappear.
01:02:11.950 --> 01:02:16.120
Disapparate is probably
the appropriate.
01:02:16.120 --> 01:02:18.870
Stuff doesn't leak
out of the world,
01:02:18.870 --> 01:02:20.490
so whatever goes
in must come out.
01:02:20.490 --> 01:02:23.760
What that means is norm of
A squared plus norm of D
01:02:23.760 --> 01:02:26.970
squared, which is the
probability density in
01:02:26.970 --> 01:02:29.030
and probability
density out, must
01:02:29.030 --> 01:02:32.000
be equal to B squared
plus C squared.
01:02:37.160 --> 01:02:39.422
Everyone agree with that?
01:02:39.422 --> 01:02:42.340
AUDIENCE: Why can't stuff
stay in the potential?
01:02:42.340 --> 01:02:44.960
PROFESSOR: Why can't stuff
stay in the potential?
01:02:44.960 --> 01:02:46.350
That's a good.
01:03:00.750 --> 01:03:03.260
So if we're looking at
fixed energy eigenstates,
01:03:03.260 --> 01:03:05.070
we know that we're in
a stationary state,
01:03:05.070 --> 01:03:09.200
so whatever the amplitude
going into the middle is,
01:03:09.200 --> 01:03:12.187
it must also be coming
out of the middle.
01:03:12.187 --> 01:03:13.770
There's another way
to say this, which
01:03:13.770 --> 01:03:16.520
is let's think about it not
in terms of individual energy
01:03:16.520 --> 01:03:17.020
eigenstates.
01:03:17.020 --> 01:03:18.936
Let's think about it in
terms of wave packets.
01:03:18.936 --> 01:03:20.590
So if we take a
wave packet of stuff
01:03:20.590 --> 01:03:23.400
and we send in that wave packet,
it has some momentum, right?
01:03:26.280 --> 01:03:28.190
This is going to get
delicate and technical.
01:03:28.190 --> 01:03:31.040
Let me just stick with
the first statement, which
01:03:31.040 --> 01:03:33.350
is that if stuff
is going in, then
01:03:33.350 --> 01:03:36.140
it has to also be
coming out by the fact
01:03:36.140 --> 01:03:38.150
that this is an
energy eigenstate.
01:03:38.150 --> 01:03:39.650
The overall probability
distribution
01:03:39.650 --> 01:03:40.810
is not changing in time.
01:03:43.324 --> 01:03:44.990
If stuff went in and
it didn't come out,
01:03:44.990 --> 01:03:45.770
that would mean
it's staying there.
01:03:45.770 --> 01:03:46.700
That would mean
that the probability
01:03:46.700 --> 01:03:48.060
density is changing in time.
01:03:48.060 --> 01:03:50.060
That's not what happens
in an energy eigenstate.
01:03:50.060 --> 01:03:53.631
The probability distribution
is time independent.
01:03:53.631 --> 01:03:54.630
Everyone cool with that?
01:04:02.380 --> 01:04:04.570
So let's think, though,
about what this is.
01:04:04.570 --> 01:04:07.080
I can write this in
the following nice way.
01:04:07.080 --> 01:04:12.440
I can write this as A
complex conjugate D complex
01:04:12.440 --> 01:04:16.522
conjugate, AD.
01:04:16.522 --> 01:04:17.980
And on the right
hand side, this is
01:04:17.980 --> 01:04:23.525
equal to B complex conjugate,
C complex conjugate, BC.
01:04:23.525 --> 01:04:25.400
I have done nothing
other than write this out
01:04:25.400 --> 01:04:27.440
in some suggested form.
01:04:27.440 --> 01:04:31.760
But B and C are equal
to the S matrix,
01:04:31.760 --> 01:04:36.380
so BC is the S matrix times
A, and B complex conjugate,
01:04:36.380 --> 01:04:39.550
C complex conjugate is the
transposed complex conjugate.
01:04:39.550 --> 01:04:44.970
So this is equal to A complex
conjugate, D complex conjugate,
01:04:44.970 --> 01:04:48.620
S transpose complex conjugate,
also known as adjoint,
01:04:48.620 --> 01:04:51.370
and this is S on AD.
01:04:55.010 --> 01:04:57.100
Yeah?
