WEBVTT
00:00:00.090 --> 00:00:02.430
The following content is
provided under a Creative
00:00:02.430 --> 00:00:03.820
Commons license.
00:00:03.820 --> 00:00:06.030
Your support will help
MIT OpenCourseWare
00:00:06.030 --> 00:00:10.120
continue to offer high-quality
educational resources for free.
00:00:10.120 --> 00:00:12.660
To make a donation or to
view additional materials
00:00:12.660 --> 00:00:16.620
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:16.620 --> 00:00:17.850
at osu.mit.edu.
00:00:21.560 --> 00:00:24.600
ROBERT FIELD: You know that
there is exam tomorrow night.
00:00:24.600 --> 00:00:31.210
It's very heavily
on the particle
00:00:31.210 --> 00:00:33.440
in a box, the
harmonic oscillator,
00:00:33.440 --> 00:00:35.900
and the time-independent
Schrodinger equation.
00:00:35.900 --> 00:00:38.610
It's really quite an
amazing amount of stuff.
00:00:38.610 --> 00:00:39.110
OK.
00:00:39.110 --> 00:00:46.160
So last time, we talked
about some time-independent
00:00:46.160 --> 00:00:47.630
Hamiltonian examples.
00:00:47.630 --> 00:00:52.620
And two that I like are the
half harmonic oscillator
00:00:52.620 --> 00:00:56.960
and the vertical excitation--
the Franck-Condon excitation.
00:00:56.960 --> 00:01:00.805
Now, when you're doing
time-dependent Hamiltonians--
00:01:00.805 --> 00:01:03.290
or when you're doing
time-dependent problems
00:01:03.290 --> 00:01:07.370
for a time-independent
Hamiltonian--
00:01:07.370 --> 00:01:12.410
You always start with the
form of the function at t
00:01:12.410 --> 00:01:15.690
equals zero and
automatically extend
00:01:15.690 --> 00:01:18.470
it using the fact that
you have a complete set
00:01:18.470 --> 00:01:22.190
of eigenfunctions of the
time-independent Hamiltonian
00:01:22.190 --> 00:01:25.290
to the time-dependent
wave function.
00:01:25.290 --> 00:01:27.800
And with the time-dependent
wave function,
00:01:27.800 --> 00:01:32.870
you're able to calculate
almost anything you want.
00:01:32.870 --> 00:01:35.759
And I did several
examples of things
00:01:35.759 --> 00:01:36.800
that you could calculate.
00:01:36.800 --> 00:01:41.260
One is the probability
density as a function of time.
00:01:41.260 --> 00:01:45.540
So the other is the
survival probability.
00:01:45.540 --> 00:01:51.050
And the survival probability
is a really neat thing,
00:01:51.050 --> 00:01:55.280
because it says, we've got some
object which is particle-like.
00:01:55.280 --> 00:01:58.820
And the time evolution
makes the wave function
00:01:58.820 --> 00:02:01.670
move away from its birthplace.
00:02:01.670 --> 00:02:05.990
And that's an easy
thing to understand,
00:02:05.990 --> 00:02:11.660
but what's surprising is that
we think about a wave packet as
00:02:11.660 --> 00:02:14.840
localized in
position, but it also
00:02:14.840 --> 00:02:18.060
has encoded in it
momentum information.
00:02:18.060 --> 00:02:20.360
And so when the wave
packet moves away
00:02:20.360 --> 00:02:21.620
from its starting point--
00:02:21.620 --> 00:02:24.920
if it starts at rest--
00:02:24.920 --> 00:02:30.010
The initial fast
changes are in momentum.
00:02:30.010 --> 00:02:33.790
And the momentum-- the
change of the momentum--
00:02:33.790 --> 00:02:37.040
is sampling the gradient
of the potential.
00:02:37.040 --> 00:02:39.250
And that's something you
usually want to know.
00:02:39.250 --> 00:02:41.440
The gradient of the
potential at a turning
00:02:41.440 --> 00:02:43.570
point-- at an energy you know.
00:02:43.570 --> 00:02:46.630
And that is a
measurable quantity.
00:02:46.630 --> 00:02:48.880
And it's really a
beautiful example
00:02:48.880 --> 00:02:53.140
of how you can find easily
observable, or easily
00:02:53.140 --> 00:02:56.980
calculable, quantum
mechanical things that
00:02:56.980 --> 00:03:00.400
reflect classical mechanics.
00:03:00.400 --> 00:03:01.750
OK.
00:03:01.750 --> 00:03:05.530
I talked about grand rephasings
for problems like the harmonic
00:03:05.530 --> 00:03:10.240
oscillator, and the rigid rotor,
and the particle in a box--
00:03:10.240 --> 00:03:14.200
You have this fantastic property
that all of the energy level
00:03:14.200 --> 00:03:21.480
differences are integer
multiples of a common factor.
00:03:21.480 --> 00:03:23.820
And that guarantees
that you will
00:03:23.820 --> 00:03:26.940
have periodic
rephasings at times
00:03:26.940 --> 00:03:30.650
related to that common factor.
00:03:30.650 --> 00:03:33.730
And so that enables
you to observe
00:03:33.730 --> 00:03:36.800
the evolution of something
for a very long time
00:03:36.800 --> 00:03:39.190
and to see whether
the rephasings are
00:03:39.190 --> 00:03:41.620
perfect or not quite perfect.
00:03:41.620 --> 00:03:44.320
And the imperfections
are telling you something
00:03:44.320 --> 00:03:46.380
beyond the simple model--
00:03:46.380 --> 00:03:50.410
They're telling you
about anharmonicity
00:03:50.410 --> 00:03:54.250
or some other thing that makes
the energy levels not quite
00:03:54.250 --> 00:03:57.500
integer multiples
of a common factor.
00:03:57.500 --> 00:04:01.960
Now, one of the nicest things is
the illustration of tunneling.
00:04:01.960 --> 00:04:05.650
And we don't observe tunneling.
00:04:05.650 --> 00:04:09.190
Tunneling is a quantum
mechanical thing.
00:04:09.190 --> 00:04:13.540
And it's encoded in what's most
easily observed-- the energy
00:04:13.540 --> 00:04:15.460
level pattern.
00:04:15.460 --> 00:04:18.130
It's encoded as a
level staggering.
00:04:18.130 --> 00:04:21.680
Now, we talked about
this problem, where we
00:04:21.680 --> 00:04:24.150
have a barrier in the middle.
00:04:24.150 --> 00:04:27.420
And with a barrier
in the middle,
00:04:27.420 --> 00:04:30.330
half of the energy levels
are almost unaffected,
00:04:30.330 --> 00:04:32.340
and the other half
are affected a lot.
00:04:34.930 --> 00:04:37.690
Now, here is a problem
that you can understand--
00:04:37.690 --> 00:04:39.750
In the same way, there's
no tunneling here,
00:04:39.750 --> 00:04:43.200
but there is some
extra stuff here.
00:04:43.200 --> 00:04:45.900
And the level diagram can
tell you the difference
00:04:45.900 --> 00:04:47.400
between these two things.
00:04:47.400 --> 00:04:49.860
Does anybody want to
tell me what's different?
00:04:49.860 --> 00:04:52.350
What is the qualitative
signature of this?
00:04:55.460 --> 00:04:56.720
Yes.
00:04:56.720 --> 00:04:58.820
AUDIENCE: The spacings
move up a little bit.
00:04:58.820 --> 00:05:01.230
The odd spacings are
erased in energy,
00:05:01.230 --> 00:05:04.280
and the even spacings are
just about unaffected.
00:05:04.280 --> 00:05:06.384
ROBERT FIELD: Even symmetry
levels are shifted up.
00:05:06.384 --> 00:05:07.050
AUDIENCE: Right.
00:05:07.050 --> 00:05:08.410
Sorry.
00:05:08.410 --> 00:05:11.120
ROBERT FIELD: And so now you're
ready to answer this one.
00:05:18.760 --> 00:05:20.830
If the even symmetry
levels are shifted up,
00:05:20.830 --> 00:05:24.700
because they feel the
barrier, what about the even
00:05:24.700 --> 00:05:25.750
symmetry levels here?
