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 ocw.mit.edu.
00:00:24.570 --> 00:00:26.820
ROBERT FIELD: So
today we're going
00:00:26.820 --> 00:00:29.190
to go from the classical
mechanical treatment
00:00:29.190 --> 00:00:33.360
of the harmonic oscillator to
a quantum mechanical treatment.
00:00:33.360 --> 00:00:37.520
And I warn you that
I intentionally
00:00:37.520 --> 00:00:41.710
am going to make this
look bad, because
00:00:41.710 --> 00:00:48.140
the semi-classical approach
at the end of this lecture
00:00:48.140 --> 00:00:50.480
will make it all really simple.
00:00:50.480 --> 00:00:52.670
And then on Monday
I'll introduce
00:00:52.670 --> 00:00:54.920
creation and
annihilation operators,
00:00:54.920 --> 00:00:57.050
which makes the harmonic
oscillator simpler
00:00:57.050 --> 00:00:59.530
than the particle in a box.
00:00:59.530 --> 00:01:02.470
You can't believe
that, but all right.
00:01:02.470 --> 00:01:10.930
So last time we treated the
harmonic oscillator classically
00:01:10.930 --> 00:01:16.150
and so we derive the equation
of motion from forces
00:01:16.150 --> 00:01:19.910
equal to mass times
acceleration, and we solved it.
00:01:19.910 --> 00:01:26.140
And we saw that we have this
quantity omega, which initially
00:01:26.140 --> 00:01:28.090
I just introduced
as a constant, which
00:01:28.090 --> 00:01:35.700
was a way of combining the
force constant and the mass.
00:01:35.700 --> 00:01:40.320
And then I showed that
the period of oscillation
00:01:40.320 --> 00:01:45.105
is 1 over the frequency,
which is 2 pi over omega.
00:01:47.980 --> 00:01:50.140
Now one of the
things that people
00:01:50.140 --> 00:01:54.160
have trouble remembering
under exam pressure
00:01:54.160 --> 00:01:55.585
is turning points.
00:02:01.620 --> 00:02:05.490
And this comes
when the energy is
00:02:05.490 --> 00:02:08.900
equal to the potential
at a turning point.
00:02:15.490 --> 00:02:18.140
And since the
potential is 1/2 kx
00:02:18.140 --> 00:02:23.540
squared we get the equation
for the turning point
00:02:23.540 --> 00:02:28.520
at a given energy, which
is equal to plus or minus
00:02:28.520 --> 00:02:30.260
the square root of 2e/k.
00:02:35.420 --> 00:02:37.940
Now when you're
drawing pictures there
00:02:37.940 --> 00:02:40.542
are certain things that
anchor the pictures,
00:02:40.542 --> 00:02:41.375
like turning points.
00:02:44.000 --> 00:02:49.120
And now at the end of
the previous lecture
00:02:49.120 --> 00:02:54.170
I calculated the classical
mechanical average,
00:02:54.170 --> 00:02:57.010
and we use we use
this kind of notation
00:02:57.010 --> 00:02:58.090
in classical mechanics.
00:02:58.090 --> 00:03:00.380
Sometimes we use
this notation, too,
00:03:00.380 --> 00:03:03.640
but this is what we mean by
the average value in quantum
00:03:03.640 --> 00:03:04.850
mechanics.
00:03:04.850 --> 00:03:12.040
And we found that this was
the total energy divided by 2.
00:03:12.040 --> 00:03:19.817
And the average momentum
is the energy divided by 2.
00:03:19.817 --> 00:03:21.400
And that's the basis
for some insight,
00:03:21.400 --> 00:03:24.370
because as a harmonic
oscillator moves
00:03:24.370 --> 00:03:28.270
it throws energy back and
forth between kinetic energy
00:03:28.270 --> 00:03:30.310
and potential energy.
00:03:30.310 --> 00:03:32.230
And then one of my
favorite things,
00:03:32.230 --> 00:03:36.340
and one of my favorite
tortures on short answers,
00:03:36.340 --> 00:03:41.800
is the average values of--
00:03:47.460 --> 00:03:53.220
And does anybody want to
remind the class the easy way,
00:03:53.220 --> 00:03:57.300
or the two easy ways
to know what this is?
00:03:57.300 --> 00:04:00.190
And the person who answered
the question last time
00:04:00.190 --> 00:04:02.040
is disqualified.
00:04:02.040 --> 00:04:02.966
Yes.
00:04:02.966 --> 00:04:04.590
AUDIENCE: There is
the symmetry method.
00:04:04.590 --> 00:04:05.020
ROBERT FIELD: What?
00:04:05.020 --> 00:04:06.350
AUDIENCE: There's symmetry.
00:04:06.350 --> 00:04:10.650
So if the system's symmetrical
that the average value will
00:04:10.650 --> 00:04:12.880
actually be that symmetry point.
00:04:12.880 --> 00:04:14.100
ROBERT FIELD: Yes.
00:04:14.100 --> 00:04:17.470
So and the other is that
the harmonic oscillator
00:04:17.470 --> 00:04:22.390
isn't moving, and so there is
no way that the coordinate--
00:04:22.390 --> 00:04:28.720
that the average value of either
the coordinate or the momentum
00:04:28.720 --> 00:04:30.730
could be different from 0.
00:04:30.730 --> 00:04:33.280
However you do it you
want to be doing it
00:04:33.280 --> 00:04:35.260
in seconds, not minutes.
00:04:35.260 --> 00:04:37.990
And certainly not by
calculating integral.
00:04:37.990 --> 00:04:43.180
And now x squared and p
squared, they're easy too.
00:04:43.180 --> 00:04:50.560
Especially if you know t and
v. And so we then use those
00:04:50.560 --> 00:04:55.820
to calculate the variance.
00:04:55.820 --> 00:04:58.990
And that's defined
as the average value
00:04:58.990 --> 00:05:03.880
of the square minus the square
of the average square root.
00:05:07.100 --> 00:05:11.690
And what we find is
that the variance
00:05:11.690 --> 00:05:19.470
of x times the variance of
p is equal to e over omega.
00:05:19.470 --> 00:05:26.160
So as you go up in energy this
joint uncertainty increases.
00:05:26.160 --> 00:05:32.040
And we'll find that that also
is true for quantum mechanics.
00:05:32.040 --> 00:05:35.550
So this is sort of
the kind of questions
00:05:35.550 --> 00:05:38.940
you want to be asking
in quantum mechanics.
00:05:38.940 --> 00:05:41.100
And you want to be
able to be guided
00:05:41.100 --> 00:05:43.340
by what you know from
classical mechanics,
00:05:43.340 --> 00:05:46.480
and you want to be
able to do it fast.
00:05:46.480 --> 00:05:49.240
Today's menu is
what I would call--
00:05:59.630 --> 00:06:03.110
This lecture is
gratuitous complexity.
00:06:03.110 --> 00:06:05.210
Does everybody know
what gratuitous means?
00:06:09.380 --> 00:06:12.200
This is one of my
favorite Bobisms.
00:06:15.410 --> 00:06:19.340
And you'll hear other
Bobisms during the course
00:06:19.340 --> 00:06:21.320
of this course.