01:04:57.100 --> 01:05:02.360
But this has to be equal
to this for any A and D.
01:05:02.360 --> 01:05:06.750
So what must be true of
S dagger S as a matrix?
01:05:06.750 --> 01:05:11.230
It's got to be the identity
as a matrix in order for this
01:05:11.230 --> 01:05:14.450
to be true for all A and D. Ah.
01:05:14.450 --> 01:05:15.170
That's cool.
01:05:15.170 --> 01:05:16.500
Stuff doesn't disappear.
01:05:16.500 --> 01:05:17.970
S is a unitary matrix.
01:05:23.640 --> 01:05:24.980
So S is a unitary matrix.
01:05:31.090 --> 01:05:33.290
Its inverse is its adjoint.
01:05:43.110 --> 01:05:47.530
You'll study this in a little
more detail on the problem set.
01:05:47.530 --> 01:05:49.190
You studied the
definition of unitary
01:05:49.190 --> 01:05:52.316
on the last problem set.
01:05:52.316 --> 01:05:53.690
So that's the
first thing about S
01:05:53.690 --> 01:05:55.500
and it turns out to
be completely general.
01:05:55.500 --> 01:05:57.990
Anytime, whether you're in one
dimension or two dimensions
01:05:57.990 --> 01:06:01.550
or three, if you send stuff
in, it should not get stuck.
01:06:01.550 --> 01:06:03.360
It should come out.
01:06:03.360 --> 01:06:06.064
And when it does come
out, the statement
01:06:06.064 --> 01:06:07.730
that it comes out for
energy eigenstates
01:06:07.730 --> 01:06:10.850
is the statement that
S is a unitary matrix.
01:06:10.850 --> 01:06:11.876
Questions on that.
01:06:17.840 --> 01:06:29.760
So a consequence of that is
that the eigenvalues of S
01:06:29.760 --> 01:06:32.510
are phases, pure phases.
01:06:35.020 --> 01:06:38.800
So I can write S--
I'll write them
01:06:38.800 --> 01:06:43.480
as S1 is equal to
e to the i of phi 1
01:06:43.480 --> 01:06:47.824
and S2 is equal
to the i of phi 2.
01:06:53.090 --> 01:06:56.970
So the statement that
S is a unitary matrix
01:06:56.970 --> 01:07:00.362
leads to constraints
on the coefficients,
01:07:00.362 --> 01:07:02.570
and you're going to derive
these on your problem set.
01:07:02.570 --> 01:07:03.910
I'm just going to list them now.
01:07:03.910 --> 01:07:07.960
The first is that
the magnitude of S11
01:07:07.960 --> 01:07:11.840
is equal to the
magnitude of S22.
01:07:11.840 --> 01:07:15.970
The magnitude of S12 is equal
to the magnitude of S21.
01:07:24.200 --> 01:07:31.880
More importantly, S12 norm
squared plus S11 squared
01:07:31.880 --> 01:07:33.990
is equal to 1.
01:07:33.990 --> 01:07:43.670
And finally, S11, S12 complex
conjugate, plus S21, S22
01:07:43.670 --> 01:07:47.010
complex conjugate is equal to 0.
01:07:47.010 --> 01:07:51.820
So what do these mean?
01:07:51.820 --> 01:07:53.453
What are these
conditions telling us?
01:07:56.877 --> 01:07:58.460
They're telling us,
of course, they're
01:07:58.460 --> 01:08:00.490
a consequence of
conservational probability,
01:08:00.490 --> 01:08:01.950
but they have another meaning.
01:08:01.950 --> 01:08:03.491
To get that other
meaning, let's look
01:08:03.491 --> 01:08:07.230
at the definition of the
transmission amplitudes.
01:08:07.230 --> 01:08:09.890
In particular, consider
the case that we send stuff
01:08:09.890 --> 01:08:13.390
in from the left and
nothing in from the right.
01:08:13.390 --> 01:08:16.830
That corresponds to D equals 0.
01:08:16.830 --> 01:08:18.710
When D equals 0, what
does this tell us?
01:08:18.710 --> 01:08:20.899
It tells us that B
is equal to-- and A
01:08:20.899 --> 01:08:22.119
equals 1 for normalization.
01:08:24.740 --> 01:08:30.810
B is equal to-- well, D is
equal to 0, so it's just S11 A,
01:08:30.810 --> 01:08:35.029
so B over A is S11.