00:05:28.420 --> 00:05:29.200
Yes?
00:05:29.200 --> 00:05:30.220
AUDIENCE: They would
be shifted down.
00:05:30.220 --> 00:05:31.240
ROBERT FIELD: Right.
00:05:31.240 --> 00:05:34.960
And so you get a level
staggering where, in this case,
00:05:34.960 --> 00:05:40.600
the lowest level is close
to the next higher one.
00:05:40.600 --> 00:05:44.320
And in this one, the lowest
level is shifted way down,
00:05:44.320 --> 00:05:45.670
and the next one is not shifted.
00:05:45.670 --> 00:05:48.810
And then we get the
doubling or the parent.
00:05:48.810 --> 00:05:53.110
So that's a kind of
intuition that you get just
00:05:53.110 --> 00:05:55.520
by looking at these problems.
00:05:55.520 --> 00:05:58.690
Now, one thing that
is really beautiful
00:05:58.690 --> 00:06:02.410
is, when you have a
barrier like this,
00:06:02.410 --> 00:06:04.750
since this part of
the potential problem
00:06:04.750 --> 00:06:07.450
is something that
is exactly solved,
00:06:07.450 --> 00:06:10.480
you propagate the wave
function in from the sides
00:06:10.480 --> 00:06:13.490
and they have the same phase.
00:06:13.490 --> 00:06:20.080
So there is no accumulation
of phase under the barrier.
00:06:20.080 --> 00:06:23.230
And that means the
levels that are trying
00:06:23.230 --> 00:06:25.810
to propagate under this
barrier are shifted up,
00:06:25.810 --> 00:06:28.840
because they have to accumulate
enough phase to satisfy
00:06:28.840 --> 00:06:31.930
the boundary conditions
at the turning points.
00:06:31.930 --> 00:06:34.120
Here, you're going to
accumulate more phase
00:06:34.120 --> 00:06:37.090
in this special region.
00:06:37.090 --> 00:06:39.670
Phase is really important.
00:06:39.670 --> 00:06:41.530
OK.
00:06:41.530 --> 00:06:44.530
So today we're going to talk--
00:06:44.530 --> 00:06:48.880
And this this lecture is
basically not on the exam,
00:06:48.880 --> 00:06:51.520
although it does connect
with topics on the exam
00:06:51.520 --> 00:06:55.300
and makes it possible to
understand them better.
00:06:55.300 --> 00:07:01.300
So instead of learning
about the postulates
00:07:01.300 --> 00:07:04.660
in a great abstract way at the
beginning, before you know what
00:07:04.660 --> 00:07:07.720
they're for, now
we're going to review
00:07:07.720 --> 00:07:09.730
what we understand about them.
00:07:09.730 --> 00:07:14.965
And so one thing is,
there is a wave function.
00:07:18.230 --> 00:07:20.710
And now we're considering
not just one dimension,
00:07:20.710 --> 00:07:24.780
but any number of dimensions.
00:07:24.780 --> 00:07:27.430
And this is the
state function that
00:07:27.430 --> 00:07:29.110
tells you everything
you're allowed
00:07:29.110 --> 00:07:33.130
to know about the system.
00:07:33.130 --> 00:07:37.030
And if you have this, you
can calculate everything.
00:07:37.030 --> 00:07:41.290
If you know how
observables relate to this,
00:07:41.290 --> 00:07:42.040
well, you're fine.
00:07:42.040 --> 00:07:45.229
You can then use that to
describe the Hamiltonian.
00:07:55.020 --> 00:07:57.480
Hermetian operators
are the only kind
00:07:57.480 --> 00:08:00.570
of operators you can have
in quantum mechanics.
00:08:00.570 --> 00:08:03.220
And they have the
wonderful property
00:08:03.220 --> 00:08:04.750
that their eigenvalues--
00:08:07.480 --> 00:08:10.120
all of them-- are real.
00:08:10.120 --> 00:08:13.960
Because they correspond to
something that's observable.
00:08:13.960 --> 00:08:17.380
And when you observe
something, it's a real number.
00:08:17.380 --> 00:08:18.920
It's not a complex number.
00:08:18.920 --> 00:08:20.180
It's not an imaginary number.
00:08:25.620 --> 00:08:30.360
So what is Hermetian?
00:08:30.360 --> 00:08:34.230
And why does that ensure that
you always get a real number?
00:08:42.990 --> 00:08:43.490
OK.
00:08:43.490 --> 00:08:47.570
You all know, in
quantum mechanics,
00:08:47.570 --> 00:08:50.630
that if you do an experiment,
and the experiment can
00:08:50.630 --> 00:08:54.520
be represented by
some operator, you
00:08:54.520 --> 00:08:58.360
get an eigenvalue
of that operator--
00:08:58.360 --> 00:09:02.080
nothing else-- one
experiment, one eigenvalue.
00:09:02.080 --> 00:09:06.310
100 experiments-- You might
get several eigenvalues.
00:09:06.310 --> 00:09:10.030
And the relative probabilities
of the different eigenvalues--
00:09:10.030 --> 00:09:12.940
I tell you something
about, what is this?
00:09:12.940 --> 00:09:14.835
What was the
initial preparation?
00:09:26.960 --> 00:09:30.140
So I've already said
something about this.
00:09:30.140 --> 00:09:32.780
The expectation value
of the wave function
00:09:32.780 --> 00:09:42.020
for a particular operator is
related to probabilities times
00:09:42.020 --> 00:09:42.830
eigenvalues.
00:09:46.960 --> 00:09:47.460
OK.
00:09:47.460 --> 00:09:50.880
The fifth postulate is the
time-dependent Schrodinger
00:09:50.880 --> 00:09:52.130
equation.
00:09:52.130 --> 00:09:56.970
And I don't need to talk
much about that, because it's
00:09:56.970 --> 00:09:59.000
all the way around us.
00:09:59.000 --> 00:10:05.070
And then I'm going to talk about
some neat stuff where we start
00:10:05.070 --> 00:10:18.820
using words that are very
instructive-- so completeness,
00:10:18.820 --> 00:10:23.990
orthogonality, commutators--
00:10:30.260 --> 00:10:37.200
simultaneous eigenfunctions.
00:10:37.200 --> 00:10:39.050
And this is really important.
00:10:39.050 --> 00:10:42.230
Suppose we have an
easy operator, which
00:10:42.230 --> 00:10:45.740
commutes with the Hamiltonian.
00:10:45.740 --> 00:10:50.520
And it's easy for us to find the
eigenvalues of that operator--
00:10:50.520 --> 00:10:53.831
eigenfunctions of that operator.
00:10:53.831 --> 00:10:56.080
Those eigenfunctions are
automatically eigenfunctions.
00:10:56.080 --> 00:10:58.770
It's a hard operator.
00:10:58.770 --> 00:11:00.850
So we like them.
00:11:00.850 --> 00:11:04.200
And so we are
interested in, can we
00:11:04.200 --> 00:11:08.470
have simultaneous eigenfunctions
of several operators?
00:11:08.470 --> 00:11:10.670
And the answer is yes, if
the operators can move.
00:11:13.480 --> 00:11:23.270
We use the term
basis set to describe
00:11:23.270 --> 00:11:26.150
the complete set
of eigenfunctions
00:11:26.150 --> 00:11:29.870
of some operator.
00:11:29.870 --> 00:11:43.120
And we use the words
mixing coefficients
00:11:43.120 --> 00:11:45.820
and mixing fraction.
00:11:49.000 --> 00:11:53.200
So here we have a
wave function that
00:11:53.200 --> 00:11:57.520
is expressed as a linear
combination of basis functions.
00:11:57.520 --> 00:12:00.400
And the coefficient
in front of each one
00:12:00.400 --> 00:12:02.270
is the mixing coefficient.
00:12:02.270 --> 00:12:04.450
Now, if we're talking
about probabilities,
00:12:04.450 --> 00:12:07.000
we care about mixing
fractions, which
00:12:07.000 --> 00:12:10.360
are basically the mixing
coefficient squared--
00:12:10.360 --> 00:12:13.390
or [INAUDIBLE] square modulus.