00:06:24.920 --> 00:06:27.550
What I want you to be
able to do for a lot
00:06:27.550 --> 00:06:32.620
of mechanical problems
is to know the answer,
00:06:32.620 --> 00:06:36.760
or know what things look like
without doing a calculation.
00:06:36.760 --> 00:06:39.670
In particular, not solving
a differential equation
00:06:39.670 --> 00:06:41.800
or evaluating in the integrals.
00:06:41.800 --> 00:06:47.050
You want to be able to draw
these pictures instantly.
00:06:47.050 --> 00:06:53.860
Now in the modern age
everyone has a cell phone,
00:06:53.860 --> 00:06:55.750
and one could have
a program in there
00:06:55.750 --> 00:06:59.720
to calculate what anything you
wanted for harmonic oscillator.
00:06:59.720 --> 00:07:03.700
But chances are you won't
be prepared for that.
00:07:03.700 --> 00:07:08.260
And if you want to have insights
into how do various things
00:07:08.260 --> 00:07:13.180
you want to know about harmonic
oscillators come about?
00:07:13.180 --> 00:07:15.640
You need the
pictures, as opposed
00:07:15.640 --> 00:07:17.710
to the computer program.
00:07:17.710 --> 00:07:23.650
Now the pictures involve an
advance investment of energy.
00:07:23.650 --> 00:07:25.174
You want to understand
every detail
00:07:25.174 --> 00:07:26.215
of these little pictures.
00:07:31.254 --> 00:07:32.920
I'm going to write
this runner equation,
00:07:32.920 --> 00:07:35.770
I'm going to clean it
up to get rid of units
00:07:35.770 --> 00:07:38.080
which makes it universal.
00:07:38.080 --> 00:07:41.690
So it becomes a
dimensionless equation.
00:07:41.690 --> 00:07:46.180
And the unit removal,
or the thing that
00:07:46.180 --> 00:07:50.740
takes you from a specific
problem where there's
00:07:50.740 --> 00:07:55.960
a particular force constant
and a particular reduced mass,
00:07:55.960 --> 00:07:57.640
and makes it into
a general problem.
00:07:57.640 --> 00:08:03.500
There is one or two constants
that combine those things.
00:08:03.500 --> 00:08:06.700
And you've taken them out
at the end of a calculation.
00:08:06.700 --> 00:08:11.350
If you need to have real units
you can put those back in.
00:08:11.350 --> 00:08:12.820
And that's a very
wonderful thing,
00:08:12.820 --> 00:08:16.630
and that enables
us to draw pictures
00:08:16.630 --> 00:08:21.180
without thinking about,
what is the problem?
00:08:21.180 --> 00:08:24.600
And then the solution of
this differential equation--
00:08:24.600 --> 00:08:28.260
which is actually quite an awful
differential equation at least
00:08:28.260 --> 00:08:30.900
for people who are
not mathematicians--
00:08:30.900 --> 00:08:33.750
and the solution
can be expressed
00:08:33.750 --> 00:08:36.090
as the product of
a Gaussian function
00:08:36.090 --> 00:08:39.990
which goes to 0 at plus and
minus infinity, so it makes
00:08:39.990 --> 00:08:42.390
the function well behaved.
00:08:42.390 --> 00:08:46.200
Times something that
produces nodes, a polynomial.
00:08:46.200 --> 00:08:48.960
Does anyone want to give me
a definition of a polynomial?
00:08:56.720 --> 00:08:58.430
Silence.
00:08:58.430 --> 00:09:01.495
OK, your turn.
00:09:01.495 --> 00:09:06.740
AUDIENCE: It's the linear
combination of some numbers
00:09:06.740 --> 00:09:08.417
taken to different power
00:09:08.417 --> 00:09:09.250
ROBERT FIELD: Right.
00:09:09.250 --> 00:09:12.185
A sum of integer
powers of a variable.
00:09:14.700 --> 00:09:20.610
And when we take a
derivative of a polynomial
00:09:20.610 --> 00:09:23.610
we reduce the order
of the polynomial.
00:09:23.610 --> 00:09:26.820
A little bit of thought,
if you have a first order
00:09:26.820 --> 00:09:29.260
polynomial there'll be one node.
00:09:29.260 --> 00:09:31.380
If there is a second order
there'll be two nodes.
00:09:33.920 --> 00:09:37.130
And nodes are very important.
00:09:37.130 --> 00:09:40.870
And so when we're going to
be dealing with cartoons
00:09:40.870 --> 00:09:44.650
of the wave function, and then
using semi-classical ideas
00:09:44.650 --> 00:09:49.450
to actually semi-calculate
things that you'd want to know,
00:09:49.450 --> 00:09:53.740
the nodes are really important.
00:09:53.740 --> 00:09:57.760
And what's going
to happen Monday
00:09:57.760 --> 00:10:00.980
is we'll throw away
all this garbage
00:10:00.980 --> 00:10:05.120
and we will replace
everything by these creation
00:10:05.120 --> 00:10:07.370
and annihilation operators.
00:10:07.370 --> 00:10:12.360
Which do have really
simple properties,
00:10:12.360 --> 00:10:16.190
which you can use
to do astonishingly
00:10:16.190 --> 00:10:19.190
complicated things
without breaking a sweat.
00:10:25.910 --> 00:10:30.380
And the final exam is in this
room on the first day of exam
00:10:30.380 --> 00:10:33.290
period, at least
it's on a Monday,
00:10:33.290 --> 00:10:34.864
and it's in the afternoon.
00:10:47.120 --> 00:10:51.980
In the non-lecture
part of the notes
00:10:51.980 --> 00:10:58.060
I replaced the mass for
one mass on a spring
00:10:58.060 --> 00:11:06.680
by the reduced mass, which is
m1 m2 over m1 plus m2 for two
00:11:06.680 --> 00:11:09.180
masses connected by a spring.
00:11:09.180 --> 00:11:16.640
And I go back and forth between
using mu and m, and that's OK.
00:11:16.640 --> 00:11:22.450
All right so in the notes
the differential equations
00:11:22.450 --> 00:11:24.160
in the first few
pages are expressed
00:11:24.160 --> 00:11:26.210
as partial
differential equations.
00:11:26.210 --> 00:11:28.780
They're total
differential equations.
00:11:28.780 --> 00:11:31.960
That'll get changed.
00:11:31.960 --> 00:11:43.610
The Hamiltonian is t plus
v, and in the usual form t
00:11:43.610 --> 00:11:45.920
is p squared over 2 mu.
00:11:45.920 --> 00:11:54.460
And so we get minus h-bar
squared over 2 mu par--
00:11:54.460 --> 00:11:56.345
not partial, I'm so
used to writing partials
00:11:56.345 --> 00:11:59.360
that I can't stop.
00:11:59.360 --> 00:12:08.390
Second derivative with respect
to x, plus 1/2 kx squared.
00:12:08.390 --> 00:12:12.080
So that's the Hamiltonian.
00:12:12.080 --> 00:12:16.310
Now that looks kind of
innocent, but it isn't.
00:12:16.310 --> 00:12:18.350
And so the first
thing we want to do
00:12:18.350 --> 00:12:22.260
is get rid of the
dimensionality, the units.
00:12:22.260 --> 00:12:25.670
So this is xi and it's
defined as square root
00:12:25.670 --> 00:12:27.880
of alpha times x.
