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ROBERT FIELD: Last
time, I talked a lot
00:00:25.120 --> 00:00:30.970
about the semiclassical
method, where we generalize
00:00:30.970 --> 00:00:35.830
on this wonderful
relationship to say, well,
00:00:35.830 --> 00:00:41.180
if the potential is not
constant, then we can say,
00:00:41.180 --> 00:00:45.550
well, the wavelength
changes with position.
00:00:45.550 --> 00:00:50.050
And we can say that the
momentum changes with position.
00:00:50.050 --> 00:00:51.970
But we're using
this as the guide.
00:00:51.970 --> 00:00:54.550
And so the basis is
really just saying,
00:00:54.550 --> 00:00:57.730
OK, we're going to take
this kind of a relationship
00:00:57.730 --> 00:00:59.200
seriously.
00:00:59.200 --> 00:01:02.066
Because v is not constant.
00:01:02.066 --> 00:01:05.470
We have V of x.
00:01:05.470 --> 00:01:12.790
And we also know that
the kinetic energy,
00:01:12.790 --> 00:01:18.180
which is an operator,
is p squared over 2m.
00:01:18.180 --> 00:01:24.780
And the energy
minus the potential
00:01:24.780 --> 00:01:27.940
is the kinetic energy.
00:01:27.940 --> 00:01:38.400
And so we can use this to get a
classical, mechanical function
00:01:38.400 --> 00:01:45.520
that this p of x is going
to be 2m E minus V of x.
00:01:48.515 --> 00:01:50.640
So why are we doing all
this when we can just solve
00:01:50.640 --> 00:01:52.840
the differential equations?
00:01:52.840 --> 00:01:55.560
And the answer is
we want insight.
00:01:55.560 --> 00:01:59.130
And we want to build our
insight on what we know.
00:01:59.130 --> 00:02:02.830
And so we have this
momentum function,
00:02:02.830 --> 00:02:05.910
which gives us a wavelength
function, which tells us
00:02:05.910 --> 00:02:08.690
how far apart the waves are.
00:02:08.690 --> 00:02:10.699
But we also have
another thing, which
00:02:10.699 --> 00:02:14.000
was demonstrated by my
running across the room,
00:02:14.000 --> 00:02:18.020
and that the momentum is
related to the velocity, which
00:02:18.020 --> 00:02:20.690
is related to the probability
of finding the system
00:02:20.690 --> 00:02:22.400
at a particular place.
00:02:22.400 --> 00:02:29.010
And so we have, if we commit
the travesty of saying, OK,
00:02:29.010 --> 00:02:32.870
we have a classical function--
it's not the momentum--
00:02:32.870 --> 00:02:37.450
but it's going to somehow
encode the classical behavior,
00:02:37.450 --> 00:02:42.130
we can determine without solving
any differential equation what
00:02:42.130 --> 00:02:46.840
the spacing between nodes in
the exact wave function is,
00:02:46.840 --> 00:02:50.320
and the amplitude
in that region.
00:02:50.320 --> 00:02:54.220
And that's a lot
because then we can
00:02:54.220 --> 00:02:59.840
use our knowledge of
classical mechanics to say,
00:02:59.840 --> 00:03:03.200
oh, this is what we expect
in quantum mechanics.
00:03:03.200 --> 00:03:05.000
And that's very powerful.
00:03:05.000 --> 00:03:10.090
And I keep stressing that
you want to draw cartoons.
00:03:10.090 --> 00:03:15.480
And this is the way you
get into those cartoons.
00:03:15.480 --> 00:03:23.190
OK, now, for the
free particle, we
00:03:23.190 --> 00:03:28.860
have these kinds of wave
functions, e to the ikx.
00:03:28.860 --> 00:03:33.210
And they're really
great because often, we
00:03:33.210 --> 00:03:40.250
want to evaluate things like the
integral of e to the ikx times
00:03:40.250 --> 00:03:43.490
e to the minus ikx dx.
00:03:43.490 --> 00:03:47.300
Well, that's one.
00:03:47.300 --> 00:03:50.330
And so instead of
remembering that we
00:03:50.330 --> 00:03:55.460
have to evaluate trigonometric
integrals, sine, sine theta,
00:03:55.460 --> 00:03:57.440
cosine, we can do this.
00:03:57.440 --> 00:04:00.440
And it really simplifies life.
00:04:00.440 --> 00:04:08.240
Now, using that insight,
I'm asking you, OK,
00:04:08.240 --> 00:04:12.200
the expectation value for
the momentum could be--
00:04:22.540 --> 00:04:27.640
OK, if this is the expectation
value for the momentum, what
00:04:27.640 --> 00:04:29.950
is psi of x?
00:04:33.400 --> 00:04:36.100
I promised I would
ask you this question.
00:04:36.100 --> 00:04:38.630
I don't know if anybody
really thought about it.
00:04:38.630 --> 00:04:44.880
But first of all, we're talking
about the free particle.
00:04:44.880 --> 00:04:48.500
And this is some
sort of eigenfunction
00:04:48.500 --> 00:04:51.540
of the Hamiltonian
for the free particle.
00:04:51.540 --> 00:04:55.305
So what can we say about k?
00:05:00.600 --> 00:05:02.940
We have two parts to
the wave function.
00:05:02.940 --> 00:05:04.580
We have an e to the ikx.
00:05:04.580 --> 00:05:06.990
And we have a e
to the minus ikx.
00:05:06.990 --> 00:05:10.470
And the k is the same
for both of them.
00:05:10.470 --> 00:05:14.840
So with all those
hints, what is this?
00:05:14.840 --> 00:05:19.130
We have e to the
ikx times something.
00:05:19.130 --> 00:05:25.354
And we have plus e
to the minus ikx.
00:05:25.354 --> 00:05:27.020
So what are the
somethings that go here?
00:05:34.291 --> 00:05:34.790
Yes?
00:05:34.790 --> 00:05:35.540
AUDIENCE: A and B?
00:05:35.540 --> 00:05:37.490
ROBERT FIELD: Right.
00:05:37.490 --> 00:05:42.080
And that didn't involve
very much mental gymnastics
00:05:42.080 --> 00:05:44.960
if you really have done a
little bit of practicing
00:05:44.960 --> 00:05:47.850
of integrals involving
these sorts of things.
00:05:47.850 --> 00:05:49.110
Now there's two things.
00:05:49.110 --> 00:05:53.900
One is when you have the
same k in e to the ikx
00:05:53.900 --> 00:05:55.340
and e to the minus ikx.
00:05:55.340 --> 00:05:58.150
The product is one.
00:05:58.150 --> 00:06:03.000
If you have different
k's and you're
00:06:03.000 --> 00:06:08.970
integrating over all space,
or over some cleverly chosen
00:06:08.970 --> 00:06:11.165
region, that integral is zero.
00:06:14.080 --> 00:06:17.830
Because these are
eigenfunctions of--
00:06:17.830 --> 00:06:20.050
these are different
eigenfunctions
00:06:20.050 --> 00:06:24.200
of an operator belonging
to different eigenvalues.
00:06:24.200 --> 00:06:27.110
And you always can count
on those things being zero.