00:12:13.390 --> 00:12:18.280
So these are words that are
part of the language of someone
00:12:18.280 --> 00:12:22.570
who uses quantum mechanics
to understand stuff.
00:12:22.570 --> 00:12:23.070
OK.
00:12:23.070 --> 00:12:26.055
So I'm going to try to develop
this in some useful tricks.
00:12:31.280 --> 00:12:34.010
So we have a state function.
00:12:44.880 --> 00:12:46.760
So we have a state
function, which
00:12:46.760 --> 00:12:50.510
is a function of
coordinates and time.
00:12:50.510 --> 00:12:53.810
And this thing is
telling you, what
00:12:53.810 --> 00:12:58.910
is the probability of finding
this system at that coordinate
00:12:58.910 --> 00:13:01.820
at the specified time?
00:13:01.820 --> 00:13:06.800
And the volume element
is dx, dy, and dz.
00:13:06.800 --> 00:13:10.730
Now, often, we use
an abbreviation--
00:13:10.730 --> 00:13:13.880
d tau-- for the volume
element, because we're
00:13:13.880 --> 00:13:15.320
going to be dealing
with problems
00:13:15.320 --> 00:13:18.900
that are not just single
particle, but many particles.
00:13:18.900 --> 00:13:23.770
And so we use this
notation to say,
00:13:23.770 --> 00:13:27.240
for the differential associated
with every coordinate
00:13:27.240 --> 00:13:29.100
associated with the
problem, and we're
00:13:29.100 --> 00:13:31.100
going to integrate over
those sorts of things--
00:13:31.100 --> 00:13:34.350
or at least over some of them.
00:13:34.350 --> 00:13:38.010
So this is telling you about
a probability within a volume
00:13:38.010 --> 00:13:42.030
element at a particular
point in space and time.
00:13:42.030 --> 00:13:48.770
Now, one of the
things that we're told
00:13:48.770 --> 00:13:59.860
is, if the wave functions
are well behaved.
00:13:59.860 --> 00:14:03.130
And that says something
about the wave functions
00:14:03.130 --> 00:14:07.270
and the derivatives
of the wave functions.
00:14:07.270 --> 00:14:08.812
So what's well-behaved?
00:14:11.650 --> 00:14:13.710
Well, the wave function
is normalize-able.
00:14:18.020 --> 00:14:20.690
Now, there are two kinds of
normalization-- normalization
00:14:20.690 --> 00:14:25.850
to one, implying that the system
is somewhere within a specified
00:14:25.850 --> 00:14:28.540
range of coordinates
where there's
00:14:28.540 --> 00:14:30.370
one particle in the system.
00:14:30.370 --> 00:14:31.690
Whatever.
00:14:31.690 --> 00:14:35.860
And there's normalization
to number density.
00:14:35.860 --> 00:14:38.710
When you have a free
particle, the free particle
00:14:38.710 --> 00:14:41.200
is not confined.
00:14:41.200 --> 00:14:44.720
And so you can't
say, I'm normalizing.
00:14:44.720 --> 00:14:46.870
So there's one
particle somewhere,
00:14:46.870 --> 00:14:51.490
because that means there
is no particle anywhere.
00:14:51.490 --> 00:14:54.850
And so we can extend the
concept of normalization to say,
00:14:54.850 --> 00:14:56.830
there's one particle
in a particular length
00:14:56.830 --> 00:14:57.420
of the system.
00:15:00.970 --> 00:15:02.930
And that was a problem
that got removed
00:15:02.930 --> 00:15:06.270
from the draft of the exam.
00:15:06.270 --> 00:15:11.540
But one thing that happens is,
if I write a beautiful problem,
00:15:11.540 --> 00:15:16.940
and it gets bumped from an exam,
it might appear in the final.
00:15:16.940 --> 00:15:17.620
OK.
00:15:17.620 --> 00:15:26.640
So normalize-able is part
of well-behaved continuous
00:15:26.640 --> 00:15:29.169
and single value--
00:15:29.169 --> 00:15:30.460
we'll talk about all of these--
00:15:33.900 --> 00:15:39.940
and square integrable
00:15:39.940 --> 00:15:40.880
OK.
00:15:40.880 --> 00:15:44.720
Continuous-- The
wave function has
00:15:44.720 --> 00:15:49.070
to be continuous everywhere.
00:15:49.070 --> 00:15:50.990
The first derivative
of the wave function,
00:15:50.990 --> 00:15:53.000
with respect to coordinate--
00:15:53.000 --> 00:15:55.730
we already know from
the particle in a box
00:15:55.730 --> 00:16:00.170
that that is not continuous
at an infinite wall.
00:16:00.170 --> 00:16:01.760
So an infinite wall--
00:16:01.760 --> 00:16:06.350
not just a vertical wall, but
one that goes to infinity--
00:16:06.350 --> 00:16:10.670
guarantees that this
guy is not continuous.
00:16:10.670 --> 00:16:14.020
But that's a pretty
dramatic thing.
00:16:14.020 --> 00:16:17.140
The second derivative
is not continuous
00:16:17.140 --> 00:16:22.460
when you have a vertical
step, which is not infinite.
00:16:22.460 --> 00:16:26.600
Now, when you have problems
where you divide space up
00:16:26.600 --> 00:16:28.940
into regions,
you're often trying
00:16:28.940 --> 00:16:32.810
to establish boundary conditions
between the different regions
00:16:32.810 --> 00:16:34.640
or at the borders.
00:16:34.640 --> 00:16:38.660
And the boundary
conditions are usually
00:16:38.660 --> 00:16:42.230
expressed in terms of
continuity of the wave function
00:16:42.230 --> 00:16:45.260
and continuity of
the first derivative.
00:16:45.260 --> 00:16:48.770
And we don't need this often.
00:16:48.770 --> 00:16:51.530
But you're entitled-- if
the problem is sufficiently
00:16:51.530 --> 00:16:52.130
well-behaved--
00:16:52.130 --> 00:16:54.140
All of these guys
are continuous,
00:16:54.140 --> 00:16:55.340
and you can use them all.
00:16:58.720 --> 00:16:59.860
OK.
00:16:59.860 --> 00:17:03.790
So normalize-able--
Well, normalize-able
00:17:03.790 --> 00:17:08.260
means it's square integrable,
and you don't get infinity
00:17:08.260 --> 00:17:13.660
unless we use this other
definition of normalize-able.
00:17:13.660 --> 00:17:19.119
So one of the things
that has to happen
00:17:19.119 --> 00:17:26.710
is, at the coordinate
plus and minus infinity,
00:17:26.710 --> 00:17:28.210
the wave function
has to go to zero.
00:17:30.850 --> 00:17:39.480
Now, the wave function can be
infinite at a very small region
00:17:39.480 --> 00:17:41.070
of space.
00:17:41.070 --> 00:17:44.650
So there are singularities
that can be dealt with.
00:17:44.650 --> 00:17:46.890
But normally, you say,
the wave function is never
00:17:46.890 --> 00:17:49.950
going to be infinite,
and it's never
00:17:49.950 --> 00:17:52.400
going to be anything
but 0 at infinity,
00:17:52.400 --> 00:17:53.540
or you're in real trouble.
00:18:01.210 --> 00:18:04.990
Now, there is a
wonderful example
00:18:04.990 --> 00:18:07.690
of a kind of a problem
called the delta function.
00:18:10.290 --> 00:18:14.280
A delta function is
basically an infinite spike--
00:18:14.280 --> 00:18:17.990
infinitely thin,
infinitely tall.
00:18:17.990 --> 00:18:21.410
And what it does
is, it causes this
00:18:21.410 --> 00:18:26.150
to be discontinuous
by an amount related
00:18:26.150 --> 00:18:30.915
to the value of the
derivative at the--
00:18:30.915 --> 00:18:33.560
by an amount determined
by the value of the wave
00:18:33.560 --> 00:18:37.070
function at the delta function.