00:12:27.880 --> 00:12:36.290
Where alpha is defined as k
mu square root over h-bar.
00:12:40.120 --> 00:12:43.870
Now it would be a perfectly
reasonable exam question
00:12:43.870 --> 00:12:49.210
for you to prove that if
I take this combination
00:12:49.210 --> 00:12:53.140
of physical quantities this
will have dimension of 1
00:12:53.140 --> 00:12:53.660
over length.
00:12:57.980 --> 00:13:01.220
That makes xi a
dimensionless quantity.
00:13:05.670 --> 00:13:09.020
I'm not even going to bother
going through the derivation.
00:13:09.020 --> 00:13:16.920
The Hamiltonian becomes
h-bar of omega times 2,
00:13:16.920 --> 00:13:27.110
minus second derivative with
respect to xi plus xi squared.
00:13:31.800 --> 00:13:42.080
So this is now dimensionless.
00:13:42.080 --> 00:13:43.380
This has units.
00:13:43.380 --> 00:13:48.060
We divide by h-bar omega to make
now everything dimensionless.
00:13:48.060 --> 00:13:52.380
And we get a
differential equation
00:13:52.380 --> 00:13:57.900
that has the form minus the
second derivative with respect
00:13:57.900 --> 00:14:07.230
xi plus xi squared, minus
2e over h-bar omega,
00:14:07.230 --> 00:14:11.700
times the wave function,
expressive function of xi,
00:14:11.700 --> 00:14:14.760
not x.
00:14:14.760 --> 00:14:16.980
This is the differential
equation we want to solve,
00:14:16.980 --> 00:14:21.291
and we don't do that in 5.61.
00:14:21.291 --> 00:14:23.540
You're never going to be
asked to solve a differential
00:14:23.540 --> 00:14:26.090
equation like this.
00:14:26.090 --> 00:14:27.830
But you're certainly
going to be asked
00:14:27.830 --> 00:14:30.960
to understand what the
solution looks like,
00:14:30.960 --> 00:14:34.030
and perhaps that it
is in fact a solution.
00:14:34.030 --> 00:14:38.450
But that's still pretty
high value stuff,
00:14:38.450 --> 00:14:41.510
though you wouldn't
really have to do that.
00:14:41.510 --> 00:14:43.820
This is the simplest way
of writing the differential
00:14:43.820 --> 00:14:46.182
equation and it's dimensionless.
00:14:48.900 --> 00:14:55.530
The standard way of dealing
with many differential equations
00:14:55.530 --> 00:14:59.640
is to say, OK, we
have some function
00:14:59.640 --> 00:15:07.500
and it's going to be written
as an exponential, a Gaussian,
00:15:07.500 --> 00:15:09.600
times some new function.
00:15:12.300 --> 00:15:16.170
And for quantum mechanics
this is perfectly reasonable
00:15:16.170 --> 00:15:20.430
because we have a
function in a well
00:15:20.430 --> 00:15:25.070
and the wave functions have to
go to 0 at plus minus infinity.
00:15:25.070 --> 00:15:27.985
And this thing goes to zero at
plus and minus infinity pretty
00:15:27.985 --> 00:15:28.485
strongly.
00:15:31.030 --> 00:15:34.440
So it's a good way
of building in some
00:15:34.440 --> 00:15:38.570
of the expected behavior
of the solution.
00:15:38.570 --> 00:15:40.410
And that's perfectly
legal, and it just
00:15:40.410 --> 00:15:43.110
then defines what is the
difference equation remaining
00:15:43.110 --> 00:15:45.494
for this?
00:15:45.494 --> 00:15:46.910
And it turns out,
well we're going
00:15:46.910 --> 00:15:52.120
to get the Hermite equation.
00:15:52.120 --> 00:15:56.140
And this will be a Hermite
polynomial, the solutions.
00:16:07.050 --> 00:16:15.050
Now one way of dealing with
this is to simply say, well,
00:16:15.050 --> 00:16:17.290
we know the solution of
this differential equation
00:16:17.290 --> 00:16:20.770
if this term weren't there.
00:16:20.770 --> 00:16:25.330
Because this is now the
equation for a Gaussian.
00:16:25.330 --> 00:16:31.390
So building in a Gaussian
as a factor in the solution,
00:16:31.390 --> 00:16:33.010
it's a perfectly
reasonable thing.
00:16:33.010 --> 00:16:35.400
And then we have to say,
what happens now when
00:16:35.400 --> 00:16:38.200
we put this term back in?
00:16:38.200 --> 00:16:43.460
And when we do we
get this thing.
00:16:43.460 --> 00:16:52.470
Second derivative with
respect to this polynomial--
00:16:52.470 --> 00:16:58.650
I mean of this polynomial
is equal to minus 2 xi times
00:16:58.650 --> 00:17:04.970
the hn d xi plus 2n hn.
00:17:08.560 --> 00:17:13.089
This is a famous differential
equation, the Hermite equation,
00:17:13.089 --> 00:17:17.030
which is of no interest to us.
00:17:17.030 --> 00:17:19.400
And it generates the
Hermite polynomials.
00:17:19.400 --> 00:17:22.339
These things are the
Hermite polynomials.
00:17:26.560 --> 00:17:29.380
And they are treated in
some kind of sacred manner
00:17:29.380 --> 00:17:31.990
in most of the
textbooks, and I think
00:17:31.990 --> 00:17:36.100
that's really an offense
because, well, we're
00:17:36.100 --> 00:17:39.950
not interested in
mathematical functions.
00:17:39.950 --> 00:17:42.430
We're interested in
insight and this is just
00:17:42.430 --> 00:17:43.690
putting up another barrier.
00:17:48.510 --> 00:17:54.510
Now with this equation, you
can derive two things called
00:17:54.510 --> 00:18:04.720
recursion relations,
and one of them
00:18:04.720 --> 00:18:10.150
is the derivative of this
polynomial with respect to xi
00:18:10.150 --> 00:18:17.050
is equal to 2n times
hn minus 1 of xi.
00:18:17.050 --> 00:18:20.630
Now that's not a surprise,
because this is a polynomial.
00:18:20.630 --> 00:18:24.920
If you take a derivative
of the variable
00:18:24.920 --> 00:18:29.720
you're going to reduce the
power of each term by 1.
00:18:29.720 --> 00:18:33.610
Now it just happens to be lucky
that when we reduce it to 1
00:18:33.610 --> 00:18:37.480
you don't get a sum of
many different lower order
00:18:37.480 --> 00:18:40.870
polynomials, you just get 1.
00:18:40.870 --> 00:18:44.050
And there is another one,
another recursive relation,
00:18:44.050 --> 00:18:47.890
where it tells you if
you want to increase
00:18:47.890 --> 00:18:56.200
the order you can do this,
you can multiply hn by xi.
00:18:56.200 --> 00:18:59.860
And that's obviously-- it is
going to increase the order,
00:18:59.860 --> 00:19:01.300
but it might not do it cleanly.
00:19:01.300 --> 00:19:02.480
And it doesn't.
00:19:02.480 --> 00:19:03.190
And so we get--
00:19:06.710 --> 00:19:14.840
We have a relationship
between these three
00:19:14.840 --> 00:19:15.950
different polynomials.