00:06:27.110 --> 00:06:31.270
Now, quantum mechanics
is full of integrals.
00:06:31.270 --> 00:06:34.410
Basically, there's an
infinite number of them.
00:06:34.410 --> 00:06:37.140
And most of them are zero.
00:06:37.140 --> 00:06:39.540
And you want to be able
to look at an integral
00:06:39.540 --> 00:06:42.035
and say, oh, I don't
need to evaluate that.
00:06:42.035 --> 00:06:44.160
And often, you want to look
at an integral and say,
00:06:44.160 --> 00:06:46.140
I do know how to evaluate that.
00:06:46.140 --> 00:06:49.900
And I know an infinite
number of those like that.
00:06:49.900 --> 00:06:53.460
And all of a sudden, it starts
to be transparent again.
00:06:53.460 --> 00:06:59.250
Because the barrier
between insight and quantum
00:06:59.250 --> 00:07:03.110
mechanics is usually a
whole bunch of integrals.
00:07:03.110 --> 00:07:05.280
And they're all yours.
00:07:05.280 --> 00:07:12.000
And so we like problems
where the wave functions
00:07:12.000 --> 00:07:14.260
have simple forms.
00:07:14.260 --> 00:07:16.710
And this is true
for free particle.
00:07:16.710 --> 00:07:18.455
It's true for the
particle in a box.
00:07:21.114 --> 00:07:22.530
We're going to
start talking about
00:07:22.530 --> 00:07:24.880
their harmonic oscillator.
00:07:24.880 --> 00:07:30.200
And it seems like those
integrals are not simple,
00:07:30.200 --> 00:07:31.580
but they are.
00:07:31.580 --> 00:07:33.690
I have to teach you
why they're not simple.
00:07:33.690 --> 00:07:39.740
So today, we're starting on the
classical mechanical treatment
00:07:39.740 --> 00:07:41.570
of the harmonic oscillator.
00:07:41.570 --> 00:07:46.020
Then we'll do the traditional
quantum mechanical treatment.
00:07:46.020 --> 00:07:49.430
And then, we'll come back
and use these creation
00:07:49.430 --> 00:07:51.600
and annililation
operators, which
00:07:51.600 --> 00:07:54.270
are the magic decoders
for essentially evaluating
00:07:54.270 --> 00:07:57.940
all the integrals trivially.
00:07:57.940 --> 00:08:00.640
And then, with all
that in hand, we're
00:08:00.640 --> 00:08:03.370
going to make our first step
into time-dependent quantum
00:08:03.370 --> 00:08:05.420
mechanics.
00:08:05.420 --> 00:08:07.400
And we're going to use
time-dependent quantum
00:08:07.400 --> 00:08:08.280
mechanics.
00:08:08.280 --> 00:08:11.860
Well, we're going to use
our facility with integrals
00:08:11.860 --> 00:08:15.700
to describe the properties of
some particle-like state we
00:08:15.700 --> 00:08:17.350
can construct.
00:08:17.350 --> 00:08:21.160
And these constructions
are really simple
00:08:21.160 --> 00:08:24.970
for the particle in a box
or the harmonic oscillator,
00:08:24.970 --> 00:08:28.020
depending on which
properties you want.
00:08:28.020 --> 00:08:33.450
So your ability to draw cartoons
and to use classical insights
00:08:33.450 --> 00:08:37.620
for the particle in a box
and the harmonic oscillator
00:08:37.620 --> 00:08:41.100
will be incredibly valuable
once we take our first step
00:08:41.100 --> 00:08:43.799
into the reality of
time-dependent quantum
00:08:43.799 --> 00:08:46.270
mechanics.
00:08:46.270 --> 00:08:50.380
Now, one of the things
that's going to happen
00:08:50.380 --> 00:08:56.260
is that we're going to
describe real situations,
00:08:56.260 --> 00:08:59.580
real situations that are not
one of the standard solved
00:08:59.580 --> 00:09:00.880
problems.
00:09:00.880 --> 00:09:09.440
The standard solved problems
are the particle in a box,
00:09:09.440 --> 00:09:13.970
or a particle in the infinite
box, the harmonic oscillator,
00:09:13.970 --> 00:09:19.970
the hydrogen atom,
and the rigid rotor.
00:09:19.970 --> 00:09:22.520
And the particle in a box--
00:09:22.520 --> 00:09:25.770
so the potential for each
of these is different.
00:09:25.770 --> 00:09:27.220
It looks like this.
00:09:27.220 --> 00:09:28.860
It looks like this.
00:09:28.860 --> 00:09:31.110
It looks like this.
00:09:31.110 --> 00:09:34.520
And it looks like that.
00:09:34.520 --> 00:09:38.230
So there is no stretching
in a rigid rotor.
00:09:38.230 --> 00:09:42.250
And so all of the complexity
is in the kinetic energy, not
00:09:42.250 --> 00:09:43.240
the potential.
00:09:43.240 --> 00:09:47.350
But each of these has a
different potential energy.
00:09:47.350 --> 00:09:51.486
And the energy levels
for the particle in a box
00:09:51.486 --> 00:09:52.735
are proportional to n squared.
00:09:57.390 --> 00:09:59.170
For the harmonic
oscillator, they're
00:09:59.170 --> 00:10:03.280
proportional to n plus 1/2.
00:10:03.280 --> 00:10:06.310
For the hydrogen atom,
they are proportional to 1
00:10:06.310 --> 00:10:09.470
over n squared.
00:10:09.470 --> 00:10:12.650
And for the rigid rotor, they're
proportional to n times n
00:10:12.650 --> 00:10:14.390
plus 1.
00:10:14.390 --> 00:10:16.640
So one of the
valuable things you
00:10:16.640 --> 00:10:21.020
get from looking at these
exactly soluble problems
00:10:21.020 --> 00:10:24.350
is that the energy level
patterns for each of them
00:10:24.350 --> 00:10:26.710
are slightly different.
00:10:26.710 --> 00:10:30.160
And you can tell what you've got
from the energy level pattern.
00:10:30.160 --> 00:10:33.210
And so when you take a spectrum,
so often you want to know
00:10:33.210 --> 00:10:34.460
what kind of spectrum is this?
00:10:34.460 --> 00:10:38.170
Sometimes you can tell just by
what frequency region it is.
00:10:38.170 --> 00:10:40.210
But usually, in a
spectrum, there's
00:10:40.210 --> 00:10:42.040
a pattern of energy levels.
00:10:42.040 --> 00:10:47.140
And that gets you focused on
well, what kind of problem--
00:10:47.140 --> 00:10:51.339
what is the nature
of the building
00:10:51.339 --> 00:10:52.630
blocks that we're going to use?
00:10:56.660 --> 00:10:58.420
So this is wonderful.
00:10:58.420 --> 00:11:03.040
Now, we're also going
to find that when
00:11:03.040 --> 00:11:05.950
we solve these problems
in quantum mechanics,
00:11:05.950 --> 00:11:11.680
we get an infinite number of
eigenfunctions and eigenvalues.