00:18:37.070 --> 00:18:41.190
And delta functions are
computationally wonderful,
00:18:41.190 --> 00:18:46.770
because they enable you to
treat certain kinds of problems
00:18:46.770 --> 00:18:49.120
in a trivial way.
00:18:49.120 --> 00:18:50.750
Like, if you have a barrier--
00:18:50.750 --> 00:18:51.550
Yes.
00:18:51.550 --> 00:18:54.790
AUDIENCE: Does it relate at all
to the integral of the spike?
00:18:54.790 --> 00:18:55.540
ROBERT FIELD: Yes.
00:18:59.530 --> 00:19:03.960
So we have an integral
of the delta function
00:19:03.960 --> 00:19:10.480
at x i times the wave
function at x, dx.
00:19:10.480 --> 00:19:13.606
And that gives you the
wave function at x i.
00:19:16.730 --> 00:19:19.070
OK.
00:19:19.070 --> 00:19:22.530
I haven't really talked
much about delta functions.
00:19:22.530 --> 00:19:25.790
But because it
acts like a barrier
00:19:25.790 --> 00:19:28.520
and is a trivial
barrier, it enables
00:19:28.520 --> 00:19:32.630
you to solve barrier problems
or at least understand them
00:19:32.630 --> 00:19:35.900
in a very quick and easy way.
00:19:35.900 --> 00:19:40.080
And vertical steps
are also not physical,
00:19:40.080 --> 00:19:42.020
but we like vertical
steps because it's
00:19:42.020 --> 00:19:45.140
easy to apply
boundary conditions.
00:19:45.140 --> 00:19:48.820
And so these are all just
computationally tricky things
00:19:48.820 --> 00:19:50.470
that are wonderful.
00:19:50.470 --> 00:19:54.700
And we don't worry
about, is there
00:19:54.700 --> 00:19:57.760
a real system that acts
like a delta function
00:19:57.760 --> 00:19:58.910
or a vertical step?
00:19:58.910 --> 00:19:59.410
No.
00:19:59.410 --> 00:20:00.700
There isn't.
00:20:00.700 --> 00:20:03.400
But everything you get
easily, mathematically,
00:20:03.400 --> 00:20:06.040
from these simple
things is great.
00:20:08.860 --> 00:20:10.700
OK.
00:20:10.700 --> 00:20:12.385
Did I satisfy you on the--
00:20:12.385 --> 00:20:13.010
AUDIENCE: Sure.
00:20:13.010 --> 00:20:14.570
I'll read into it.
00:20:14.570 --> 00:20:15.450
ROBERT FIELD: OK.
00:20:15.450 --> 00:20:18.300
And there's also
a notation where
00:20:18.300 --> 00:20:26.880
you have x, xi, or x minus
x i, and they're basically
00:20:26.880 --> 00:20:28.340
all the same sort of thing.
00:20:30.850 --> 00:20:31.860
If you have this--
00:20:31.860 --> 00:20:34.260
the argument-- when
the argument is 0,
00:20:34.260 --> 00:20:36.170
you get the infinite spike.
00:20:36.170 --> 00:20:38.771
And there's just lots of things.
00:20:38.771 --> 00:20:39.270
OK.
00:20:43.880 --> 00:20:54.240
So for every classical
mechanical observable,
00:20:54.240 --> 00:20:58.313
there is a quantum mechanical
operator, which is Hermetian.
00:21:04.470 --> 00:21:08.090
And the main point of
Hermetian, as I said,
00:21:08.090 --> 00:21:11.570
is that its
eigenvalues are real.
00:21:11.570 --> 00:21:13.850
And so what is the
thing that assures
00:21:13.850 --> 00:21:15.950
that we get real eigenvalues?
00:21:15.950 --> 00:21:20.930
Well, here is the definition
in a peculiar form.
00:21:20.930 --> 00:21:25.010
So we have an integral from
minus infinity to infinity
00:21:25.010 --> 00:21:28.400
of some function--
complex conjugate--
00:21:28.400 --> 00:21:32.675
a times some other function dx.
00:21:32.675 --> 00:21:35.300
So this could be a wave function
and a different wave function.
00:21:38.990 --> 00:21:46.130
And the definition of Hermetian
is this abstract definition.
00:21:46.130 --> 00:21:48.770
We have, say, interval
from minus infinity
00:21:48.770 --> 00:21:53.840
to infinity g a--
00:21:53.840 --> 00:21:56.830
Let's put a hat on it.
00:21:56.830 --> 00:22:01.280
Star, f-star, dx.
00:22:04.530 --> 00:22:06.770
Well, this is kind of fancy.
00:22:06.770 --> 00:22:07.820
So we have an operator.
00:22:07.820 --> 00:22:10.130
We can take the complex
conjugate of the operator.
00:22:10.130 --> 00:22:11.780
We have functions.
00:22:11.780 --> 00:22:15.110
We can take the complex
conjugates of the function.
00:22:15.110 --> 00:22:17.930
But here, what we're
seeing is, the operator
00:22:17.930 --> 00:22:21.680
is operating on the g
function-- the function
00:22:21.680 --> 00:22:25.190
that started out on the right.
00:22:25.190 --> 00:22:29.930
And here, what we have is
this operator operating
00:22:29.930 --> 00:22:35.630
on the f function, which
was initially on the left.
00:22:35.630 --> 00:22:38.830
And so this is prescription
for operating on the left.
00:22:42.040 --> 00:22:46.870
And it's also an
invitation to use
00:22:46.870 --> 00:22:50.140
a really convenient
and compact notation.
00:22:50.140 --> 00:22:52.590
And that is this--
00:22:52.590 --> 00:22:56.190
Put two subscripts on a.
00:22:56.190 --> 00:23:00.700
A subscript says the first
guy is the function over here,
00:23:00.700 --> 00:23:02.810
which is complex conjugated.
00:23:02.810 --> 00:23:04.740
And the second one
is the function
00:23:04.740 --> 00:23:07.710
over here, which is
not complex conjugated.
00:23:07.710 --> 00:23:19.200
And so this equation reduces
to a g f star, where, now, this
00:23:19.200 --> 00:23:22.470
is a wonderful shorthand.
00:23:22.470 --> 00:23:27.471
And this is another
way of saying that A
00:23:27.471 --> 00:23:29.565
has to be equal to A dagger.
00:23:32.370 --> 00:23:34.830
Where now we're talking
about operators,
00:23:34.830 --> 00:23:38.130
and matrix representations
of operators.
00:23:38.130 --> 00:23:42.620
Because here we have a
number with two indices,
00:23:42.620 --> 00:23:45.010
and that's how we represent
elements of a matrix.
00:23:45.010 --> 00:23:47.730
And we're soon going to be
playing with linear algebra,
00:23:47.730 --> 00:23:50.460
and talking about matrices.
00:23:50.460 --> 00:23:53.910
And so this is just saying,
well, we can take one matrix,
00:23:53.910 --> 00:23:56.820
and it's equal to
the complex conjugate
00:23:56.820 --> 00:23:58.530
of every term in the matrix.
00:23:58.530 --> 00:24:00.075
And the order switched.
00:24:03.090 --> 00:24:07.420
So this is a warning
that we're going
00:24:07.420 --> 00:24:10.630
to be using a notation, which
is way simpler than taking
00:24:10.630 --> 00:24:12.920
these integrals.
00:24:12.920 --> 00:24:17.037
And so once you recognize
what this symbol means,
00:24:17.037 --> 00:24:18.620
you're never going
to want to go back.
00:24:24.760 --> 00:24:25.260
OK.
00:24:33.870 --> 00:24:35.370
Why real?
00:24:40.700 --> 00:24:43.100
So let's look at a
specific example,
00:24:43.100 --> 00:24:47.390
where instead of having two
different functions, let's just
00:24:47.390 --> 00:24:48.770
look at one.
00:24:48.770 --> 00:24:55.595
So we have this
integral f star A f d x.
00:24:59.490 --> 00:25:00.600
So that's a f f.