01:08:35.029 --> 01:08:40.601
And similarly, C over A is S21.
01:08:40.601 --> 01:08:42.100
But C over A is the
thing that we've
01:08:42.100 --> 01:08:45.720
been calling the transmission
amplitude, little t,
01:08:45.720 --> 01:08:48.109
and this is the reflection
amplitude, little r.
01:08:48.109 --> 01:08:51.140
So this is the reflection
amplitude, if we send in stuff
01:08:51.140 --> 01:08:53.479
and it bounces off,
reflects to the left,
01:08:53.479 --> 01:08:55.869
and this is the transmission
amplitude for transmitting
01:08:55.869 --> 01:08:56.410
to the right.
01:08:58.529 --> 01:08:59.529
Everyone cool with that?
01:09:02.279 --> 01:09:04.380
This is reflection
to the left, this
01:09:04.380 --> 01:09:06.085
is transmission to the right.
01:09:06.085 --> 01:09:08.210
By the same token, this is
going to be transmission
01:09:08.210 --> 01:09:10.060
to the left and
reflection to the right.
01:09:12.680 --> 01:09:14.840
So now let's look
at these conditions.
01:09:14.840 --> 01:09:17.350
S11 is reflection and this
is going to be reflection.
01:09:17.350 --> 01:09:19.350
This says that the
reflection to the left
01:09:19.350 --> 01:09:22.875
is equal to the reflection
to the right in magnitude.
01:09:25.910 --> 01:09:30.149
Before, what we saw was
for the simple step,
01:09:30.149 --> 01:09:32.660
the reflection amplitude was
equal from left to right, not
01:09:32.660 --> 01:09:34.430
just the magnitude,
but the actual value
01:09:34.430 --> 01:09:36.100
was equal from the
left and the right.
01:09:36.100 --> 01:09:38.350
It was a little bit of a
cheat because they were real.
01:09:40.217 --> 01:09:41.800
And we saw that that
was a consequence
01:09:41.800 --> 01:09:42.870
of just being the step.
01:09:42.870 --> 01:09:43.970
We didn't know
anything more about it.
01:09:43.970 --> 01:09:45.595
But now we see that
on general grounds,
01:09:45.595 --> 01:09:47.309
on conservation of
probability grounds,
01:09:47.309 --> 01:09:49.100
the magnitude of the
reflection to the left
01:09:49.100 --> 01:09:52.510
and to the right for any
potential had better be equal.
01:09:52.510 --> 01:09:57.490
And similarly, the magnitude of
the transmission for the left
01:09:57.490 --> 01:10:03.120
and the transmission to the
right had better be equal,
01:10:03.120 --> 01:10:06.540
all other things being-- if
you're sending in from the left
01:10:06.540 --> 01:10:08.140
and then transmitting
to the left,
01:10:08.140 --> 01:10:10.598
or sending in from the right
and transmitting to the right.
01:10:10.598 --> 01:10:12.170
And what does this one tell us?
01:10:12.170 --> 01:10:18.130
Well, S12 and S11, that tells
us that little t squared, which
01:10:18.130 --> 01:10:23.890
is the total probability to
transmit, plus little r squared
01:10:23.890 --> 01:10:27.040
is the total probability
to reflect, is equal to 1,
01:10:27.040 --> 01:10:28.350
and we saw this last time, too.
01:10:32.360 --> 01:10:35.630
This was the earlier definition
of nothing gets stuck.
01:10:35.630 --> 01:10:39.771
And this one you'll study
on your problem set.
01:10:39.771 --> 01:10:41.254
It's a little more subtle.
01:10:45.440 --> 01:10:46.935
Questions?
01:10:46.935 --> 01:10:47.435
Yeah?
01:10:47.435 --> 01:10:50.345
AUDIENCE: Can you explain one
more time why is S unitary?
01:10:50.345 --> 01:10:51.777
How did you get that?
01:10:51.777 --> 01:10:54.110
PROFESSOR: So the way we got
S was unitary is first off,
01:10:54.110 --> 01:10:55.740
this is just the
definition of S. S
01:10:55.740 --> 01:10:57.800
is the matrix that,
for any energy,
01:10:57.800 --> 01:11:00.500
relates A and D, the
ingoing amplitudes,
01:11:00.500 --> 01:11:03.830
to the outgoing amplitudes B
and C. Just the definition.