00:11:11.680 --> 00:11:16.450
And often, we get presented
to us a lot of integrals
00:11:16.450 --> 00:11:23.230
involving the operators between
different wave functions.
00:11:23.230 --> 00:11:24.790
And one of the
beautiful things is
00:11:24.790 --> 00:11:27.670
when the theory gives
you an infinite number
00:11:27.670 --> 00:11:29.230
of those integrals.
00:11:29.230 --> 00:11:31.750
So now we would
have a collection
00:11:31.750 --> 00:11:33.340
of all sorts of
integrals that are
00:11:33.340 --> 00:11:39.360
evaluated for us in a simple
dependence on quantum numbers.
00:11:39.360 --> 00:11:41.350
So what do we do with them?
00:11:41.350 --> 00:11:44.780
Well, there's two main things we
do with these infinite numbers
00:11:44.780 --> 00:11:46.310
of integrals.
00:11:46.310 --> 00:11:50.360
One is we say, well,
this problem is not
00:11:50.360 --> 00:11:51.800
the standard problem.
00:11:51.800 --> 00:11:53.150
There are defects.
00:11:53.150 --> 00:11:56.230
Instead of having the potential
for the harmonic oscillator
00:11:56.230 --> 00:11:59.450
that looked like this,
it might look like that.
00:11:59.450 --> 00:12:02.950
There might be
some anharmonicity.
00:12:02.950 --> 00:12:07.560
Or there can be all
sorts of defects.
00:12:07.560 --> 00:12:10.160
And these defects can be
expressed by integrals
00:12:10.160 --> 00:12:12.310
of some other operator.
00:12:12.310 --> 00:12:15.530
And that goes into
perturbation theory.
00:12:15.530 --> 00:12:18.470
Or if we're going to want
to look at wavepackets,
00:12:18.470 --> 00:12:20.810
creating a particle-like
state and asking
00:12:20.810 --> 00:12:26.310
how it propagates in time,
those integrals are useful.
00:12:26.310 --> 00:12:29.180
So there's all sorts
of fantastic stuff,
00:12:29.180 --> 00:12:32.690
once we kill the
standard problems.
00:12:32.690 --> 00:12:36.350
It's not just some
thing you have to study,
00:12:36.350 --> 00:12:38.250
and it's over with.
00:12:38.250 --> 00:12:39.620
You're going to use this.
00:12:39.620 --> 00:12:42.440
As long as you're involved
in physical chemistry,
00:12:42.440 --> 00:12:45.170
you're going to be using
perturbation theory
00:12:45.170 --> 00:12:50.130
to understand what's going
on beyond the simple stuff.
00:12:50.130 --> 00:12:56.595
OK, so let's get started.
00:12:59.480 --> 00:13:02.680
So why is the harmonic
oscillator so special?
00:13:02.680 --> 00:13:06.910
If we look at any potential
energy curve, any one
00:13:06.910 --> 00:13:11.200
dimensional problem,
it's typically
00:13:11.200 --> 00:13:13.960
harmonic at the bottom,
no matter what it does.
00:13:13.960 --> 00:13:18.330
I mean, this is a typical
molecular diatomic molecule
00:13:18.330 --> 00:13:19.420
potential.
00:13:19.420 --> 00:13:22.000
We have a hard inner wall.
00:13:22.000 --> 00:13:25.570
And we have bond breaking
at the outer wall.
00:13:25.570 --> 00:13:28.040
And so this is an
anharmonic potential.
00:13:28.040 --> 00:13:33.120
But the bottom part is harmonic.
00:13:33.120 --> 00:13:37.800
And so we can use
everything we learn
00:13:37.800 --> 00:13:39.660
about the harmonic
oscillator to begin
00:13:39.660 --> 00:13:44.780
to draw a picture of
arbitrary potentials.
00:13:44.780 --> 00:13:54.695
So this is hard
wall bond breaking.
00:13:57.530 --> 00:13:59.300
Now, we're mostly chemists.
00:13:59.300 --> 00:14:02.540
And breaking a bond is--
00:14:02.540 --> 00:14:04.440
how much energy does it
take to break a bond?
00:14:04.440 --> 00:14:06.023
Well, is that encoded
in the spectrum?
00:14:06.023 --> 00:14:07.650
Yeah.
00:14:07.650 --> 00:14:10.860
So this is the sort of
thing we would care about.
00:14:10.860 --> 00:14:18.580
And now let me put
some notation here.
00:14:18.580 --> 00:14:21.360
So this is the potential.
00:14:21.360 --> 00:14:26.440
And this is the potential of--
the internuclear resistance,
00:14:26.440 --> 00:14:30.000
which is traditionally called R.
We're going to switch notations
00:14:30.000 --> 00:14:32.160
really quickly from R to x.
00:14:32.160 --> 00:14:36.460
But this, the minimum of the
potential, is the equilibrium
00:14:36.460 --> 00:14:38.834
internuclear distance R sub e.
00:14:42.810 --> 00:14:52.680
So for R near R sub e
the potential V of R
00:14:52.680 --> 00:14:59.730
looks simply like k over 2, a
forced constant, R minus R sub
00:14:59.730 --> 00:15:00.345
e squared.
00:15:05.980 --> 00:15:09.580
So this is harmonic oscillator.
00:15:09.580 --> 00:15:12.920
And now we're going to
change notation a little bit.
00:15:12.920 --> 00:15:18.010
We're going to use lowercase
x to be R minus Re.
00:15:18.010 --> 00:15:21.040
In other words, it's
the distortion away
00:15:21.040 --> 00:15:23.550
from equilibrium.
00:15:23.550 --> 00:15:28.830
And we can take any potential
and write it as a power series.
00:15:51.262 --> 00:15:51.762
Sorry.
00:16:00.670 --> 00:16:01.290
And so on.
00:16:04.320 --> 00:16:10.035
So if we know these derivatives,
we know what to do with them.
00:16:10.035 --> 00:16:10.535
OK.
00:16:20.900 --> 00:16:27.870
So one standard potential is
called the Morse potential.
00:16:27.870 --> 00:16:29.710
Because it looks like this.
00:16:29.710 --> 00:16:33.530
It looks like what you need
for a diatomic molecule.
00:16:33.530 --> 00:16:36.200
And the Morse potential
has an analytic form,
00:16:36.200 --> 00:16:41.400
where the potential v
of x is equal to D sub
00:16:41.400 --> 00:16:46.185
e, the dissociation energy,
1 minus e to the minus ax.
00:16:49.050 --> 00:16:52.640
Well, that doesn't look
like polynomial of x.
00:16:52.640 --> 00:16:56.930
But we power series expand
this, and we get a polynomial x.
00:16:56.930 --> 00:16:59.610
OK, now, what is this D sub e?
00:16:59.610 --> 00:17:03.890
So if x is equal to 0--
00:17:03.890 --> 00:17:05.810
this is 1 0.
00:17:05.810 --> 00:17:08.490
And so we get the energy.
00:17:08.490 --> 00:17:12.859
The potential at
x equals 0 is 0.