00:25:03.940 --> 00:25:09.170
And the definition says,
well, we're going to get the--
00:25:17.100 --> 00:25:25.080
So it says, replace the
original thing by moving this--
00:25:25.080 --> 00:25:27.330
anyway, yes.
00:25:27.330 --> 00:25:32.760
And this is a f f star.
00:25:32.760 --> 00:25:37.230
If you know how this
notation translates into--
00:25:37.230 --> 00:25:39.690
now you can see, oh,
well, what is this?
00:25:39.690 --> 00:25:43.750
It's just taking the
complex conjugate.
00:25:43.750 --> 00:25:46.270
And these two guys are equal.
00:25:46.270 --> 00:25:50.400
And so a number is real if it's
equal to its complex conjugate.
00:25:53.100 --> 00:25:54.960
And this is just a special case.
00:25:54.960 --> 00:25:58.490
It's-- the Hermitian property is
a little bit more powerful than
00:25:58.490 --> 00:26:03.440
that, but the important thing is
that it guarantees that if you
00:26:03.440 --> 00:26:08.960
calculate the expectation value
of an observable quantity,
00:26:08.960 --> 00:26:11.470
you're going to
get a real number,
00:26:11.470 --> 00:26:13.580
if the operator is Hermitian.
00:26:13.580 --> 00:26:16.299
And it has to be Hermitian,
if it's observable.
00:26:19.233 --> 00:26:27.010
Now, it's often
useful if you have
00:26:27.010 --> 00:26:30.550
a classical mechanical
operator, and you translate it
00:26:30.550 --> 00:26:33.730
into a quantum
mechanical operator
00:26:33.730 --> 00:26:37.125
by doing the usual replacement
of x with x, and p with i
00:26:37.125 --> 00:26:40.540
h bar, the derivative
with respect to x.
00:26:40.540 --> 00:26:44.040
If you do all that sort of
stuff, you might be unlucky
00:26:44.040 --> 00:26:47.390
and you might get a
non-Hermitian operator.
00:26:47.390 --> 00:26:50.020
And so you can generate
a Hermitian operator
00:26:50.020 --> 00:26:52.990
if you write--
00:26:57.590 --> 00:27:01.510
that's guaranteed
to be Hermitian.
00:27:01.510 --> 00:27:05.470
So you take classic mechanics,
you generate something
00:27:05.470 --> 00:27:08.230
following some simple rules,
and you have bad luck,
00:27:08.230 --> 00:27:09.730
it doesn't come out
to be Hermitian.
00:27:09.730 --> 00:27:11.831
This is the way we
make it Hermitian.
00:27:19.060 --> 00:27:24.180
So if this is not Hermitian,
and this is not Hermitian,
00:27:24.180 --> 00:27:27.560
but that we're defining
something that is Hermitian,
00:27:27.560 --> 00:27:30.790
so let's just put a
little twiddle over A,
00:27:30.790 --> 00:27:31.896
that's Hermitian.
00:27:36.360 --> 00:27:43.420
OK then we're now talking
about the third postulate,
00:27:43.420 --> 00:27:55.470
and each measurement of a
gives an eigenvalue of a.
00:27:58.560 --> 00:28:01.360
We've talked about this enough.
00:28:01.360 --> 00:28:07.640
But your first encounter of this
was the Two-Slit experiment.
00:28:07.640 --> 00:28:13.560
In the Two-Slit experiment,
this experiment is an operator.
00:28:13.560 --> 00:28:19.125
And you have some initial
photons entering this operator
00:28:19.125 --> 00:28:23.600
, and you get dots,
on the screen,
00:28:23.600 --> 00:28:27.350
those are eigenvalues
of this operator.
00:28:27.350 --> 00:28:34.970
Now it may be that the operator
has continuous eigenvalues,
00:28:34.970 --> 00:28:37.760
but they don't have
uniform probabilities.
00:28:37.760 --> 00:28:40.880
And so what you observe is a
whole distribution of dots that
00:28:40.880 --> 00:28:43.460
doesn't look like
anything special,
00:28:43.460 --> 00:28:46.430
and it's not reproducible from
one experiment to the other,
00:28:46.430 --> 00:28:49.310
but you have this periodic
structure that's appearing,
00:28:49.310 --> 00:28:53.580
which is related to the
properties of the operator.
00:28:53.580 --> 00:28:56.810
And so there is not
uniform probabilities
00:28:56.810 --> 00:29:00.470
of each eigenvalues,
and so you get that.
00:29:00.470 --> 00:29:04.360
OK, these are simple,
but really beautiful.
00:29:11.090 --> 00:29:26.170
OK, if we have a normalized
state then we can say, OK,
00:29:26.170 --> 00:29:28.440
and we never use this notation.
00:29:28.440 --> 00:29:32.370
But whenever you
see this symbol,
00:29:32.370 --> 00:29:36.300
it means the expectation value
of an operator for some wave
00:29:36.300 --> 00:29:38.940
functions, so we could
actually symbolically put
00:29:38.940 --> 00:29:42.480
that wave function down here,
or some symbols saying, OK,
00:29:42.480 --> 00:29:45.120
which one?
00:29:45.120 --> 00:29:55.900
And that this is equal to
psi star A psi d tau, or dx.
00:29:55.900 --> 00:29:57.600
And if the wave
function is normalized,
00:29:57.600 --> 00:30:01.070
we don't need to divide by
a normalization integral.
00:30:01.070 --> 00:30:05.960
If it's not normalized, like
if it's a free particle,
00:30:05.960 --> 00:30:08.670
we divide by some kind of
free particle integral.
00:30:18.330 --> 00:30:24.450
So now the next topic,
which is related to this,
00:30:24.450 --> 00:30:36.567
is completeness
and orthogonality.
00:30:45.340 --> 00:30:48.830
So we have a
particular operator.
00:30:48.830 --> 00:30:53.510
There exists some complete
set of eigenfunctions
00:30:53.510 --> 00:30:55.490
of that operator.
00:30:55.490 --> 00:30:59.500
Usually that complete
set is infinite,
00:30:59.500 --> 00:31:03.220
but they're related to
each other in a simple way.
00:31:03.220 --> 00:31:05.140
You have some
class of functions,
00:31:05.140 --> 00:31:07.210
and you change an integer
to get a new function.
00:31:09.760 --> 00:31:15.370
And orthogonality is, well, if
you have all the eigenfunctions
00:31:15.370 --> 00:31:20.830
of an operator, if they belong
to different eigenvalues,
00:31:20.830 --> 00:31:25.230
they're guaranteed
to be orthogonal.
00:31:25.230 --> 00:31:27.990
Which is convenient, because
that means you get lots of 0s
00:31:27.990 --> 00:31:30.600
from integrals, and we
like that, because we
00:31:30.600 --> 00:31:32.880
don't have to worry about them.
00:31:32.880 --> 00:31:36.664
And you want to be able to
recognize the zero integrals,
00:31:36.664 --> 00:31:38.830
so that you can move very
quickly through a problem.
00:31:43.510 --> 00:31:48.670
Completeness means, take
any function defined
00:31:48.670 --> 00:31:51.520
on the space of the operator
you're interested in.
00:31:51.520 --> 00:31:52.900
You might have an
operator that's
00:31:52.900 --> 00:31:58.390
only operating on a particular
coordinate of a many electron
00:31:58.390 --> 00:32:01.000
atom or molecule.
00:32:01.000 --> 00:32:04.450
There's lots of ways of saying,
it's not over all space,
00:32:04.450 --> 00:32:07.900
but for each operator you
know what space the operator
00:32:07.900 --> 00:32:10.290
in question is dealing with.
00:32:10.290 --> 00:32:12.730
And then it's always
possible to write.
00:32:19.920 --> 00:32:22.020
So this is some
general function,
00:32:22.020 --> 00:32:24.670
defined in the space
of the operator,
00:32:24.670 --> 00:32:27.300
and this is the equation
that says, well, completeness
00:32:27.300 --> 00:32:31.350
tells us that we can
take the sum over all
00:32:31.350 --> 00:32:35.850
of the eigenfunctions
with mixing coefficients.
00:32:35.850 --> 00:32:38.560
c j.