01:11:03.830 --> 01:11:06.640
Meanwhile, I claim that
stuff doesn't go away,
01:11:06.640 --> 01:11:09.010
nothing gets stuck,
nothing disappears,
01:11:09.010 --> 01:11:11.350
so the total probability
density of stuff going in
01:11:11.350 --> 01:11:12.550
must be equal to the
total probability
01:11:12.550 --> 01:11:13.675
density of stuff going out.
01:11:15.826 --> 01:11:17.200
The probability
of stuff going in
01:11:17.200 --> 01:11:19.970
can be expressed as this
row vector times this column
01:11:19.970 --> 01:11:23.020
vector, and on the out, this
row vector times this column
01:11:23.020 --> 01:11:25.560
vector, and then we use the
definition of the S matrix.
01:11:25.560 --> 01:11:28.590
This column vector is equal
to S times this row vector.
01:11:31.800 --> 01:11:34.090
BC is equal to S times AD.
01:11:34.090 --> 01:11:36.470
And when we take the
transposed complex conjugate,
01:11:36.470 --> 01:11:39.050
I get AD transposed
complex conjugate,
01:11:39.050 --> 01:11:42.040
S transposed complex conjugate,
but that's the S adjoint.
01:11:42.040 --> 01:11:46.730
But in order for this to be
true for any vectors A and D,
01:11:46.730 --> 01:11:50.340
it must be that S dagger S
is unitary is the identity,
01:11:50.340 --> 01:11:53.340
but that's the definition
of a unitary matrix.
01:11:53.340 --> 01:11:55.270
Cool?
01:11:55.270 --> 01:11:55.770
Others.
01:11:59.240 --> 01:12:01.100
You're going to prove
a variety of things
01:12:01.100 --> 01:12:07.370
on the problem set
about the scattering
01:12:07.370 --> 01:12:08.930
matrix and its
coefficients, but I
01:12:08.930 --> 01:12:12.600
want to show you two
properties of it.
01:12:12.600 --> 01:12:17.630
The first is reasonably
tame, and it'll
01:12:17.630 --> 01:12:21.135
make a little sharper the
step result we got earlier
01:12:21.135 --> 01:12:23.260
that the reflection in both
directions off the step
01:12:23.260 --> 01:12:25.170
potential was in fact
exactly the same.
01:12:34.974 --> 01:12:36.890
Suppose our system is
time reversal invariant.
01:12:40.390 --> 01:12:43.990
So if t goes to minus
t, nothing changes.
01:12:43.990 --> 01:12:45.490
This would not be
true, for example,
01:12:45.490 --> 01:12:47.365
if we had electric
currents in our system
01:12:47.365 --> 01:12:50.797
because as we take t to minus
t, then the current reverses.
01:12:50.797 --> 01:12:52.630
So if the current shows
up in the potential,
01:12:52.630 --> 01:12:54.140
or if a magnetic
field due to a current
01:12:54.140 --> 01:12:55.620
shows up in the
potential energy,
01:12:55.620 --> 01:12:57.090
then as we change
t to minus t, we
01:12:57.090 --> 01:12:58.590
change the direction
of the current,
01:12:58.590 --> 01:13:01.250
we change the direction
of the magnetic field.
01:13:01.250 --> 01:13:04.879
In simple systems where we
have time reversal invariance,
01:13:04.879 --> 01:13:06.420
for example,
electrostatics, but not,
01:13:06.420 --> 01:13:09.590
for example, magnetostatics,
suppose we have time reversal
01:13:09.590 --> 01:13:13.560
invariance, then what you've
shown on a previous problem
01:13:13.560 --> 01:13:19.060
set is that psi is a
solution, then psi star,
01:13:19.060 --> 01:13:20.820
psi complex conjugate
is also a solution.
01:13:23.920 --> 01:13:26.882
And using these, what you'll
see what you can show--
01:13:26.882 --> 01:13:28.590
and I'm not going to
go through the steps
01:13:28.590 --> 01:13:31.750
for this-- well, that's easy.