00:17:12.859 --> 00:17:17.359
And if x is equal
to infinity, v of x
00:17:17.359 --> 00:17:21.630
equals D sub e,
so basically, this
00:17:21.630 --> 00:17:26.099
is the energy between
the bottom of the well
00:17:26.099 --> 00:17:27.490
and where it breaks.
00:17:27.490 --> 00:17:30.050
This is D sub e.
00:17:30.050 --> 00:17:30.550
OK?
00:17:30.550 --> 00:17:34.890
So this is well interpreted.
00:17:34.890 --> 00:17:36.630
All the rest of the
action is in here.
00:17:36.630 --> 00:17:41.300
AUDIENCE: [INAUDIBLE]
00:17:41.300 --> 00:17:42.300
ROBERT FIELD: I'm sorry?
00:17:42.300 --> 00:17:44.170
AUDIENCE: Do you
just square the x?
00:17:44.170 --> 00:17:46.400
ROBERT FIELD: Yep!
00:17:46.400 --> 00:17:47.000
Thank you.
00:17:50.150 --> 00:17:51.580
It's an innocent factor.
00:17:51.580 --> 00:17:54.010
But it turns out to
be very important.
00:17:54.010 --> 00:17:57.300
OK, so let's square this.
00:17:57.300 --> 00:18:02.740
We have v of x is
equal to De times 1
00:18:02.740 --> 00:18:12.245
minus 2e to the minus ax
plus e to the minus 2ax.
00:18:16.190 --> 00:18:19.640
OK, so we're going to do power
series expansions of these.
00:18:19.640 --> 00:18:21.910
And you can do that.
00:18:21.910 --> 00:18:32.950
And so we have
plus some function
00:18:32.950 --> 00:18:43.660
of x plus some function
of x squared, et cetera.
00:18:43.660 --> 00:18:47.270
Now, what is the function of x?
00:18:47.270 --> 00:18:53.320
We're taking our derivatives at
the equilibrium, at x equals 0.
00:18:53.320 --> 00:18:58.110
And we have a potential,
which has a minimum.
00:18:58.110 --> 00:19:02.050
So what's the derivative of
the potential at equilibrium?
00:19:02.050 --> 00:19:02.550
Yes?
00:19:02.550 --> 00:19:03.150
AUDIENCE: Zero.
00:19:03.150 --> 00:19:03.983
ROBERT FIELD: Right.
00:19:03.983 --> 00:19:06.650
And so the x term goes away.
00:19:06.650 --> 00:19:08.730
And so we have a constant
term, which we usually
00:19:08.730 --> 00:19:11.630
choose to be zero.
00:19:11.630 --> 00:19:16.420
And we have this quadratic term.
00:19:16.420 --> 00:19:18.440
Looks like a
harmonic oscillator.
00:19:18.440 --> 00:19:19.960
And then there are
other terms which
00:19:19.960 --> 00:19:24.310
express the personality
of the potential.
00:19:24.310 --> 00:19:25.240
This is universal.
00:19:25.240 --> 00:19:30.290
And the rest becomes
a special case.
00:19:30.290 --> 00:19:39.850
So we have v of
x is De a squared
00:19:39.850 --> 00:19:44.770
x squared plus other stuff.
00:19:44.770 --> 00:19:46.530
OK, we're going
to call this 1/2k.
00:19:49.350 --> 00:19:49.850
why?
00:19:49.850 --> 00:19:53.210
Because we'd like it to look
like a harmonic oscillator.
00:19:53.210 --> 00:19:55.430
I mean, we know
that the potential
00:19:55.430 --> 00:19:59.930
for a harmonic oscillator
is described in this way.
00:20:02.810 --> 00:20:07.510
So let's just draw a picture.
00:20:15.840 --> 00:20:19.740
So we have a spring and a mass.
00:20:19.740 --> 00:20:24.960
And I should have drawn
this up a little higher.
00:20:24.960 --> 00:20:29.340
OK, so at equilibrium
there is no force.
00:20:29.340 --> 00:20:33.850
If the mass is down here,
there is a force pulling it up.
00:20:33.850 --> 00:20:36.740
And if it's up here, there's
a force pushing it down.
00:20:36.740 --> 00:20:48.390
Hooke's Law says
that the force is
00:20:48.390 --> 00:20:56.380
equal to minus k x minus x 0.
00:20:56.380 --> 00:21:01.420
OK, and now we're going to
switch to just the lowercase x
00:21:01.420 --> 00:21:02.710
in a second.
00:21:02.710 --> 00:21:09.860
But now, the force
according to Newton
00:21:09.860 --> 00:21:17.530
is minus gradient
of the potential.
00:21:17.530 --> 00:21:20.970
So the potential
for this problem
00:21:20.970 --> 00:21:28.382
is v of little x is
equal to 1/2k x squared.
00:21:32.380 --> 00:21:39.310
OK, so we have what we expect
for a harmonic oscillator.
00:21:39.310 --> 00:21:44.700
And we're going to say the small
displacement part of the Morse
00:21:44.700 --> 00:21:46.950
oscillator looks like 1/2k.
00:21:46.950 --> 00:21:48.910
x squared looks harmonic.
00:21:48.910 --> 00:21:50.550
And so what do we do now?
00:21:54.970 --> 00:22:02.250
Well, one of
Newton's laws, force
00:22:02.250 --> 00:22:04.770
is equal to the mass
times the acceleration.
00:22:04.770 --> 00:22:14.490
And so we can say, oh,
well, the acceleration
00:22:14.490 --> 00:22:22.190
is the second derivative
of x with respect to t.
00:22:22.190 --> 00:22:29.850
And the force is a minus
gradient of the potential.
00:22:29.850 --> 00:22:32.300
So it's minus 1/2kx squared.
00:22:37.620 --> 00:22:38.370
Minus kx.
00:22:41.680 --> 00:22:45.900
OK, so this is the
gradient of the potential.
00:22:45.900 --> 00:22:48.790
This is the mass--
00:22:48.790 --> 00:22:50.570
well, I need the mass--
00:22:50.570 --> 00:22:52.116
times the acceleration.
00:22:52.116 --> 00:22:53.990
And so this is the
equation we have to solve.
00:23:00.590 --> 00:23:05.680
And so we want to find
the solution x of t.
00:23:05.680 --> 00:23:08.080
And this is a second order
differential equation
00:23:08.080 --> 00:23:10.760
because we have a
second derivative.
00:23:10.760 --> 00:23:12.830
So there's going
to be two terms.
00:23:12.830 --> 00:23:21.530
And they'll be-- we'll have some
sine and some cosine function.
00:23:21.530 --> 00:23:24.940
And we're going to
want a derivative,
00:23:24.940 --> 00:23:30.460
a second derivative, that
brings down a constant k.
00:23:30.460 --> 00:23:32.780
And so we know that
these are going
00:23:32.780 --> 00:23:45.500
to be things that have the form
a sine kx k or m square root
00:23:45.500 --> 00:23:56.840
x plus b cosine k m x.
00:23:56.840 --> 00:23:58.620
So this is the solution.
00:23:58.620 --> 00:24:00.560
And we have to find A and B.
00:24:00.560 --> 00:24:02.870
Now, why do we have k over m?