00:32:38.560 --> 00:32:41.290
And this set of all
of the eigenfunctions
00:32:41.290 --> 00:32:43.510
is called the basis set.
00:32:43.510 --> 00:32:45.040
It's a complete basis set.
00:32:48.070 --> 00:32:50.300
So it's always true,
you can do this.
00:32:50.300 --> 00:32:53.090
So you know from other
problems, that if you
00:32:53.090 --> 00:32:59.630
have a finite region of space,
you can represent anything
00:32:59.630 --> 00:33:06.770
within that finite region via
sum over Fourier components.
00:33:06.770 --> 00:33:08.930
That's a discrete sum.
00:33:08.930 --> 00:33:10.430
If you have an
infinite space, you
00:33:10.430 --> 00:33:13.490
have to do a Fourier
integral, but it's basically
00:33:13.490 --> 00:33:14.120
the same thing.
00:33:14.120 --> 00:33:18.170
You're expressing a
function, anything you want,
00:33:18.170 --> 00:33:21.920
in terms of simple,
manipulable objects.
00:33:24.730 --> 00:33:29.410
So sometimes the these things
are Fourier components,
00:33:29.410 --> 00:33:33.028
and sometimes they're just
simple wave functions.
00:33:36.020 --> 00:33:41.380
OK, now suppose you have
two operators, a and b.
00:33:45.130 --> 00:33:47.780
If they operate
over the same space,
00:33:47.780 --> 00:33:55.140
the question is, can we
take eigenfunctions of one
00:33:55.140 --> 00:33:57.450
and be sure that they're
eigenfunctions of the other?
00:34:09.840 --> 00:34:13.719
OK, but let's deal
with something simpler.
00:34:13.719 --> 00:34:37.650
So suppose we have psi i and
psi j, both belong to a sub i.
00:34:37.650 --> 00:34:39.784
So they both have
the same eigenvalue.
00:34:43.659 --> 00:34:48.310
Well, in that case, we cannot
be sure that these two functions
00:34:48.310 --> 00:34:50.850
are orthogonal.
00:34:50.850 --> 00:34:53.960
So there is a handy
dandy procedure
00:34:53.960 --> 00:35:01.580
called Schmidt
Orthogonalization that
00:35:01.580 --> 00:35:08.000
says, take any two
functions, and let's
00:35:08.000 --> 00:35:11.650
construct an orthogonal pair.
00:35:11.650 --> 00:35:15.010
This is amazingly
useful when you're
00:35:15.010 --> 00:35:19.330
trying to understand
a problem using
00:35:19.330 --> 00:35:21.520
a complete
orthonormal basis set.
00:35:24.470 --> 00:35:27.890
We know, I'm not
going to prove it,
00:35:27.890 --> 00:35:30.950
because I don't know where
it is in my notes, what
00:35:30.950 --> 00:35:35.040
the sequence is, I'm
just going to forget it--
00:35:35.040 --> 00:35:37.100
I think I'm going to do it.
00:35:37.100 --> 00:35:41.300
If you have functions belonging
to different eigenvalues,
00:35:41.300 --> 00:35:43.010
they are automatically
orthogonal.
00:35:43.010 --> 00:35:48.140
That's also really valuable,
because you have to check.
00:35:48.140 --> 00:35:50.480
So you have harmonic
oscillator functions,
00:35:50.480 --> 00:35:53.690
and they're all
orthogonal to each other.
00:35:53.690 --> 00:35:57.110
You have, perhaps, one
harmonic oscillator,
00:35:57.110 --> 00:35:59.250
and a different
harmonic oscillator,
00:35:59.250 --> 00:36:02.044
and the functions for these two
different harmonic oscillators
00:36:02.044 --> 00:36:03.210
don't know about each other.
00:36:03.210 --> 00:36:05.960
They're not guaranteed
to be orthogonal.
00:36:05.960 --> 00:36:11.075
But any two eigenvalues of
this guy are orthogonal,
00:36:11.075 --> 00:36:12.770
and any of those are orthogonal.
00:36:12.770 --> 00:36:14.680
But not here.
00:36:14.680 --> 00:36:21.840
OK, so let's just
say we have now two
00:36:21.840 --> 00:36:24.390
eigenfunctions of
an operator that
00:36:24.390 --> 00:36:28.250
belong to the same eigenvalues.
00:36:28.250 --> 00:36:29.280
And that happens.
00:36:29.280 --> 00:36:33.630
There are often very many,
very high degeneracies.
00:36:33.630 --> 00:36:35.880
But we want to make
sure that we've got two.
00:36:35.880 --> 00:36:40.770
So let's say, here is a
number, the overlap integral
00:36:40.770 --> 00:36:45.600
between psi 1 and psi 2 dx.
00:36:45.600 --> 00:36:47.580
This is a calculable number.
00:36:47.580 --> 00:36:50.220
So you have two
original functions,
00:36:50.220 --> 00:36:52.920
which are not guaranteed
to be orthogonal,
00:36:52.920 --> 00:36:55.590
because they belong to
the same eigenvalue,
00:36:55.590 --> 00:36:58.290
and you can calculate
this number.
00:36:58.290 --> 00:37:02.760
And then we can say, let us
now construct something up psi
00:37:02.760 --> 00:37:08.610
2, which is guaranteed to be a
psi 2 prime which is guaranteed
00:37:08.610 --> 00:37:10.680
to be orthogonal to psi 1.
00:37:10.680 --> 00:37:11.430
How do we do that?
00:37:14.300 --> 00:37:19.310
Well, we define psi 2
to be a normalization
00:37:19.310 --> 00:37:22.700
factor, times psi
2 the original,
00:37:22.700 --> 00:37:28.250
plus some constant times psi 1.
00:37:28.250 --> 00:37:32.900
And then we say, let us
calculate the overlap integral
00:37:32.900 --> 00:37:36.305
between psi 1 and psi 2 prime.
00:37:39.780 --> 00:37:42.240
OK, so we do that.
00:37:42.240 --> 00:37:49.980
And so we have the integral, and
we have psi 1 star times psi 2
00:37:49.980 --> 00:37:54.320
plus a psi 1 dx.
00:37:54.320 --> 00:37:57.910
Psi 1 on psi 1 is 1.
00:37:57.910 --> 00:37:59.250
So we get an a.
00:37:59.250 --> 00:38:05.310
And psi 1 star
times psi 2 gives s.
00:38:05.310 --> 00:38:10.110
So this integral,
which is supposed
00:38:10.110 --> 00:38:14.970
to be 0, because
we want psi 1 to be
00:38:14.970 --> 00:38:21.270
orthogonal to psi 2 prime, is
going to be n times s plus a.
00:38:23.970 --> 00:38:25.230
Well, how do we satisfy this?
00:38:25.230 --> 00:38:29.770
We just make a to be minus s.
00:38:29.770 --> 00:38:30.414
Guaranteed.
00:38:30.414 --> 00:38:32.080
Now this is one of
the tricks that I use
00:38:32.080 --> 00:38:35.230
the most when I do derivations.
00:38:35.230 --> 00:38:38.020
I want orthogonal functions,
and this little Schmidt
00:38:38.020 --> 00:38:41.740
Orthogonalization enables
me to take my simple idea
00:38:41.740 --> 00:38:44.180
and propagate it into
a complicated problem.
00:38:44.180 --> 00:38:45.460
It's very valuable.
00:38:45.460 --> 00:38:49.150
You'll probably never
use it, but I love it.
00:38:49.150 --> 00:38:53.890
So if a is equal to minus
s, you've got orthogonality.
00:38:53.890 --> 00:38:59.830
And the general formula
is psi 2 prime is equal
00:38:59.830 --> 00:39:07.770
to 1 minus x squared,
the square root of that,
00:39:07.770 --> 00:39:11.810
times psi 2 minus s psi 1.
00:39:15.460 --> 00:39:20.140
So this is a normalized function
which is orthogonal to psi 1.
00:39:23.040 --> 00:39:24.290
And it doesn't take much work.
00:39:24.290 --> 00:39:27.880
You calculate one
interval, and it's done.