01:13:31.750 --> 01:13:35.010
If we do the time reversal,
the wave function,
01:13:35.010 --> 01:13:39.160
looking on the left or on the
right, so comparing these guys,
01:13:39.160 --> 01:13:40.650
what changes?
01:13:40.650 --> 01:13:43.244
Under time reversal,
we get a solution.
01:13:43.244 --> 01:13:45.160
Given this solution, we
have another solution,
01:13:45.160 --> 01:13:49.990
A star e to the minus ikx,
B star e to the plus ikx.
01:13:49.990 --> 01:13:55.082
Minus star, star, plus.
01:13:55.082 --> 01:13:56.540
So we can run
exactly the same game
01:13:56.540 --> 01:13:58.168
but now with this amplitude.
01:14:02.760 --> 01:14:05.680
And when you put the
conditions together,
01:14:05.680 --> 01:14:08.850
which you'll do on the problem
set, another way to say this
01:14:08.850 --> 01:14:12.220
is the same solution with k
to minus k and with A and B
01:14:12.220 --> 01:14:13.780
replaced by their
complex conjugates
01:14:13.780 --> 01:14:17.480
and C and D replaced by each
other's complex conjugates.
01:14:17.480 --> 01:14:19.810
Then this implies
that it must be true
01:14:19.810 --> 01:14:25.840
that A and D, which are now
the outgoing guys because we've
01:14:25.840 --> 01:14:29.780
time reversed, is equal
to S-- and I'll write this
01:14:29.780 --> 01:14:39.440
out explicitly-- S11, S12,
S21, S22, B star, and C star.
01:14:42.530 --> 01:14:44.440
So these together give you that.
01:14:48.950 --> 01:14:53.500
Therefore, S complex
conjugate S is equal to 1.
01:14:57.460 --> 01:14:59.780
If S complex conjugate
S is equal to 1,
01:14:59.780 --> 01:15:05.160
then S inverse is
equal to S transpose,
01:15:05.160 --> 01:15:06.840
just putting this on the right.
01:15:06.840 --> 01:15:12.027
So S transpose is equal to S
inverse is equal to S adjoint
01:15:12.027 --> 01:15:13.110
because S is also unitary.
01:15:15.910 --> 01:15:26.960
So time reversal invariance
implies, for example,
01:15:26.960 --> 01:15:28.590
that S dagger is
equal to S star,
01:15:28.590 --> 01:15:29.673
or S equal to S transpose.
01:15:36.380 --> 01:15:37.880
This is what I
wanted to write here.
01:15:37.880 --> 01:15:40.390
Gives us that S is
equal to S transpose.
01:15:40.390 --> 01:15:46.110
And in particular, this tells
us that S21 is equal to S12.
01:15:46.110 --> 01:15:48.950
The off diagonal
terms are equal,
01:15:48.950 --> 01:15:52.650
not just in magnitude, which
was insured by unitary,
01:15:52.650 --> 01:15:55.457
but if, in addition to
being a unitary system,
01:15:55.457 --> 01:15:57.540
which of course, it should
be, if in addition it's
01:15:57.540 --> 01:16:00.580
time reversal invariant, then we
see that the off diagonal terms
01:16:00.580 --> 01:16:03.104
are equal not just in
magnitude but also in phase.
01:16:03.104 --> 01:16:04.770
And as you know, the
phase is important.
01:16:04.770 --> 01:16:06.480
The phase contains physics.
01:16:06.480 --> 01:16:09.930
It tells you about time delays
and shifts in the scattering
01:16:09.930 --> 01:16:10.576
process.
01:16:10.576 --> 01:16:11.700
So the phases are the same.
01:16:11.700 --> 01:16:13.200
That statement is
not a trivial one.
01:16:13.200 --> 01:16:16.080
It contains physics.
01:16:16.080 --> 01:16:18.960
So when the system is time
reversal invariant, the phases
01:16:18.960 --> 01:16:20.890
as well as the
amplitudes are the same.
01:16:20.890 --> 01:16:23.120
And you'll derive a series
of related conditions
01:16:23.120 --> 01:16:27.482
or consequences for the S
matrix from various properties
01:16:27.482 --> 01:16:28.940
of the system, for
example, parity,
01:16:28.940 --> 01:16:31.670
if you could have a
symmetric potential.