00:19:19.740 --> 00:19:23.490
Now it turns out that
these two equations are
00:19:23.490 --> 00:19:27.630
going to reappear, or
at least their progeny
00:19:27.630 --> 00:19:32.700
will reappear, on Monday in
terms of raising and lowering
00:19:32.700 --> 00:19:34.770
operators.
00:19:34.770 --> 00:19:38.800
And what you intuit about
what happens if you multiply
00:19:38.800 --> 00:19:42.790
polynomial by the variable?
00:19:42.790 --> 00:19:44.940
Or what happens if you
take its derivative?
00:19:44.940 --> 00:19:47.006
And it is very
simple and beautiful,
00:19:47.006 --> 00:19:49.380
but I don't think this is very
beautiful for our purposes
00:19:49.380 --> 00:19:52.500
as chemists.
00:19:52.500 --> 00:19:57.300
And one of the things that these
recursive relationships do,
00:19:57.300 --> 00:20:01.470
which also hints at
what's to come on Monday,
00:20:01.470 --> 00:20:05.925
is that we can calculate
integrals like this.
00:20:11.242 --> 00:20:12.200
That's not what I want.
00:20:17.880 --> 00:20:20.460
This is a quantum
number, it's an integer.
00:20:20.460 --> 00:20:28.236
It's v, not nu, and
multiply it by x to the n,
00:20:28.236 --> 00:20:34.010
p to the m, psi that
should be complex
00:20:34.010 --> 00:20:41.050
conjugated, v plus l, dx.
00:20:44.630 --> 00:20:47.480
It turns out for
almost everything
00:20:47.480 --> 00:20:50.400
we want to do with
harmonic oscillators
00:20:50.400 --> 00:20:54.210
we're going to want to know
a lot of integrals like this.
00:20:58.570 --> 00:21:01.770
And one of the things
we like is when
00:21:01.770 --> 00:21:03.919
an integral is promised
to be 0 so we don't ever
00:21:03.919 --> 00:21:04.710
have to look at it.
00:21:07.370 --> 00:21:09.640
And so there are
selection rules.
00:21:09.640 --> 00:21:13.720
And the selection rules
for this kind of integral
00:21:13.720 --> 00:21:22.950
is l is equal to m plus
n, m plus n minus 2,
00:21:22.950 --> 00:21:27.470
down to minus m plus n.
00:21:27.470 --> 00:21:33.260
So the only possible non-zero
integrals of this form
00:21:33.260 --> 00:21:37.280
are for the change
in quantum number
00:21:37.280 --> 00:21:44.550
by this l, which goes from
m plus n down to minus m
00:21:44.550 --> 00:21:47.780
plus n in steps of two.
00:21:47.780 --> 00:21:49.910
The two shouldn't
be too surprising,
00:21:49.910 --> 00:21:52.850
because there is
symmetry and we have
00:21:52.850 --> 00:21:55.640
odd functions for
odd quantum numbers
00:21:55.640 --> 00:21:58.340
and even functions for
even quantum numbers.
00:21:58.340 --> 00:22:00.680
And so something
like this is going
00:22:00.680 --> 00:22:02.390
to have a definite
symmetry and it's
00:22:02.390 --> 00:22:06.360
going to change things
within a symmetry,
00:22:06.360 --> 00:22:11.400
and so it's going to change the
selection rule in steps two.
00:22:11.400 --> 00:22:13.400
Now you don't know what
selection rules are for,
00:22:13.400 --> 00:22:16.220
or why you should get excited
about these sorts of things,
00:22:16.220 --> 00:22:20.180
but it's really nice to know
that almost all the integrals
00:22:20.180 --> 00:22:23.990
you are ever going to face for
a particular problem are zero.
00:22:23.990 --> 00:22:27.170
And you can focus on a small
number of non-zero ones,
00:22:27.170 --> 00:22:29.120
and it just turns out
that the non-zero ones
00:22:29.120 --> 00:22:30.360
have really simple value.
00:22:35.950 --> 00:22:45.440
There also exists what's called
a generating function, which
00:22:45.440 --> 00:22:51.850
is the Rodriguez formula.
00:22:51.850 --> 00:23:00.740
And that is the hn of xi is
equal to minus 1 to the n,
00:23:00.740 --> 00:23:11.830
e to the psi squared, the
derivative with respect to xi,
00:23:11.830 --> 00:23:13.992
e to minus xi squared.
00:23:16.500 --> 00:23:19.397
We have one that has
a positive exponent,
00:23:19.397 --> 00:23:21.480
and one has a negative
exponent, and we have this.
00:23:21.480 --> 00:23:25.080
So we could calculate
any Hermite polynomial
00:23:25.080 --> 00:23:30.160
using this formula,
which you will never do.
00:23:30.160 --> 00:23:36.870
But it's treated with
great fanfare in textbooks.
00:23:36.870 --> 00:23:43.640
Now the solution to the harmonic
oscillator wave function
00:23:43.640 --> 00:23:49.980
in real units, as opposed to
dimensionless quantities, is--
00:23:49.980 --> 00:23:54.880
and I'm just writing this down
because I never would ever
00:23:54.880 --> 00:23:59.900
think about it this way, but
I have to at least provide you
00:23:59.900 --> 00:24:01.910
with guidance--
00:24:01.910 --> 00:24:07.580
so we have a factor 2 to
the v, again this is v.
00:24:07.580 --> 00:24:15.320
Now the reason I'm emphasizing
this is that in all texts
00:24:15.320 --> 00:24:19.590
v quantum numbers
are italicized.
00:24:19.590 --> 00:24:24.170
And if you've thought about
it for a minute, an italic v--
00:24:24.170 --> 00:24:27.250
for mortals-- looks like a nu.
00:24:27.250 --> 00:24:28.270
It isn't quite.
00:24:28.270 --> 00:24:29.770
I don't know what
the difference is,
00:24:29.770 --> 00:24:33.010
but if you have them side
by side they are different.
00:24:33.010 --> 00:24:37.120
And so a large number of
people who should know better
00:24:37.120 --> 00:24:41.540
refer to the vibrational
quantum number as nu, which
00:24:41.540 --> 00:24:44.660
marks that person
as, well, I won't
00:24:44.660 --> 00:24:48.350
say but it's not complimentary!
00:24:48.350 --> 00:24:56.880
We have this factor
to the square root,
00:24:56.880 --> 00:25:00.180
and that's a normalization--
oh we got another part of it.
00:25:00.180 --> 00:25:06.390
Alpha over pi to the 1/4 power.
00:25:06.390 --> 00:25:08.240
So that's normalization.
00:25:08.240 --> 00:25:09.980
Then we have the
Hermite polynomial.
00:25:13.560 --> 00:25:16.020
And you notice I've
got xi back in here,
00:25:16.020 --> 00:25:17.460
which is really a shame.
00:25:17.460 --> 00:25:18.150
And we have--
00:25:21.290 --> 00:25:25.520
So this is the
general solution, we
00:25:25.520 --> 00:25:30.320
have the exponentially damped
function, we have polynomials.
00:25:30.320 --> 00:25:31.820
These are all the
actors that we're
00:25:31.820 --> 00:25:32.944
going to have to deal with.
00:25:36.310 --> 00:25:40.050
And I promise you,
you will never
00:25:40.050 --> 00:25:43.710
use this unless you want
to program a computer
00:25:43.710 --> 00:25:46.230
to calculate the wave function
for God knows what reason.