00:24:02.870 --> 00:24:05.550
Well, you can look at this
differential equation.
00:24:05.550 --> 00:24:07.385
And you can see that
we have an m here.
00:24:07.385 --> 00:24:08.510
We have a k here.
00:24:08.510 --> 00:24:11.180
And that's what you need
in order to solve it.
00:24:11.180 --> 00:24:14.340
So now our job is
simply to find A and B.
00:24:14.340 --> 00:24:19.740
AUDIENCE: Should these be
functions of t and not x?
00:24:19.740 --> 00:24:21.560
ROBERT FIELD: You
know, sometimes
00:24:21.560 --> 00:24:27.480
I go onto automatic pilot
because it's so familiar to me.
00:24:27.480 --> 00:24:31.030
I'm just writing what comes
up in my subconscious.
00:24:31.030 --> 00:24:34.860
But yes, that's a good point.
00:24:34.860 --> 00:24:37.350
All right, so we want x of t.
00:24:37.350 --> 00:24:40.470
And it is this combination.
00:24:40.470 --> 00:24:43.635
OK, so now we put
in some insights.
00:24:52.920 --> 00:24:56.340
We want to know what the
period of oscillation is.
00:24:56.340 --> 00:25:04.860
And so x t plus tau has
to be equal to x of t.
00:25:07.520 --> 00:25:12.210
And so when we do
that, we discover--
00:25:12.210 --> 00:25:14.700
and I'm just going to
skip a lot of steps--
00:25:14.700 --> 00:25:26.210
that k over m square root times
tau has to be equal to 2 pi
00:25:26.210 --> 00:25:27.350
in order to satisfy that.
00:25:35.490 --> 00:25:42.230
We call k over m
square root omega, just
00:25:42.230 --> 00:25:47.090
to simplify the equations.
00:25:47.090 --> 00:25:49.610
But we also discover
that this actually
00:25:49.610 --> 00:25:51.650
is an angular frequency.
00:25:55.980 --> 00:26:04.390
So if we say omega
tau is equal to 2 pi,
00:26:04.390 --> 00:26:09.690
then tau equals 2 pi
over omega, which is
00:26:09.690 --> 00:26:13.140
equal to 1 over the frequency.
00:26:13.140 --> 00:26:14.680
Just exactly what we expect.
00:26:19.150 --> 00:26:26.430
So we have the beginning of
a solution and omega, tau,
00:26:26.430 --> 00:26:32.190
and frequency make sense.
00:26:35.640 --> 00:26:38.790
Everything is what
we sort of expect.
00:26:38.790 --> 00:26:41.460
OK, so now the next
step is to determine
00:26:41.460 --> 00:26:43.560
the values of the constants.
00:26:48.950 --> 00:26:52.810
So normally, when we have
a differential equation,
00:26:52.810 --> 00:27:01.530
after we find the general form,
we apply boundary conditions.
00:27:01.530 --> 00:27:06.550
And so we're going to apply
some boundary conditions.
00:27:06.550 --> 00:27:09.130
So here we have the potential.
00:27:09.130 --> 00:27:26.450
And-- so this is the potential
as a function of coordinate.
00:27:26.450 --> 00:27:30.740
And this is the
turning point, this
00:27:30.740 --> 00:27:33.160
is the inner turning
point at energy E.
00:27:33.160 --> 00:27:36.860
This is the outer turning point.
00:27:36.860 --> 00:27:39.380
Well, what's true at
the turning point?
00:27:39.380 --> 00:27:42.020
The turning point,
the potential is
00:27:42.020 --> 00:27:49.100
equal to the energy at
the two turning points.
00:27:49.100 --> 00:27:58.790
So 1/2k x plus or
minus of E squared.
00:27:58.790 --> 00:28:02.900
So if we know E, we know
where the turning points are.
00:28:02.900 --> 00:28:12.030
And so x of plus minus
of E, we can solve this.
00:28:12.030 --> 00:28:19.740
And so we have 2E over k.
00:28:19.740 --> 00:28:20.240
Yeah.
00:28:27.200 --> 00:28:28.602
So this is the turning points.
00:28:28.602 --> 00:28:29.810
These are the turning points.
00:28:32.750 --> 00:28:36.810
Now, this is a fairly frequent
exercise in quantum mechanics.
00:28:36.810 --> 00:28:41.180
You're going to want to know
where are the turning points.
00:28:41.180 --> 00:28:44.420
Because this is how you impose
boundary conditions easily.
00:28:44.420 --> 00:28:47.780
And so knowing that a
turning point corresponds
00:28:47.780 --> 00:28:50.570
to where the potential is
equal to the total energy
00:28:50.570 --> 00:28:54.820
is enough to be able
to solve for this.
00:28:54.820 --> 00:29:07.600
OK, so suppose we start
at x equals x plus.
00:29:07.600 --> 00:29:14.620
And so x of 0 is
equal to x plus.
00:29:14.620 --> 00:29:19.730
And that determines one
of the coefficients.
00:29:19.730 --> 00:29:22.690
So x of 0--
00:29:22.690 --> 00:29:29.500
so we have at t equals
0 the sine term is 0
00:29:29.500 --> 00:29:32.580
and the cosine term is 1.
00:29:32.580 --> 00:29:39.440
And so the first
thing we get at t
00:29:39.440 --> 00:29:48.307
equals 0 is that B
is equal to x plus.
00:29:48.307 --> 00:29:48.806
Right?
00:29:52.070 --> 00:29:57.224
The next step is to find a.
00:29:57.224 --> 00:29:58.640
There are several
ways to do this.
00:29:58.640 --> 00:30:03.220
But it's useful to
draw a little picture.
00:30:03.220 --> 00:30:06.550
So x plus is here.
00:30:06.550 --> 00:30:12.190
And that occurs at t equals 0.
00:30:12.190 --> 00:30:17.880
And x minus occurs
at tau over 2.
00:30:17.880 --> 00:30:20.020
And in the middle
we have tau over 4.
00:30:22.710 --> 00:30:29.190
So let's ask for, what is the
value of the wave function
00:30:29.190 --> 00:30:33.150
when x is equal to 0?
00:30:33.150 --> 00:30:38.010
So how do we make the
x be 0 at tau over 4?
00:30:38.010 --> 00:30:45.750
And that determines the value
of B. I mean, the value of A.
00:30:45.750 --> 00:30:49.906
So we have everything we need.
00:30:49.906 --> 00:30:54.480
And now, before just
rushing on, let me just say,
00:30:54.480 --> 00:31:02.160
well, this just gives
A is equal to zero.
00:31:02.160 --> 00:31:05.650
OK, there's a
different approach.
00:31:05.650 --> 00:31:10.230
When we have a sine
plus a cosine term,
00:31:10.230 --> 00:31:14.100
we can always
re-express it as x of t
00:31:14.100 --> 00:31:22.550
is equal to some other constant
times sine omega t plus phi.
00:31:28.620 --> 00:31:31.480
And so the same sort
of analysis gives
00:31:31.480 --> 00:31:39.330
c is equal to E
over k square root.
00:31:39.330 --> 00:31:44.900
And phi is equal
to minus pi over 2.