01:16:31.670 --> 01:16:33.375
But now in the last
few minutes, I just
01:16:33.375 --> 01:16:35.920
want to tell you a
really lovely thing.
01:16:35.920 --> 01:16:37.730
So it should be pretty
clear at this point
01:16:37.730 --> 01:16:40.170
that all the information
about scattering
01:16:40.170 --> 01:16:42.806
is contained in the S matrix and
its dependence on the energy.
01:16:42.806 --> 01:16:44.680
If you know what the
incident amplitudes are,
01:16:44.680 --> 01:16:47.236
you know what the
outgoing amplitudes are,
01:16:47.236 --> 01:16:49.110
and that's cool because
you can measure this.
01:16:49.110 --> 01:16:51.191
You can take a potential,
you can literally just
01:16:51.191 --> 01:16:52.690
send in a beam of
particles, and you
01:16:52.690 --> 01:16:54.970
can ask, how likely
are they to get out.
01:16:54.970 --> 01:16:59.180
And more importantly, if I
build a wave packet, on average,
01:16:59.180 --> 01:17:02.049
what's the time
delay or acceleration
01:17:02.049 --> 01:17:03.340
of the transmitted wave packet?
01:17:03.340 --> 01:17:04.890
And that way, I can
measure the phase as well.
01:17:04.890 --> 01:17:06.630
I can measure both the
transmission probabilities
01:17:06.630 --> 01:17:09.110
and the phases, or at least
the gradient of the phase
01:17:09.110 --> 01:17:10.800
with energy.
01:17:10.800 --> 01:17:11.300
Go ahead.
01:17:11.300 --> 01:17:12.883
AUDIENCE: Is there
a special condition
01:17:12.883 --> 01:17:16.330
that we can [? pose ?]
to see the resonance?
01:17:16.330 --> 01:17:17.580
PROFESSOR: Excellent question.
01:17:17.580 --> 01:17:20.569
Hold onto your
question for a second.
01:17:20.569 --> 01:17:22.860
There's an enormous amount
of the physics of scattering
01:17:22.860 --> 01:17:25.529
contained in the S matrix, and
you can measure the S matrix,
01:17:25.529 --> 01:17:27.570
and you can measure its
dependence on the energy.
01:17:27.570 --> 01:17:31.070
You can measure the coefficients
S12 and S22, their phases
01:17:31.070 --> 01:17:33.780
and their amplitudes,
as a function of energy,
01:17:33.780 --> 01:17:35.970
and you can plot them.
01:17:35.970 --> 01:17:37.660
Here's what I want
to convince you of.
01:17:37.660 --> 01:17:41.480
If you plot those and look
at how the functions behave
01:17:41.480 --> 01:17:45.560
as functions of energy and ask,
how do those functions extend
01:17:45.560 --> 01:17:48.680
to negative energy by
just drawing the line,
01:17:48.680 --> 01:17:51.260
continuing the lines,
you can derive the energy
01:17:51.260 --> 01:17:55.320
of any bound states
in the system, too.
01:17:55.320 --> 01:17:59.950
Knowledge of the scattering is
enough to determine the bound
01:17:59.950 --> 01:18:02.952
state energies of a system,
and let me show you that.
01:18:02.952 --> 01:18:05.410
And this is one of the coolest
things in quantum mechanics.
01:18:05.410 --> 01:18:07.760
Here's how this works.
01:18:07.760 --> 01:18:09.740
We have, from the
definition of the S matrix,
01:18:09.740 --> 01:18:13.930
that BC is equal to
the S matrix on AD,
01:18:13.930 --> 01:18:15.790
where the wave
function-- let me just
01:18:15.790 --> 01:18:20.930
put this back in the original
form-- is CD, AB, e to the ikx,
01:18:20.930 --> 01:18:26.812
e to the minus ikx, and e to the
plus ikx, e to the minus ikx.
01:18:26.812 --> 01:18:28.520
So that's the definition
of the S matrix.
01:18:28.520 --> 01:18:31.730
The S matrix, at
a given energy e,
01:18:31.730 --> 01:18:34.060
is a coefficient relation
matrix between the ingoing
01:18:34.060 --> 01:18:36.820
and outgoing, or, more
to the point, A and D.