00:25:49.840 --> 00:25:56.840
The quantum numbers
we are 0, 1, 2.
00:25:56.840 --> 00:26:00.630
And for a harmonic oscillator,
which goes to infinity,
00:26:00.630 --> 00:26:01.950
v goes to infinity, too.
00:26:01.950 --> 00:26:05.100
There's an infinite
number of eigenfunctions
00:26:05.100 --> 00:26:08.990
of the quantum
mechanical Hamiltonian--
00:26:08.990 --> 00:26:12.940
of quantum mechanical
harmonic oscillator.
00:26:12.940 --> 00:26:21.200
We have-- and the
functions are normalized,
00:26:21.200 --> 00:26:26.690
we have psi plus and
minus infinity goes to 0,
00:26:26.690 --> 00:26:31.370
we have psi v of 0.
00:26:31.370 --> 00:26:32.300
So this for all the--
00:26:32.300 --> 00:26:33.400
I put a v there.
00:26:36.400 --> 00:26:50.050
Psi v is 0 for odd v, derivative
of psi with respect to x, at x
00:26:50.050 --> 00:26:58.900
equals 0 is 0 per even v. So
we have symmetric functions.
00:27:03.040 --> 00:27:11.140
And we have the energy levels
is equal to h-bar omega, v
00:27:11.140 --> 00:27:13.810
plus 1/2.
00:27:13.810 --> 00:27:16.960
Now this is h over
2 pi, and this
00:27:16.960 --> 00:27:20.680
is nu times 2 pi, the
frequency times 2 pi.
00:27:20.680 --> 00:27:23.050
So it could also be h nu.
00:27:23.050 --> 00:27:27.700
I have trouble remembering
when there is a 2 pi involved.
00:27:34.280 --> 00:27:37.360
And we have this
wonderful thing.
00:27:47.620 --> 00:27:51.750
It's as if v prime is equal
to v this integral is 1,
00:27:51.750 --> 00:27:53.520
it's normalized.
00:27:53.520 --> 00:27:57.500
And a v prime is not
equal to v, it's zero.
00:27:57.500 --> 00:28:00.630
And that stems from a
theorem I mentioned before,
00:28:00.630 --> 00:28:10.000
is if you have two eigenvalues
of the same Hamiltonian,
00:28:10.000 --> 00:28:12.300
eigenfunctions of
the same Hamiltonian,
00:28:12.300 --> 00:28:15.710
and they belong to
different eigenvalues
00:28:15.710 --> 00:28:19.520
their overlap integral is 0.
00:28:19.520 --> 00:28:21.890
We like zeros.
00:28:21.890 --> 00:28:25.490
We like normalization because
the integral is just 1,
00:28:25.490 --> 00:28:26.930
it goes away.
00:28:26.930 --> 00:28:31.470
Or the integral is 0, the
whole thing goes away.
00:28:31.470 --> 00:28:34.330
So that's really good.
00:28:34.330 --> 00:28:39.990
So we call this set
of v's is orthonormal.
00:28:39.990 --> 00:28:41.760
Orthogonal and normalized.
00:28:41.760 --> 00:28:44.250
And the orthonormal
terminology is used a lot.
00:28:44.250 --> 00:28:47.580
And in almost all quantum
mechanical problems
00:28:47.580 --> 00:28:52.800
we like using an
orthonormal set of functions
00:28:52.800 --> 00:28:54.348
to solve everything.
00:28:57.340 --> 00:29:00.300
Sometimes we have to do a
little work to establish that,
00:29:00.300 --> 00:29:02.460
and I'll show much
later in the course
00:29:02.460 --> 00:29:06.080
how when you have functions
that are not orthogonal,
00:29:06.080 --> 00:29:08.280
and not normalized,
you can create
00:29:08.280 --> 00:29:11.320
a set of functions which are.
00:29:11.320 --> 00:29:13.070
And this is something
that a computer will
00:29:13.070 --> 00:29:14.380
do without breaking a sweat.
00:29:20.150 --> 00:29:27.190
Now we're back to my favorite
topic, semi-classical.
00:29:32.100 --> 00:29:35.790
Because it's really
easy to understand.
00:29:35.790 --> 00:29:38.220
Not just to understand
the harmonic oscillator,
00:29:38.220 --> 00:29:40.050
but to use it in many problems.
00:29:43.560 --> 00:29:48.590
So in classic mechanics
the kinetic energy
00:29:48.590 --> 00:29:58.540
is e minus v of x, or
p squared over 2 mu.
00:29:58.540 --> 00:30:01.770
And so we can derive
an equation for p
00:30:01.770 --> 00:30:09.330
of x classical mechanically,
which is 2 mu e
00:30:09.330 --> 00:30:13.598
minus v of x square root.
00:30:16.670 --> 00:30:20.000
This is an extremely
useful function.
00:30:20.000 --> 00:30:22.720
It's not an operator.
00:30:22.720 --> 00:30:24.240
It's a thing that
we're going to use
00:30:24.240 --> 00:30:29.410
to make sense of everything,
but it's not an operator.
00:30:29.410 --> 00:30:31.380
And so this is classic
mechanics, and then
00:30:31.380 --> 00:30:36.900
in quantum mechanics, we know
that Mr. de Broglie told us
00:30:36.900 --> 00:30:41.730
that the wavelength
is equal to h over p.
00:30:41.730 --> 00:30:46.350
And we can generalize and say,
well, maybe the wavelength
00:30:46.350 --> 00:30:49.390
is a function of
x for potential,
00:30:49.390 --> 00:30:50.640
which is not constant.
00:30:57.710 --> 00:31:02.740
And even though this is not an
operator in quantum mechanics
00:31:02.740 --> 00:31:04.740
this is true.
00:31:04.740 --> 00:31:09.870
That you can say the distance
between consecutive nodes
00:31:09.870 --> 00:31:12.560
is lambda over 2.
00:31:12.560 --> 00:31:15.650
We can use this
node relationship
00:31:15.650 --> 00:31:16.670
to great advantage.
00:31:21.830 --> 00:31:25.940
For the pair of
nodes closest to x
00:31:25.940 --> 00:31:29.600
we can use this to calculate
the distance between them.
00:31:32.710 --> 00:31:34.180
Very valuable.
00:31:34.180 --> 00:31:36.760
Because I also want
to mention something.
00:31:36.760 --> 00:31:41.140
If you have an integrand
which is rapidly oscillating,
00:31:41.140 --> 00:31:43.270
or if you have two rapidly
oscillating functions
00:31:43.270 --> 00:31:46.120
and you're multiplying
them together,
00:31:46.120 --> 00:31:55.040
that integral will accumulate to
its final value at the position
00:31:55.040 --> 00:31:57.710
where the two oscillating
functions are oscillating
00:31:57.710 --> 00:31:59.390
at the same frequency.
00:31:59.390 --> 00:32:01.460
That's the stationary
phase point.
00:32:01.460 --> 00:32:03.980
And this is also
a wonderful thing,
00:32:03.980 --> 00:32:06.560
because if you can
figure out where
00:32:06.560 --> 00:32:08.540
the things you're
multiplying together
00:32:08.540 --> 00:32:10.920
are oscillating at
the same frequency,
00:32:10.920 --> 00:32:14.640
your integral becomes a number.