00:31:44.900 --> 00:31:46.377
OK, I'm just writing this.
00:31:46.377 --> 00:31:47.210
You want to do that.
00:31:54.740 --> 00:32:05.000
OK, so now we're
going to be preparing
00:32:05.000 --> 00:32:09.830
to do quantum mechanics, the
quantum mechanical solution
00:32:09.830 --> 00:32:12.300
of the harmonic oscillator.
00:32:12.300 --> 00:32:15.020
And so there are going to
be other things that we care
00:32:15.020 --> 00:32:18.570
about, and one is
the kinetic energy
00:32:18.570 --> 00:32:20.400
and one is the potential energy.
00:32:23.420 --> 00:32:27.230
And in particular, we'd like to
know the kinetic energy, which
00:32:27.230 --> 00:32:28.670
we call t.
00:32:28.670 --> 00:32:33.700
And we'd like to know the
expectation value of t
00:32:33.700 --> 00:32:37.280
as a function of time,
or just T bar of t.
00:32:40.120 --> 00:32:46.940
And similarly, we'd like
to know the expectation
00:32:46.940 --> 00:32:51.450
value of the potential
energy as a function of time.
00:32:51.450 --> 00:32:58.190
And that's going
to be V bar of t.
00:32:58.190 --> 00:33:00.680
And so from classical
mechanics, we
00:33:00.680 --> 00:33:03.560
should be able to determine
what these average values
00:33:03.560 --> 00:33:06.420
of the kinetic and
potential energy.
00:33:06.420 --> 00:33:10.870
So what do we know?
00:33:10.870 --> 00:33:17.580
We know that the frequency
is omega over 2 pi.
00:33:17.580 --> 00:33:22.360
We know the period
is 1 over omega.
00:33:22.360 --> 00:33:35.280
OK, and so T of t is
1/2 mv squared of t,
00:33:35.280 --> 00:33:37.800
or p squared over 2m.
00:33:43.550 --> 00:33:50.120
But all right, v
is the derivative
00:33:50.120 --> 00:33:52.670
of x with respect to t.
00:33:52.670 --> 00:33:55.500
And we have the
solutions over here.
00:33:55.500 --> 00:34:07.480
And so we know that we can
write x of t and v of t
00:34:07.480 --> 00:34:08.800
just by taking derivatives.
00:34:08.800 --> 00:34:17.710
And so we have x
of t is 2e over k
00:34:17.710 --> 00:34:23.934
square root sine
omega t plus phi.
00:34:26.780 --> 00:34:41.370
And v of t is omega times 2e
over k square root cosine omega
00:34:41.370 --> 00:34:42.604
t plus phi.
00:34:45.510 --> 00:34:48.790
So we want v squared
to be able to calculate
00:34:48.790 --> 00:34:50.550
the kinetic energy.
00:34:50.550 --> 00:34:51.659
And so we do that.
00:34:54.250 --> 00:35:01.860
And so the kinetic
energy T of t is
00:35:01.860 --> 00:35:16.170
1/2m m omega squared 2e
over k cosine squared
00:35:16.170 --> 00:35:18.430
omega t plus phi.
00:35:22.290 --> 00:35:27.780
This here, omega is k over m,
the square root of k over m.
00:35:27.780 --> 00:35:30.740
S this is m times k over m.
00:35:30.740 --> 00:35:32.100
So we just get k out there.
00:35:40.070 --> 00:35:43.840
And so now we would like
to know the average value
00:35:43.840 --> 00:35:45.460
of the kinetic energy.
00:35:55.210 --> 00:36:00.380
OK, so we have m omega squared.
00:36:00.380 --> 00:36:05.190
And so that's k over k.
00:36:05.190 --> 00:36:12.980
And so we just get
E integral from 0
00:36:12.980 --> 00:36:27.470
to tau bt of cosine square
omega t plus phi over tau.
00:36:27.470 --> 00:36:29.160
We want the time average.
00:36:29.160 --> 00:36:32.670
And so we calculate this
integral and we divide by tau.
00:36:32.670 --> 00:36:36.000
That's how we take an average.
00:36:36.000 --> 00:36:47.870
And what we discover is
that this integral is
00:36:47.870 --> 00:36:55.010
the numerator is 1/2 tau
times E. No, I'm sorry.
00:36:55.010 --> 00:36:58.340
1/2 tau, we have E times
1/2 tau divided by tau.
00:36:58.340 --> 00:37:06.560
And this becomes E over
2, an important result.
00:37:06.560 --> 00:37:11.000
And we're going to discover
that the average value
00:37:11.000 --> 00:37:13.820
of the momentum for
a harmonic oscillator
00:37:13.820 --> 00:37:17.360
is E over 2 in
quantum mechanics.
00:37:17.360 --> 00:37:21.400
We do the same thing
for the potential.
00:37:21.400 --> 00:37:24.290
And we discover that
it is also E over 2.
00:37:29.890 --> 00:37:40.240
So we know that E is equal
to T of t plus V of t.
00:37:40.240 --> 00:37:44.470
But it's also true that
the average is equal--
00:37:44.470 --> 00:38:00.870
so T bar is equal to V bar
which is equal to E over 2.
00:38:00.870 --> 00:38:04.080
So now we have an
important interpretation
00:38:04.080 --> 00:38:06.130
of the harmonic oscillator.
00:38:06.130 --> 00:38:09.090
The harmonic
oscillator is moving
00:38:09.090 --> 00:38:12.630
from turning point to the middle
to the other turning point.
00:38:12.630 --> 00:38:15.690
And what's happening is
energy is being exchanged
00:38:15.690 --> 00:38:21.570
between all potential
energy at a turning point
00:38:21.570 --> 00:38:24.750
to all kinetic
energy in the middle.
00:38:24.750 --> 00:38:27.420
So energy is going
back and forth
00:38:27.420 --> 00:38:33.750
between kinetic energy
and potential energy.
00:38:33.750 --> 00:38:40.490
We can solve for the
relationship between T of t
00:38:40.490 --> 00:38:41.240
and V of t.
00:38:41.240 --> 00:38:53.770
And we can find that V of t
is equal to T minus pi over 4.
00:38:58.730 --> 00:38:59.610
tau over 4.
00:39:04.170 --> 00:39:08.040
So this is telling us
just what I said before,
00:39:08.040 --> 00:39:11.790
that energy is being exchanged
between potential and kinetic
00:39:11.790 --> 00:39:12.630
energy.
00:39:12.630 --> 00:39:16.470
And that the potential energy
is lagging by tau over 4
00:39:16.470 --> 00:39:20.220
behind the momentum.
00:39:24.734 --> 00:39:25.650
This is all very fast.
00:39:25.650 --> 00:39:29.210
But it's all classical
mechanics, which you know.
00:39:29.210 --> 00:39:31.100
And we're going to
be rediscovering all
00:39:31.100 --> 00:39:33.840
of this in quantum mechanics.
00:39:33.840 --> 00:39:36.080
And so we have to
know what are we
00:39:36.080 --> 00:39:39.270
aiming for in quantum mechanics?