01:18:36.820 --> 01:18:39.850
And in all of this, I've assumed
that the energy was positive,
01:18:39.850 --> 01:18:42.840
that the k1 and k2
are positive and real.
01:18:42.840 --> 01:18:44.340
But now let's ask
the question, what
01:18:44.340 --> 01:18:46.215
would have happened if,
in the whole process,
01:18:46.215 --> 01:18:53.490
I had taken the
energy less than 0?
01:18:53.490 --> 01:18:57.280
If the energy were less
than 0, instead of k,
01:18:57.280 --> 01:18:59.490
k would be replaced by i alpha.
01:18:59.490 --> 01:19:01.240
Let's think about
what that does.
01:19:01.240 --> 01:19:07.540
If k is i alpha, this is
e to the ik is minus alpha
01:19:07.540 --> 01:19:11.350
and minus ik is plus alpha.
01:19:11.350 --> 01:19:15.890
Similarly, ik times i,
that gives me a minus alpha
01:19:15.890 --> 01:19:18.730
and this gives me a plus alpha.
01:19:18.730 --> 01:19:19.770
Yeah?
01:19:19.770 --> 01:19:21.800
So as equations, they're
the same equations
01:19:21.800 --> 01:19:26.800
with k replaced by i alpha.
01:19:26.800 --> 01:19:30.030
And now what must be
true for these states
01:19:30.030 --> 01:19:32.870
to be normalizable?
01:19:32.870 --> 01:19:34.890
What must be true,
for example, of A?
01:19:38.070 --> 01:19:40.770
A must be 0 because at minus
infinity, there's divergence.
01:19:40.770 --> 01:19:42.416
Not normalizable.
01:19:42.416 --> 01:19:43.790
So in order to be
at bound state,
01:19:43.790 --> 01:19:47.160
in order to have a physical
state, A must be 0.
01:19:47.160 --> 01:19:49.416
What about D?
01:19:49.416 --> 01:19:50.220
Same reason.
01:19:50.220 --> 01:19:51.930
It's got to be 0 at
positive infinity.
01:19:51.930 --> 01:19:54.470
These guys are convergent,
so C and B can be non-zero.
01:19:54.470 --> 01:19:57.800
So now here's my question.
01:19:57.800 --> 01:19:59.760
We know that these
relations must be true
01:19:59.760 --> 01:20:04.690
because all these relations are
encoding is how a solution here
01:20:04.690 --> 01:20:07.150
matches to a solution
here through a potential
01:20:07.150 --> 01:20:10.140
in between with continuity of
the derivative and anything
01:20:10.140 --> 01:20:11.900
else that's true of
that potential inside.
01:20:11.900 --> 01:20:13.810
All that S is doing
from that point of view
01:20:13.810 --> 01:20:15.610
is telling me how
these coefficients
01:20:15.610 --> 01:20:17.150
match onto these coefficients.
01:20:17.150 --> 01:20:18.480
Yes?
01:20:18.480 --> 01:20:21.960
Now what, for a bound state--
if we have e less than 0,
01:20:21.960 --> 01:20:23.510
what must be true?
01:20:23.510 --> 01:20:30.200
It must be true that AD is equal
to 0, and in particular, 00.
01:20:30.200 --> 01:20:31.310
So what are B and C?
01:20:34.090 --> 01:20:37.000
Well, a matrix times
0 is equal to--
01:20:37.000 --> 01:20:37.790
AUDIENCE: 0.
01:20:37.790 --> 01:20:40.508
PROFESSOR: Unless?
01:20:40.508 --> 01:20:41.867
AUDIENCE: [INAUDIBLE].
01:20:41.867 --> 01:20:45.170
PROFESSOR: Unless the
matrix itself is diverging,
01:20:45.170 --> 01:20:47.300
and then you have to be
more careful, but let's
01:20:47.300 --> 01:20:49.340
be naive for the moment.
01:20:49.340 --> 01:20:56.720
If A is 00, then in order
for B and C to be non-zero,
01:20:56.720 --> 01:20:57.650
S must have a pole.
01:21:01.580 --> 01:21:05.120
S must go like 1 over 0.
01:21:05.120 --> 01:21:08.560
S must diverge at some
special value of the energy.
01:21:08.560 --> 01:21:09.389
Well, that's easy.