00:32:14.640 --> 00:32:17.170
No work ever.
00:32:17.170 --> 00:32:18.800
And that's a useful thing.
00:32:18.800 --> 00:32:22.200
OK, so the stationary
phase method
00:32:22.200 --> 00:32:26.040
enables you to use this
in a really fantastic way.
00:32:26.040 --> 00:32:29.690
And it's a little bit like
Feynman's path integral idea,
00:32:29.690 --> 00:32:36.660
that you can calculate a
complicated thing by evaluating
00:32:36.660 --> 00:32:40.410
an integral over
a convenient path
00:32:40.410 --> 00:32:42.750
as opposed to integrating
overall space,
00:32:42.750 --> 00:32:45.750
because everything
that you care about
00:32:45.750 --> 00:32:47.790
comes from a stationary phase.
00:32:47.790 --> 00:32:50.400
Quantum mechanics is
full of oscillations,
00:32:50.400 --> 00:32:53.220
classical mechanics
doesn't have oscillations,
00:32:53.220 --> 00:32:56.190
and the two meet at the
stationary phase point.
00:32:58.760 --> 00:33:02.120
Now we're going to use
these ideas to calculate
00:33:02.120 --> 00:33:07.760
useful stuff for quantum
mechanical vibrational wave
00:33:07.760 --> 00:33:08.570
functions.
00:33:08.570 --> 00:33:20.460
The shapes of psi of x
gets exponentially damped,
00:33:20.460 --> 00:33:31.930
but it extends into the
classically forbidden
00:33:31.930 --> 00:33:34.120
e less than v of x regions.
00:33:38.220 --> 00:33:42.250
The wave function,
if we have potential
00:33:42.250 --> 00:33:45.650
and we have a wave
function, that wave function
00:33:45.650 --> 00:33:48.620
is going to not go
to 0 at the edge
00:33:48.620 --> 00:33:50.930
but it's going to have a tail.
00:33:50.930 --> 00:33:53.620
And that tail goes
to 0 at infinity.
00:33:53.620 --> 00:33:58.400
And so there is some amplitude
where the particle isn't
00:33:58.400 --> 00:34:00.400
allowed to be, classically.
00:34:00.400 --> 00:34:04.760
And that's where
tunneling comes in.
00:34:04.760 --> 00:34:07.730
But the important thing,
the important insight
00:34:07.730 --> 00:34:12.260
is that there are no nodes
in the classically forbidden
00:34:12.260 --> 00:34:13.280
region.
00:34:13.280 --> 00:34:17.449
There is only exponential
decay towards 0,
00:34:17.449 --> 00:34:22.290
and if you've chosen the
wrong value of the energy,
00:34:22.290 --> 00:34:26.870
in other words a place where
there is no eigenfunction,
00:34:26.870 --> 00:34:29.780
the wave function in the
classically forbidden region
00:34:29.780 --> 00:34:31.730
will usually go to infinity.
00:34:31.730 --> 00:34:34.020
Either over here or
over here, and says,
00:34:34.020 --> 00:34:37.969
well, it's clearly
not a good function.
00:34:37.969 --> 00:34:39.370
But there are no 0 crossings.
00:34:44.340 --> 00:34:52.170
It's oscillating
in e greater than v
00:34:52.170 --> 00:34:53.840
of x, the classically
allowed region.
00:34:58.240 --> 00:35:05.340
The number of nodes
is v. So we can
00:35:05.340 --> 00:35:07.860
have a v equals 0 function that
just goes up and goes down,
00:35:07.860 --> 00:35:09.220
no internal nodes.
00:35:09.220 --> 00:35:12.060
v equals one, it crosses
zero right in the middle.
00:35:15.980 --> 00:35:17.510
And we have the even oddness.
00:35:17.510 --> 00:35:26.116
Even v, even function.
00:35:26.116 --> 00:35:30.110
Odd v, odd function.
00:35:30.110 --> 00:35:35.390
For an even function you have a
relative maximum at x equals 0,
00:35:35.390 --> 00:35:37.670
and for an odd
function you have a 0.
00:35:37.670 --> 00:35:40.175
And the opposite
further derivatives.
00:36:05.360 --> 00:36:08.150
The outer lobes,
the ones on the ends
00:36:08.150 --> 00:36:16.300
just before the particle
encounters the classical wall,
00:36:16.300 --> 00:36:18.560
you get the maximum amplitude.
00:36:21.260 --> 00:36:31.110
And so you can draw cartoons
which look sort of like that.
00:36:31.110 --> 00:36:35.600
Most of the valuable stuff is
at the other turning point.
00:36:35.600 --> 00:36:38.400
And there's
oscillations in between,
00:36:38.400 --> 00:36:41.730
but often you really care
about these two outer lobes.
00:36:44.880 --> 00:36:47.200
That's a pretty
good simplification.
00:36:47.200 --> 00:36:53.630
Now there's a nice picture
in McQuarrie on page 226
00:36:53.630 --> 00:36:57.390
which shows, especially
for psi squared,
00:36:57.390 --> 00:36:59.870
that the nodes are pretty big.
00:36:59.870 --> 00:37:03.860
But they're not as big for
relatively low quantum numbers
00:37:03.860 --> 00:37:07.110
as I've implied.
00:37:07.110 --> 00:37:08.640
But at really high
quantum numbers
00:37:08.640 --> 00:37:12.930
we have a thing called the
correspondence principle.
00:37:12.930 --> 00:37:18.450
And the corresponding principle
says that quantum mechanics
00:37:18.450 --> 00:37:20.929
will do what classical
mechanics does
00:37:20.929 --> 00:37:22.470
in the limit of high
quantum numbers.
00:37:22.470 --> 00:37:24.136
And in the limit of
high quantum numbers
00:37:24.136 --> 00:37:28.470
essentially all the amplitude
is at the turning points.
00:37:28.470 --> 00:37:31.280
And in classic
mechanics the particle
00:37:31.280 --> 00:37:35.570
is moving fast in the middle,
and stops and turns around,
00:37:35.570 --> 00:37:37.340
and essentially all
of the amplitude
00:37:37.340 --> 00:37:40.060
is at the turning point.
00:37:40.060 --> 00:37:41.640
So this is nice.
00:37:44.830 --> 00:37:48.610
Now we're getting
into Bobism territory,
00:37:48.610 --> 00:37:50.110
because I'm really
going to show you
00:37:50.110 --> 00:37:57.610
how to calculate
whatever you need using
00:37:57.610 --> 00:37:58.780
these semi-classical ideas.
00:38:04.340 --> 00:38:09.420
We have the
probability envelope.
00:38:14.220 --> 00:38:22.056
Psi star of x psi
of x, and we're
00:38:22.056 --> 00:38:24.180
going to have both of these
having the same quantum
00:38:24.180 --> 00:38:26.550
number, dx.
00:38:26.550 --> 00:38:32.490
So this is the probability of
finding the particle near x
00:38:32.490 --> 00:38:36.030
in a region with dx.
00:38:44.330 --> 00:38:54.490
And this is the same thing
as dx over v classical.
00:38:54.490 --> 00:38:58.480
It's not the same thing, there
is a constant here, sorry.