00:39:39.270 --> 00:39:41.270
So that we can
completely say, yes, it's
00:39:41.270 --> 00:39:45.450
consistent with
classical mechanics.
00:39:45.450 --> 00:39:46.870
And there's some really--
00:39:46.870 --> 00:39:48.510
now, there's another
really neat thing.
00:39:52.030 --> 00:40:07.480
So if we look at X of t,
X of t is oscillating.
00:40:07.480 --> 00:40:13.929
And suppose we start
out at a turning
00:40:13.929 --> 00:40:15.220
point, the outer turning point.
00:40:19.450 --> 00:40:23.970
So we can tell from this, the
derivative of x with respect
00:40:23.970 --> 00:40:29.786
to t at x equals x
plus is going to be 0.
00:40:33.440 --> 00:40:35.510
So at a turning
point, the particle
00:40:35.510 --> 00:40:38.090
is hardly moving, not moving.
00:40:41.080 --> 00:40:44.470
And so what about the momentum?
00:40:44.470 --> 00:40:49.245
Well, the momentum is
going to look like this.
00:40:53.270 --> 00:40:54.810
Yeah, it's going
to look like that.
00:40:57.820 --> 00:41:07.020
And so at the time that we are
starting at a turning point,
00:41:07.020 --> 00:41:08.910
the time derivative
of the momentum
00:41:08.910 --> 00:41:10.260
is at its maximum value.
00:41:13.570 --> 00:41:17.310
So this is going to be
really important when
00:41:17.310 --> 00:41:21.030
we start looking at properties
of time-evolving wave
00:41:21.030 --> 00:41:21.540
functions.
00:41:21.540 --> 00:41:24.090
Because what we're
going to discover is,
00:41:24.090 --> 00:41:30.420
suppose we start our system
here, at a turning point.
00:41:30.420 --> 00:41:33.750
And that's actually something
that we can do very easily
00:41:33.750 --> 00:41:35.670
in an experiment.
00:41:35.670 --> 00:41:43.830
Because when we excite from one
electronic state to another,
00:41:43.830 --> 00:41:47.760
you automatically
create a wave packet,
00:41:47.760 --> 00:41:50.220
which is localized at a
particular internuclear
00:41:50.220 --> 00:41:50.720
distance.
00:41:53.460 --> 00:41:57.400
And so you go typically,
to a turning point.
00:41:57.400 --> 00:42:00.630
So then, well, what's
going to happen?
00:42:00.630 --> 00:42:02.910
Well, there is a thing in
quantum mechanics, which
00:42:02.910 --> 00:42:05.880
you will become familiar with,
called the autocorrelation
00:42:05.880 --> 00:42:07.320
function.
00:42:07.320 --> 00:42:07.910
No, it's not.
00:42:07.910 --> 00:42:09.570
It's called the
survival probability.
00:42:18.550 --> 00:42:23.400
And that's going
to be the product
00:42:23.400 --> 00:42:32.760
of the time-dependent
wave function at x and T,
00:42:32.760 --> 00:42:37.380
the time-dependent wave
function at x and 0.
00:42:37.380 --> 00:42:40.620
So this is expressing
somehow how
00:42:40.620 --> 00:42:43.620
the wave function
that is created at t
00:42:43.620 --> 00:42:47.740
equals 0 gets away from itself.
00:42:47.740 --> 00:42:49.440
It's a very important idea.
00:42:49.440 --> 00:42:51.450
Because you make something.
00:42:51.450 --> 00:42:53.580
And it evolves.
00:42:53.580 --> 00:42:57.300
And for a harmonic oscillator,
if you make it at a turning
00:42:57.300 --> 00:43:00.360
point, this thing changes.
00:43:00.360 --> 00:43:03.320
Because the momentum changes.
00:43:03.320 --> 00:43:08.670
And the contribution
of the coordinate
00:43:08.670 --> 00:43:15.120
to the decay of the
survival probability is--
00:43:15.120 --> 00:43:19.950
it's all due to the momentum,
and not the coordinate change.
00:43:19.950 --> 00:43:22.380
That's actually, a
very important insight.
00:43:22.380 --> 00:43:27.210
Because the momentum, the time
rate of change of the momentum,
00:43:27.210 --> 00:43:30.610
is minus the gradient
of the potential.
00:43:30.610 --> 00:43:34.380
So this is one way we
learn about the potential
00:43:34.380 --> 00:43:37.350
simply by starting
at a turning point
00:43:37.350 --> 00:43:40.380
and knowing that this
thing, which we can measure,
00:43:40.380 --> 00:43:43.980
is measuring the
thing we want to know.
00:43:43.980 --> 00:43:45.981
Now, we will get
to this very soon.
00:43:45.981 --> 00:43:48.480
Because I haven't even told you
about time-dependent quantum
00:43:48.480 --> 00:43:49.410
mechanics.
00:43:49.410 --> 00:43:52.664
But those are the things
that we expect to encounter.
00:43:58.100 --> 00:44:02.440
OK, now I want to give you--
00:44:02.440 --> 00:44:04.280
I'm going to throw at you--
00:44:04.280 --> 00:44:09.770
some useful stuff, which
turns out to be really easy.
00:44:09.770 --> 00:44:14.450
Suppose we want to
know the expectation
00:44:14.450 --> 00:44:20.030
values, or the average
values, of x, x squared,
00:44:20.030 --> 00:44:21.620
p, and p squared.
00:44:25.310 --> 00:44:29.490
OK, so we have a
harmonic oscillator.
00:44:29.490 --> 00:44:34.400
And do we know what this is?
00:44:34.400 --> 00:44:37.868
Do we know the expectation
value of the coordinate?
00:44:37.868 --> 00:44:38.740
AUDIENCE: 0.
00:44:38.740 --> 00:44:39.740
ROBERT FIELD: We have 0.
00:44:39.740 --> 00:44:41.330
Why is it 0?
00:44:41.330 --> 00:44:44.130
There's two ways of
answering that question.
00:44:44.130 --> 00:44:47.660
But these are easy
questions, which on an exam,
00:44:47.660 --> 00:44:49.310
you don't want to
evaluate an integral.
00:44:49.310 --> 00:44:51.530
You want to know why is it 0.
00:44:51.530 --> 00:44:53.030
And there's two answers.
00:44:53.030 --> 00:44:54.950
You said 0, right?
00:44:54.950 --> 00:44:56.328
Why did you say 0?
00:44:56.328 --> 00:44:58.240
AUDIENCE: Because
it's symmetric.
00:44:58.240 --> 00:44:59.430
ROBERT FIELD: Yes.
00:44:59.430 --> 00:45:03.100
OK, so there is a
symmetry argument.
00:45:03.100 --> 00:45:06.930
Another is well, is
the potential moving?
00:45:06.930 --> 00:45:08.440
The particle is in a potential.
00:45:08.440 --> 00:45:09.523
It's going back and forth.
00:45:09.523 --> 00:45:11.390
The potential is stationary.
00:45:11.390 --> 00:45:13.530
So there's no way
that x could move.
00:45:13.530 --> 00:45:14.910
X could be time dependent.