01:21:09.389 --> 01:21:11.930
That tells you that if you look
at any particular coefficient
01:21:11.930 --> 01:21:15.440
in S, any of the
matrix elements of S,
01:21:15.440 --> 01:21:17.940
the numerator can you whatever
you want, some finite number,
01:21:17.940 --> 01:21:19.740
but the denominator
had better be?
01:21:19.740 --> 01:21:20.600
AUDIENCE: 0.
01:21:20.600 --> 01:21:21.590
PROFESSOR: 0.
01:21:21.590 --> 01:21:23.830
So let's look at
the denominator.
01:21:23.830 --> 01:21:30.466
If I compute S21-- actually,
let me do this over here.
01:21:30.466 --> 01:21:32.890
No, it's all filled.
01:21:32.890 --> 01:21:35.280
Let's do it over here.
01:21:35.280 --> 01:21:39.060
If I look at S21 for
the potential well,
01:21:39.060 --> 01:21:40.530
scattering off
the potential well
01:21:40.530 --> 01:21:44.610
that we looked at at the
beginning of today's lecture,
01:21:44.610 --> 01:21:46.840
this guy, and now
I'm going to look
01:21:46.840 --> 01:21:51.420
at S21, one of the
coefficients of this guy,
01:21:51.420 --> 01:21:55.990
also known as the scattering
amplitude t for the well.
01:21:58.870 --> 01:22:05.020
This is equal to-- and it's a
godawful expression-- 2 k1 k2
01:22:05.020 --> 01:22:12.210
e to the i k2 L over
2 k1 k2 cos of k
01:22:12.210 --> 01:22:18.920
prime L minus i k squared plus
k1 squared plus k2 squared
01:22:18.920 --> 01:22:24.910
times sine of k2 L. This
is some horrible thing.
01:22:24.910 --> 01:22:30.220
But now I ask the condition,
when does this have a pole?
01:22:30.220 --> 01:22:32.810
When the energy has
continued to be negative,
01:22:32.810 --> 01:22:34.650
for what values does
this have a pole
01:22:34.650 --> 01:22:36.410
or does the
denominator have a 0?
01:22:36.410 --> 01:22:38.170
And the answer is
if you take this
01:22:38.170 --> 01:22:39.780
and you massage
the equation, this
01:22:39.780 --> 01:22:43.753
is equal to 0 a little bit, you
get the following expression,
01:22:43.753 --> 01:22:53.870
k2 L upon 2 tangent of k2 L
upon 2 is equal to k1 L upon 2.
01:22:53.870 --> 01:22:56.310
This is the condition
for the bound state
01:22:56.310 --> 01:22:58.980
energies of the
square well, and we
01:22:58.980 --> 01:23:03.080
computed it using knowledge
only of the scattering states.
01:23:03.080 --> 01:23:05.320
If you took particles
and a square well
01:23:05.320 --> 01:23:08.180
and roll the particles across
the square well potential
01:23:08.180 --> 01:23:10.857
and measure it as a function
of energy, the scattering
01:23:10.857 --> 01:23:12.440
amplitude, the
transmission amplitude,
01:23:12.440 --> 01:23:15.720
and in particular S21, an
element of the S matrix,
01:23:15.720 --> 01:23:17.910
and you plotted it as
a function of energy,
01:23:17.910 --> 01:23:23.860
and then you approximated that
by a function of energy that
01:23:23.860 --> 01:23:27.196
satisfies the basic properties
of unitarity, what you would
01:23:27.196 --> 01:23:28.570
find is that when
you then extend
01:23:28.570 --> 01:23:31.276
that function in
mathematica to minus
01:23:31.276 --> 01:23:32.650
a particular value
of the energy,
01:23:32.650 --> 01:23:36.000
the denominator
diverges at that energy.
01:23:36.000 --> 01:23:38.912
You know that there
will be a bound state.
01:23:38.912 --> 01:23:40.620
And so from scattering,
you've determined
01:23:40.620 --> 01:23:42.236
the existence of a bound state.
01:23:42.236 --> 01:23:44.610
This is how we find an awful
lot of the particles that we
01:23:44.610 --> 01:23:49.050
actually deduce must
exist in the real world.
01:23:49.050 --> 01:23:52.050
We'll pick up next time.
01:23:52.050 --> 01:23:53.600
[APPLAUSE]