00:38:58.480 --> 00:39:02.380
This probability
density that you want
00:39:02.380 --> 00:39:05.890
is basically 1 over
the classical velocity.
00:39:05.890 --> 00:39:08.649
And I demonstrated
that when I walked
00:39:08.649 --> 00:39:10.690
across the room, when I
walked fast in the middle
00:39:10.690 --> 00:39:13.110
and slow the outside.
00:39:13.110 --> 00:39:16.380
And you get the probability,
you get this constant,
00:39:16.380 --> 00:39:19.140
by saying, OK, how long did it
take for me to go from one end
00:39:19.140 --> 00:39:20.880
to the other?
00:39:20.880 --> 00:39:22.890
And comparing that,
how long it took
00:39:22.890 --> 00:39:27.270
for me to go in some
differential position.
00:39:27.270 --> 00:39:32.450
You get this constant
in a simple way.
00:39:37.450 --> 00:39:50.490
v classical is equal to p
classical over the mass,
00:39:50.490 --> 00:39:51.190
over the mu.
00:39:53.900 --> 00:39:56.820
But we know the function
for p classical.
00:39:56.820 --> 00:40:05.857
We have 1 over mu times
2 mu e, minus v of x.
00:40:10.150 --> 00:40:12.340
So we know the
velocity everywhere,
00:40:12.340 --> 00:40:16.930
and there's nothing terribly
hard about figuring that out.
00:40:19.854 --> 00:40:22.270
And now we want to know what
this proportionality constant
00:40:22.270 --> 00:40:23.140
is.
00:40:23.140 --> 00:40:33.140
And so for that we say the
time to go from x to x plus dx,
00:40:33.140 --> 00:40:40.957
over the time to go
from x minus to x plus.
00:40:40.957 --> 00:40:42.540
Because what's
happening, the particle
00:40:42.540 --> 00:40:44.180
is going back and
forth inside this
00:40:44.180 --> 00:40:49.310
well and so this is the time
it takes to go one pass,
00:40:49.310 --> 00:40:51.770
and this is the
time it takes to go
00:40:51.770 --> 00:40:53.350
through the region of interest.
00:40:53.350 --> 00:40:55.564
And so this ratio
is the probability.
00:41:22.410 --> 00:41:27.920
And so we have the probability
moving from left turning point
00:41:27.920 --> 00:41:30.270
the right turning
point, and we want
00:41:30.270 --> 00:41:33.210
to know the probability
in that interval.
00:41:33.210 --> 00:41:45.450
And so that's just dx over
v classical at x, over tau,
00:41:45.450 --> 00:41:46.530
over 2.
00:41:46.530 --> 00:41:50.970
Because tau is the period, and
we have half of the period,
00:41:50.970 --> 00:41:54.860
and so it's all together.
00:41:54.860 --> 00:41:58.020
I'm going to skip a little step,
because it's taking too long.
00:41:58.020 --> 00:42:12.890
Psi star psi dx is equal
to k over 2 pi squared,
00:42:12.890 --> 00:42:19.590
e minus v of x square of dx.
00:42:19.590 --> 00:42:24.290
Now so if we know the potential,
and we know the energy,
00:42:24.290 --> 00:42:26.760
and we know the force
constant we can say,
00:42:26.760 --> 00:42:30.600
well, this is the
probability and--
00:42:33.890 --> 00:42:41.330
but this is the probability
of the semi-classical
00:42:41.330 --> 00:42:46.254
representation of
psi star psi dx at x.
00:42:48.960 --> 00:42:52.250
Now this is oscillating,
and this is not.
00:42:57.000 --> 00:42:58.470
So what we really want to know--
00:43:05.490 --> 00:43:17.000
here's the-- if we have
psi star whoops, slow down.
00:43:21.870 --> 00:43:28.980
This is oscillating and
what we've calculated before
00:43:28.980 --> 00:43:32.290
is something that looks
sort of like that.
00:43:32.290 --> 00:43:35.410
And if we multiply
by 2 we have a curve
00:43:35.410 --> 00:43:38.210
that goes to the maximum
of all these oscillations.
00:43:44.130 --> 00:43:49.920
The envelope psi star
psi has the form.
00:43:49.920 --> 00:43:52.770
We've multiplied by
2, and so we end up
00:43:52.770 --> 00:44:08.677
getting 2k over pi squared,
e minus v of x square root.
00:44:08.677 --> 00:44:11.010
Now you might say, well these
are complicated functions,
00:44:11.010 --> 00:44:12.840
why should I bother with them?
00:44:12.840 --> 00:44:16.690
But if you wanted
anything starting
00:44:16.690 --> 00:44:19.480
from the correct solution
to the harmonic oscillator,
00:44:19.480 --> 00:44:21.220
using the Hermite
polynomials, there's
00:44:21.220 --> 00:44:22.555
a whole lot more overhead.
00:44:25.620 --> 00:44:27.930
Notice also this is a
function of x, not xi.
00:44:38.440 --> 00:44:40.960
This is the overlap
function, it's
00:44:40.960 --> 00:44:44.440
the curve that touches the
maximum of all these things
00:44:44.440 --> 00:44:47.470
and it's very useful if you
want to know the probability
00:44:47.470 --> 00:44:49.233
of finding the system anywhere.
00:44:55.980 --> 00:45:03.696
And we get the node spacing
from the equation h over p of x.
00:45:03.696 --> 00:45:05.700
And now here comes
something really nice.
00:45:08.970 --> 00:45:12.040
It's called the semi-classical
quantization interval.
00:45:14.560 --> 00:45:18.500
If we have any one
dimensional potential,
00:45:18.500 --> 00:45:21.510
and we're at some
energy, we'd like
00:45:21.510 --> 00:45:27.360
to be able to know how many
levels are at that energy,
00:45:27.360 --> 00:45:28.710
or below.
00:45:28.710 --> 00:45:31.240
Or where are the energy levels?
00:45:31.240 --> 00:45:33.870
And we get that from this
really incredible thing.
00:45:40.860 --> 00:45:43.140
We want to know that--
00:45:50.710 --> 00:45:53.785
this is the difference
between nodes.
00:45:53.785 --> 00:45:56.160
Right?
00:45:56.160 --> 00:45:59.160
And now if we would
like to know x
00:45:59.160 --> 00:46:14.050
minus to x plus and some energy,
we can replace lambda of x by h
00:46:14.050 --> 00:46:15.010
over p of x.
00:46:15.010 --> 00:46:24.790
And so we get p of x at
that energy over h dx.
00:46:24.790 --> 00:46:28.920
pdx, that's called
an action integral.
00:46:28.920 --> 00:46:30.960
Now I have to tell
a little story.
00:46:30.960 --> 00:46:34.850
When I was a senior
at Amherst College
00:46:34.850 --> 00:46:44.810
we had a oral exam for
whether my thesis was
00:46:44.810 --> 00:46:47.310
going to be accepted or not.
00:46:47.310 --> 00:46:55.020
And one of my examiners asked
me, what is the unit of h?
00:46:55.020 --> 00:46:58.340
Well, it's energy times time.
00:46:58.340 --> 00:46:59.990
And he wouldn't stop.
00:46:59.990 --> 00:47:02.720
He said no, I want
something else.
00:47:02.720 --> 00:47:04.040
It's called action.