00:39:27.880 --> 00:39:30.160
Now later in the
course, we're going
00:39:30.160 --> 00:39:34.060
to talk we're going to talk
about a secular determinant
00:39:34.060 --> 00:39:37.570
that you use to solve
for all the eigenvalues
00:39:37.570 --> 00:39:40.480
and eigenfunctions of
a complicated problem.
00:39:40.480 --> 00:39:44.410
And often, when you do that,
you use convenient functions
00:39:44.410 --> 00:39:47.020
which are not guaranteed
to be orthogonal.
00:39:47.020 --> 00:39:52.570
And there is a procedure you
apply to this secular matrix,
00:39:52.570 --> 00:39:55.240
which orthogonalizes it first.
00:39:55.240 --> 00:39:57.760
Then you diagonalize
a simple thing,
00:39:57.760 --> 00:40:00.170
and then you undiagonalize,
if you need to.
00:40:00.170 --> 00:40:03.640
Anyway, this is terribly
valuable, especially
00:40:03.640 --> 00:40:05.710
when you're doing quantum
chemistry, which you're
00:40:05.710 --> 00:40:08.511
going to see towards the end.
00:40:08.511 --> 00:40:09.010
OK.
00:40:17.630 --> 00:40:22.750
Often, we would like to express
a particular eigenfunction
00:40:22.750 --> 00:40:29.890
expectation value as
a sum over P i a i.
00:40:29.890 --> 00:40:32.720
So this is a probability.
00:40:32.720 --> 00:40:35.260
And so how do we do that?
00:40:35.260 --> 00:40:41.320
Certainly, the average
of this operator over psi
00:40:41.320 --> 00:40:46.270
can be written as the
eigenvalues of a times
00:40:46.270 --> 00:40:48.281
the probability
of each operator.
00:40:48.281 --> 00:40:49.030
And what are they?
00:40:49.030 --> 00:40:52.480
Well, you can show that
this is equal to c i--
00:40:56.230 --> 00:40:58.892
I better be careful here.
00:40:58.892 --> 00:40:59.850
Well, let's just do it.
00:41:04.030 --> 00:41:05.320
Well, I'll just write it.
00:41:09.110 --> 00:41:11.300
Where this is the
mixing coefficient
00:41:11.300 --> 00:41:17.750
of the eigenfunction of a
in the original function.
00:41:17.750 --> 00:41:22.130
So we get probabilities
mixing fractions,
00:41:22.130 --> 00:41:24.950
and we have mixing coefficients.
00:41:24.950 --> 00:41:26.550
And this is the
language we're going
00:41:26.550 --> 00:41:29.336
to use to describe many things.
00:41:29.336 --> 00:41:30.520
AUDIENCE: [INAUDIBLE]
00:41:30.520 --> 00:41:31.520
ROBERT FIELD: I'm sorry?
00:41:31.520 --> 00:41:32.895
AUDIENCE: The
probability doesn't
00:41:32.895 --> 00:41:35.502
have an a i, the average does.
00:41:35.502 --> 00:41:39.950
You just remove the a i.
00:41:39.950 --> 00:41:41.615
ROBERT FIELD: So
the probability--
00:41:41.615 --> 00:41:42.740
AUDIENCE: The probability--
00:41:42.740 --> 00:41:44.750
ROBERT FIELD: Oh, yes.
00:41:44.750 --> 00:41:49.390
You see, when I start
lecturing from what's
00:41:49.390 --> 00:41:50.395
in my addled brain--
00:41:52.900 --> 00:41:55.210
OK, thank you.
00:42:02.640 --> 00:42:04.440
OK.
00:42:04.440 --> 00:42:06.150
Now let's do some
really neat things.
00:42:19.930 --> 00:42:23.800
So we have a commutator
of two operators.
00:42:26.730 --> 00:42:34.490
If that commutator is 0,
then all non-degenerate
00:42:34.490 --> 00:42:45.770
eigenfunctions of A are
eigenfunctions of B.
00:42:45.770 --> 00:42:52.670
If this is not equal to 0,
then we can say something about
00:42:52.670 --> 00:43:02.130
the variances of A and B. So
these quantities are greater
00:43:02.130 --> 00:43:11.450
than or equal to minus
1/4 times the integral--
00:43:11.450 --> 00:43:13.620
I better write this
on the board below.
00:43:34.530 --> 00:43:37.820
And this is greater
than 0, and real.
00:43:37.820 --> 00:43:39.705
This is the
uncertainty principle.
00:43:42.620 --> 00:43:48.740
So it's possible to prove
this, and it's really strange,
00:43:48.740 --> 00:43:50.990
because we have a
square of a number here.
00:43:54.010 --> 00:43:58.230
We think the square of a
number is going to be real,
00:43:58.230 --> 00:44:01.200
but not if it's imaginary.
00:44:01.200 --> 00:44:07.740
Most non-zero commutators
are imaginary.
00:44:07.740 --> 00:44:11.340
And so this thing is
negative, and it's canceled.
00:44:11.340 --> 00:44:16.710
So the joint uncertainty is
related to the expectation
00:44:16.710 --> 00:44:19.590
value of a commutator.
00:44:19.590 --> 00:44:26.850
And this is all traced back
to x P x is equal to ih bar.
00:44:30.360 --> 00:44:35.090
And this commutator
is imaginary.
00:44:35.090 --> 00:44:40.350
And everything that
appears in here,
00:44:40.350 --> 00:44:42.840
it comes from the
non-commutation
00:44:42.840 --> 00:44:45.360
of coordinate and momentum.
00:44:45.360 --> 00:44:50.070
And this is why this
commutator is often
00:44:50.070 --> 00:44:55.840
regarded as the foundation
of quantum mechanics.
00:44:55.840 --> 00:44:58.610
Because all of the
strangeness comes from it.
00:45:03.740 --> 00:45:06.500
So yes, this is surprising.
00:45:06.500 --> 00:45:09.110
It's saying that, when we
have a non-zero commutator,
00:45:09.110 --> 00:45:12.500
this is what determines the
joint uncertainty of two
00:45:12.500 --> 00:45:13.220
properties.
00:45:16.220 --> 00:45:18.050
This commutator is
always imaginary,
00:45:18.050 --> 00:45:21.830
that is a big, big surprise.
00:45:21.830 --> 00:45:27.390
And as a result the joint
uncertainty is greater than 0,
00:45:27.390 --> 00:45:29.580
If the two operators
don't commute,
00:45:29.580 --> 00:45:31.230
it's because x and
P don't commute.
00:45:33.870 --> 00:45:34.820
It's really scary.
00:45:39.100 --> 00:45:40.440
OK, what time is it?
00:45:40.440 --> 00:45:45.680
We've got five minutes left,
and I can do one more thing.
00:45:45.680 --> 00:45:49.700
Well, I guess I'm going
to be just talking about--
00:45:49.700 --> 00:45:53.730
So the uncertainty principle.
00:46:01.820 --> 00:46:12.900
If we know operator A and B, we
can calculate their commutator.
00:46:12.900 --> 00:46:15.150
This is a property
of the operators.
00:46:15.150 --> 00:46:16.890
It doesn't have to
do with constructing
00:46:16.890 --> 00:46:19.560
some clever experiment,
where you tried to measure
00:46:19.560 --> 00:46:23.170
two things simultaneously.
00:46:23.170 --> 00:46:29.980
It says, all of the problems
with simultaneous observations
00:46:29.980 --> 00:46:33.940
of two operators, two things,
comes from the structure
00:46:33.940 --> 00:46:34.780
of the operators.
00:46:34.780 --> 00:46:37.060
Comes from their
commutation rule.
00:46:37.060 --> 00:46:49.030
Which traces all the way back to
the commutator between x and P.
00:46:49.030 --> 00:46:52.330
So at the beginning, I said I
don't like these experiments,
00:46:52.330 --> 00:46:59.620
where we try to confine the
coordinate of the photon
00:46:59.620 --> 00:47:04.060
or the electron, and it
results in uncertainty
00:47:04.060 --> 00:47:06.410
in the measurement of
the conjugate property,
00:47:06.410 --> 00:47:09.550
the momentum, or
something like that.