00:45:14.910 --> 00:45:19.110
So we know this is 0.
00:45:19.110 --> 00:45:21.510
What about p?
00:45:21.510 --> 00:45:27.540
Same thing, whether you
use symmetry or just
00:45:27.540 --> 00:45:29.784
physical insight.
00:45:29.784 --> 00:45:30.825
But what about x squared?
00:45:36.150 --> 00:45:44.050
Well, x squared is equal
to V of x over k over 2.
00:45:48.650 --> 00:45:52.370
So expectation
value of x squared
00:45:52.370 --> 00:46:00.800
is equal to the expectation
value of V over k over 2.
00:46:00.800 --> 00:46:04.850
But we know the expectation
value of V. It's E over 2.
00:46:13.920 --> 00:46:17.880
So we know without
doing any integrals what
00:46:17.880 --> 00:46:21.300
the expectation value of x
squared is, and similarly,
00:46:21.300 --> 00:46:22.050
for p squared.
00:46:26.980 --> 00:46:32.300
And that's just going to
be m times E. OK, so why
00:46:32.300 --> 00:46:33.990
do I care about these things?
00:46:33.990 --> 00:46:37.360
Well, we have a little
thing called the uncertainty
00:46:37.360 --> 00:46:39.190
principle.
00:46:39.190 --> 00:46:44.920
And we'd like to know the
uncertainty in the coordinate
00:46:44.920 --> 00:46:46.090
and the momentum.
00:46:46.090 --> 00:46:48.230
And this is our
classical view of it.
00:46:48.230 --> 00:46:52.510
But it's going to
remind us of what
00:46:52.510 --> 00:46:54.400
we find quantum mechanically.
00:46:54.400 --> 00:46:59.530
So the uncertainty
in x can be defined
00:46:59.530 --> 00:47:08.650
as the average value of x
squared minus the average value
00:47:08.650 --> 00:47:11.910
of x squared square root.
00:47:11.910 --> 00:47:14.460
That's just the variance.
00:47:14.460 --> 00:47:17.085
And well, this is 0.
00:47:17.085 --> 00:47:19.090
And we know what this is.
00:47:19.090 --> 00:47:27.670
So we know that the uncertainty
in x is E over k square root.
00:47:27.670 --> 00:47:30.400
And the uncertainty
in p is going
00:47:30.400 --> 00:47:36.970
to be p squared average
value minus p squared.
00:47:36.970 --> 00:47:38.330
This is still 0.
00:47:38.330 --> 00:47:44.060
And this one is then
m times E square root.
00:47:47.010 --> 00:47:55.310
So delta x delta p, it looks
like the uncertainty principle.
00:47:55.310 --> 00:48:01.235
This is classic mechanics,
is just E over omega.
00:48:01.235 --> 00:48:02.710
But what's E over omega?
00:48:06.034 --> 00:48:06.534
AUDIENCE: h?
00:48:09.890 --> 00:48:10.950
h bar.
00:48:10.950 --> 00:48:12.450
ROBERT FIELD: Right.
00:48:12.450 --> 00:48:16.470
So it's all going
to come around.
00:48:16.470 --> 00:48:21.360
This is related to the
uncertainty principle
00:48:21.360 --> 00:48:22.500
in quantum mechanics.
00:48:22.500 --> 00:48:26.340
There is a minimum joint
uncertainty between x and p.
00:48:26.340 --> 00:48:29.620
And it's just related
to this constant.
00:48:29.620 --> 00:48:32.080
Now, this doesn't say
the uncertainty grows
00:48:32.080 --> 00:48:34.510
as you go to higher
and higher energy,
00:48:34.510 --> 00:48:36.280
as it will in quantum mechanics.
00:48:36.280 --> 00:48:39.080
But this is really
a neat thing to see.
00:48:39.080 --> 00:48:41.020
OK, I've got two minutes left.
00:48:44.010 --> 00:48:49.760
So if we wanted to know
the probability of finding
00:48:49.760 --> 00:49:03.279
x as a function of the
coordinate, the turning points.
00:49:03.279 --> 00:49:04.570
So here are the turning points.
00:49:04.570 --> 00:49:07.330
And we can calculate
what is going
00:49:07.330 --> 00:49:13.131
to be the probability of finding
the particle at one turning
00:49:13.131 --> 00:49:13.630
point.
00:49:13.630 --> 00:49:15.640
Well, it comes
down from infinity,
00:49:15.640 --> 00:49:18.620
goes back up to infinity.
00:49:18.620 --> 00:49:28.660
So here at the turning points,
the particle is stopped.
00:49:28.660 --> 00:49:32.410
And so the probability of
finding it at a turning point
00:49:32.410 --> 00:49:35.530
is infinite, times dx.
00:49:35.530 --> 00:49:39.636
So we don't have
an infinite number.
00:49:39.636 --> 00:49:41.010
In quantum mechanics,
we're going
00:49:41.010 --> 00:49:44.204
to discover that quantum
mechanics is smarter than that.
00:49:44.204 --> 00:49:45.870
And what quantum
mechanics does, suppose
00:49:45.870 --> 00:49:47.325
we have this is the energy.
00:49:51.040 --> 00:49:54.100
What quantum mechanics does
is that the probability
00:49:54.100 --> 00:49:57.940
at a turning point is not 0.
00:49:57.940 --> 00:50:02.600
And there's something--
what happens
00:50:02.600 --> 00:50:08.490
is there's tunneling tails that
instead of going to infinity,
00:50:08.490 --> 00:50:09.950
it has a finite value.
00:50:09.950 --> 00:50:14.270
And it reaches out into
the forbidden region
00:50:14.270 --> 00:50:17.300
in an exponentially
decreasing way.
00:50:17.300 --> 00:50:19.610
And that's basically
the difference
00:50:19.610 --> 00:50:21.980
between classical mechanics
and quantum mechanics.
00:50:21.980 --> 00:50:24.470
And there's an awful
lot of important stuff
00:50:24.470 --> 00:50:25.780
that happens there.
00:50:25.780 --> 00:50:27.920
Now, I've been very
fast through all of this
00:50:27.920 --> 00:50:32.084
because it's built on stuff
that you're supposed to know.
00:50:32.084 --> 00:50:33.500
And it's built on
stuff that we're
00:50:33.500 --> 00:50:36.840
going to work hard to understand
from a quantum mechanical point
00:50:36.840 --> 00:50:37.340
of view.
00:50:37.340 --> 00:50:39.740
But this sets the
stage for, what
00:50:39.740 --> 00:50:43.700
are the things we have to
look for in quantum mechanics?
00:50:43.700 --> 00:50:47.880
So I do recommend that
you look at the notes
00:50:47.880 --> 00:50:51.330
and make sure you can follow
all the steps, which I went
00:50:51.330 --> 00:50:52.855
through incredibly rapidly.
00:50:56.260 --> 00:50:57.730
And it'll be really helpful.
00:50:57.730 --> 00:51:02.680
Because your job will
be to draw cartoons.
00:51:02.680 --> 00:51:04.770
And these will guide
you through it.
00:51:04.770 --> 00:51:09.989
OK, so I'll see you on Friday.