00:47:04.040 --> 00:47:07.400
Momentum times position.
00:47:07.400 --> 00:47:09.110
This is an action integral.
00:47:09.110 --> 00:47:11.150
And so anyway
that's just a story.
00:47:11.150 --> 00:47:14.900
I spent a half an hour,
and I was damn stubborn.
00:47:14.900 --> 00:47:19.520
I was not-- you know, it
was energy times time.
00:47:19.520 --> 00:47:22.130
But that is much
more insight here
00:47:22.130 --> 00:47:24.230
and that's maybe
why I got so excited
00:47:24.230 --> 00:47:26.540
about this sort of an integral.
00:47:31.060 --> 00:47:36.250
If we want to know if
we have an eigenvalue
00:47:36.250 --> 00:47:41.980
this integral has to
be equal to h over
00:47:41.980 --> 00:47:47.180
2 times the number of nodes.
00:47:47.180 --> 00:47:49.440
Well it's pretty simple.
00:47:49.440 --> 00:47:55.690
So we can adjust
e to satisfy this.
00:47:55.690 --> 00:48:00.100
Or if we wanted to know
how many energy levels are
00:48:00.100 --> 00:48:02.470
at an energy below
the energy we've
00:48:02.470 --> 00:48:05.870
chosen we evaluate
this integral,
00:48:05.870 --> 00:48:10.420
and we get a number like 13.5.
00:48:10.420 --> 00:48:14.670
Well, it means there are 13
energy levels below that.
00:48:14.670 --> 00:48:19.400
Now often you want to know
the density of states,
00:48:19.400 --> 00:48:23.120
the number of energy
levels per unit energy,
00:48:23.120 --> 00:48:25.460
because that turns out to
be the critical quantity
00:48:25.460 --> 00:48:28.190
in calculating many
things you want to know.
00:48:28.190 --> 00:48:31.321
And you can get that from the
semi-classical quantization.
00:48:35.480 --> 00:48:43.100
We're close to the end, I just
want to say where we're going.
00:48:43.100 --> 00:48:47.750
We have classical pictures
and I really, really
00:48:47.750 --> 00:48:50.900
want you to think about
these classical pictures
00:48:50.900 --> 00:48:53.990
and use them rather
than thinking, well,
00:48:53.990 --> 00:48:57.050
I'm going to have my cell
phone program to evaluate
00:48:57.050 --> 00:48:59.619
all the necessary stuff.
00:48:59.619 --> 00:49:01.160
And there are certain
things you want
00:49:01.160 --> 00:49:05.170
to remember about this
semi-classical picture.
00:49:05.170 --> 00:49:10.420
And now we have the ability to
calculate an infinite number
00:49:10.420 --> 00:49:13.270
of integrals involving
harmonic oscillator
00:49:13.270 --> 00:49:17.230
functions in certain operators.
00:49:17.230 --> 00:49:18.980
Well, la-di-da.
00:49:18.980 --> 00:49:20.170
Why do we want them?
00:49:20.170 --> 00:49:22.000
Well one of the
things we want is
00:49:22.000 --> 00:49:23.740
to be able to calculate
the probability
00:49:23.740 --> 00:49:26.960
of a vibrational transition.
00:49:26.960 --> 00:49:29.060
That's called a
transition element,
00:49:29.060 --> 00:49:30.680
and that's an easy
thing to calculate.
00:49:34.000 --> 00:49:37.580
Another thing we want
to do is to say, well,
00:49:37.580 --> 00:49:39.320
nature screwed up.
00:49:39.320 --> 00:49:43.790
This oscillator isn't harmonic,
there's anharmonic term,
00:49:43.790 --> 00:49:46.080
and I would like to know
what is the contribution
00:49:46.080 --> 00:49:51.450
of a constant times x
cubed in the potential
00:49:51.450 --> 00:49:53.700
to the energy levels.
00:49:53.700 --> 00:49:55.710
And that's called
perturbation theory.
00:49:55.710 --> 00:49:59.100
Or I want to have many
harmonic oscillators
00:49:59.100 --> 00:50:01.800
in a polyatomic molecule,
they talk to each other,
00:50:01.800 --> 00:50:06.360
and I want to calculate
the interactions
00:50:06.360 --> 00:50:08.460
between these
harmonic oscillators
00:50:08.460 --> 00:50:09.990
affect the energy level.
00:50:09.990 --> 00:50:13.470
Remember, when we have
a separable Hamiltonian
00:50:13.470 --> 00:50:15.510
we can just write the
energy levels as the sum
00:50:15.510 --> 00:50:16.670
of the individual Hamilton.
00:50:16.670 --> 00:50:19.080
And then there's
coupling terms, and we
00:50:19.080 --> 00:50:22.950
deal with those but
perturbation theory.
00:50:22.950 --> 00:50:26.680
There's all sorts of
wonderful things we do.
00:50:26.680 --> 00:50:35.320
But we're going to consider
these magical operators,
00:50:35.320 --> 00:50:39.430
a creation and annihilation
operator, where a star--
00:50:39.430 --> 00:50:49.010
a dagger operating
on psi v gives
00:50:49.010 --> 00:50:54.570
the square root of v plus
one, times psi v plus 1.
00:50:54.570 --> 00:50:58.960
And a dagger operating on v
gives the square root of v--
00:50:58.960 --> 00:51:00.220
let's write it--
00:51:00.220 --> 00:51:04.540
I mean a non-dagger
gives v square root times
00:51:04.540 --> 00:51:07.060
psi, v minus 1.
00:51:07.060 --> 00:51:13.210
And that x is equal to a plus
a dagger times a constant.
00:51:13.210 --> 00:51:17.980
And so all of a sudden we
can evaluate all integrals
00:51:17.980 --> 00:51:23.440
involving x, or powers of x, or
momenta, or powers of momenta
00:51:23.440 --> 00:51:24.640
without thinking.
00:51:24.640 --> 00:51:28.420
Without ever looking
at a function.
00:51:28.420 --> 00:51:32.550
And I guarantee you
that this is embodied
00:51:32.550 --> 00:51:35.310
in the incredible amount
of work done wherever
00:51:35.310 --> 00:51:39.510
we pretend almost every
problem is a harmonic
00:51:39.510 --> 00:51:41.680
oscillator in disguise.
00:51:41.680 --> 00:51:44.710
Because these a's
and a daggers enable
00:51:44.710 --> 00:51:47.500
you to generate
everything without ever
00:51:47.500 --> 00:51:52.600
converting from x to xi, without
ever looking at an integral.
00:51:52.600 --> 00:51:57.190
It's all just a
manipulation of algebra.
00:51:57.190 --> 00:52:01.380
And it's not just convenient,
but there's insight
00:52:01.380 --> 00:52:04.950
and so this is what
I want to convey.
00:52:04.950 --> 00:52:08.370
That you will get
tremendous insight.
00:52:08.370 --> 00:52:11.270
Maybe, maybe I sold
you on semi-classical,
00:52:11.270 --> 00:52:16.340
and I don't apologize for that
because that's very useful.
00:52:16.340 --> 00:52:20.630
But the next lecture, when
you get the a's and a daggers,
00:52:20.630 --> 00:52:23.290
it'll just knock your socks off.
00:52:23.290 --> 00:52:25.430
OK, that's it.