00:47:09.550 --> 00:47:13.771
These experiments depend on your
cleverness, but this doesn't.
00:47:13.771 --> 00:47:16.970
This Is fundamental.
00:47:16.970 --> 00:47:18.160
So I like that a lot better.
00:47:21.930 --> 00:47:24.340
OK, the last thing I
want to talk about,
00:47:24.340 --> 00:47:27.070
which I have just barely
enough time to do,
00:47:27.070 --> 00:47:41.770
is suppose we have
a wave function.
00:47:41.770 --> 00:47:49.510
Let's call it psi 2 in quotes
of x for the particle in a box.
00:47:49.510 --> 00:47:52.140
Particle in a infinite box.
00:47:52.140 --> 00:47:56.290
This is a wave function, which
is not the eigenfunction,
00:47:56.290 --> 00:47:59.790
but it is constructed to
look like the eigenfunction.
00:47:59.790 --> 00:48:04.350
Mainly because it has the
right number of nodes.
00:48:04.350 --> 00:48:07.890
And so suppose we
call this thing
00:48:07.890 --> 00:48:19.050
some normalization factor x x
minus a and x minus a over 2.
00:48:19.050 --> 00:48:23.610
This guarantees you have
a node at x equals 0.
00:48:23.610 --> 00:48:27.990
This guarantees you have
a node at x equals a.
00:48:27.990 --> 00:48:32.400
This guarantees you have a
node at x equals a over 2.
00:48:32.400 --> 00:48:36.360
So this is that the
generic property
00:48:36.360 --> 00:48:40.320
of the second eigenfunction
for the particle in a box.
00:48:43.140 --> 00:48:47.080
And this is a very clever
guess, and often you
00:48:47.080 --> 00:48:49.030
want to make a guess.
00:48:49.030 --> 00:48:53.140
And so, how well
you do with a guess?
00:48:53.140 --> 00:49:01.470
And so this function,
this part of it
00:49:01.470 --> 00:49:08.620
looks sort of like this between
x and a, and this part of it
00:49:08.620 --> 00:49:11.110
looks like this.
00:49:11.110 --> 00:49:13.390
And we multiply
these two together
00:49:13.390 --> 00:49:19.220
and we get something
that looks like this.
00:49:19.220 --> 00:49:22.600
Which is, at least
a sketch, looks
00:49:22.600 --> 00:49:26.689
like the n equals
2 eigenfunction
00:49:26.689 --> 00:49:27.730
with a particle in a box.
00:49:27.730 --> 00:49:33.970
So let's just go through
and see how well we can do.
00:49:33.970 --> 00:49:37.390
So first of all, we
have to determine n.
00:49:37.390 --> 00:49:40.625
So we do the
normalization integral.
00:49:40.625 --> 00:49:49.310
Psi 2 star psi 2 dx.
00:49:49.310 --> 00:49:50.400
That has to be 1.
00:49:55.590 --> 00:49:59.490
So what we do, now this
is kind of a cheat,
00:49:59.490 --> 00:50:03.270
because we do this because we
don't know an eigenfunction.
00:50:03.270 --> 00:50:04.980
But we do know these
eigenfunctions,
00:50:04.980 --> 00:50:06.840
so we can expand
these functions,
00:50:06.840 --> 00:50:13.890
in terms of the particle
in a box eigenfunctions.
00:50:13.890 --> 00:50:19.920
So we use these things,
which you know very well.
00:50:28.090 --> 00:50:31.130
Now we have a sine
function here.
00:50:31.130 --> 00:50:33.250
That's because I've
chosen the box that
00:50:33.250 --> 00:50:39.310
doesn't have 0 left edge, it's
symmetric about x equals 0.
00:50:39.310 --> 00:50:41.700
And that would be appropriate
for this kind of function.
00:50:46.610 --> 00:50:51.560
So anyway, when we do this, we
find the mixing coefficients.
00:50:51.560 --> 00:50:55.630
And I know I just
said something wrong
00:50:55.630 --> 00:50:58.870
and that's, we don't
have time to correct it.
00:50:58.870 --> 00:51:03.210
Because I said it the wave
function is 0 at x equals 0,
00:51:03.210 --> 00:51:04.192
and at x equals a.
00:51:04.192 --> 00:51:05.650
And I'm now saying,
all of a sudden
00:51:05.650 --> 00:51:09.400
I'm using symmetric box--
00:51:09.400 --> 00:51:12.520
this does not matter,
because the calculation
00:51:12.520 --> 00:51:13.900
is done correctly.
00:51:13.900 --> 00:51:19.270
And what we end up
getting, is that the mixing
00:51:19.270 --> 00:51:27.690
coefficients for these functions
of the general form, 800--
00:51:27.690 --> 00:51:34.640
I'm sorry, 840--
this is algebra!--
00:51:34.640 --> 00:51:44.520
Square root a to the
minus 7/2 2 over n--
00:51:44.520 --> 00:51:47.112
2 over a.
00:51:47.112 --> 00:51:47.820
I think that's a.
00:51:51.850 --> 00:51:53.890
Well, I'm not sure
whether that's a or n,
00:51:53.890 --> 00:52:00.190
but let's just say it's
2 over n square root--
00:52:00.190 --> 00:52:02.320
oh, it's going to be 2 over a.
00:52:02.320 --> 00:52:12.340
And times the
integral-- anyway, so we
00:52:12.340 --> 00:52:29.830
get c 2n is equal to 1680 square
root over 6 over 2n pi cubed.
00:52:33.270 --> 00:52:43.230
And that becomes equal to
0.9914n to the minus 3.
00:52:43.230 --> 00:52:46.950
So this is a general formula
which you can derive.
00:52:46.950 --> 00:52:48.420
I don't recommend
it, and I don't
00:52:48.420 --> 00:52:49.650
think it's really important.
00:52:49.650 --> 00:52:52.540
The important thing is
what I'm about to say.
00:52:52.540 --> 00:52:54.570
And I have no time.
00:52:54.570 --> 00:52:55.710
This is almost 1.
00:53:01.140 --> 00:53:10.450
So when we calculate the
energy using these functions,
00:53:10.450 --> 00:53:17.290
we get that the energy
of this 2n function
00:53:17.290 --> 00:53:31.160
is equal to 4E1 times
0.983 integral from n
00:53:31.160 --> 00:53:39.900
equals 1 to infinity
times n to the minus 4.
00:53:43.180 --> 00:53:47.050
OK, the first term
and this is 1.
00:53:47.050 --> 00:53:54.040
And so we have something
that looks like 4 times E1.
00:53:54.040 --> 00:53:58.740
Now the sum is larger than one,
and the product of these two
00:53:58.740 --> 00:54:01.140
things is larger than 1.
00:54:01.140 --> 00:54:10.080
And so what we get is that, E2n
is larger, but only slightly
00:54:10.080 --> 00:54:17.660
larger, than the exact results.
00:54:17.660 --> 00:54:22.630
So this is sort of a taste
of a variation calculation.
00:54:22.630 --> 00:54:30.060
We can solve for the
form of a function
00:54:30.060 --> 00:54:36.780
by doing a minimization of
the energy of that function.
00:54:36.780 --> 00:54:41.040
And that function will look
like the true function,
00:54:41.040 --> 00:54:45.570
but its energy will always be
larger than the true function.
00:54:48.450 --> 00:54:51.460
But it's great, because
the bigger the calculation,
00:54:51.460 --> 00:54:52.750
the better you do.
00:54:52.750 --> 00:54:55.390
And that's how most of
the money, the computer
00:54:55.390 --> 00:54:58.330
time in the world, is expended.
00:54:58.330 --> 00:55:00.970
Doing large variational
calculations
00:55:00.970 --> 00:55:06.110
to find eigenfunctions
of complicated problems.
00:55:06.110 --> 00:55:08.720
OK, good luck on the exam.
00:55:08.720 --> 00:55:14.130
I hope you find it fun,
and I meant it to be fun.