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PROFESSOR: Begin
with a new topic,
00:00:24.010 --> 00:00:28.260
which is breakdown of classical
statistical mechanics.
00:00:43.750 --> 00:00:47.990
So we developed a formalism
to probabilistically describe
00:00:47.990 --> 00:00:50.590
collections of large particles.
00:00:50.590 --> 00:00:56.560
And once we have that, from that
formalism calculate properties
00:00:56.560 --> 00:01:00.060
of matter that have to do
with heat, temperature,
00:01:00.060 --> 00:01:02.210
et cetera, and things
coming to equilibrium.
00:01:05.870 --> 00:01:14.190
So question is, is this
formalism always successful?
00:01:14.190 --> 00:01:21.110
And by the time you come to
the end of the 19th Century,
00:01:21.110 --> 00:01:25.240
there were several things
that were hanging around that
00:01:25.240 --> 00:01:28.090
had to do with thermal
properties of the matter
00:01:28.090 --> 00:01:33.810
where this formalism
was having difficulties.
00:01:33.810 --> 00:01:36.500
And the difficulties
ultimately pointed out
00:01:36.500 --> 00:01:39.220
to emergence of
quantum mechanics.
00:01:39.220 --> 00:01:42.210
So essentially, understanding
the relationship
00:01:42.210 --> 00:01:46.030
between thermodynamics,
statistical mechanics,
00:01:46.030 --> 00:01:50.700
and properties of matter was
very important to development
00:01:50.700 --> 00:01:52.020
of quantum mechanics.
00:01:52.020 --> 00:01:58.180
And in particular, I will
mention three difficulties.
00:01:58.180 --> 00:02:03.820
The most important one
that really originally set
00:02:03.820 --> 00:02:06.490
the first stone for
quantum mechanics
00:02:06.490 --> 00:02:08.834
is the spectrum of
black body radiation.
00:02:21.010 --> 00:02:23.320
And it's basically
the observation
00:02:23.320 --> 00:02:25.840
that you heat something.
00:02:25.840 --> 00:02:30.110
And when it becomes hot,
it starts to radiate.
00:02:30.110 --> 00:02:34.720
And typically, the color of
the radiation that you get
00:02:34.720 --> 00:02:36.960
is a function of
temperature, but does not
00:02:36.960 --> 00:02:40.850
depend on the properties of the
material that you are heating.
00:02:40.850 --> 00:02:43.020
So that has to do with heat.
00:02:43.020 --> 00:02:45.770
And you should be
able to explain
00:02:45.770 --> 00:02:50.570
that using
statistical mechanics.
00:02:50.570 --> 00:02:53.650
Another thing that we
have already mentioned
00:02:53.650 --> 00:02:57.450
has to do with the third
law of thermodynamics.
00:02:57.450 --> 00:03:04.270
And let's say the heat capacity
of materials such as solids.
00:03:08.960 --> 00:03:14.780
We mentioned this
Nernst theorem that
00:03:14.780 --> 00:03:20.160
was the third law
of thermodynamics
00:03:20.160 --> 00:03:22.200
based on observation.
00:03:22.200 --> 00:03:26.150
Consequence of it was that
heat capacity of most things
00:03:26.150 --> 00:03:31.650
that you can measure go to 0
as you go to 0 temperature.
00:03:31.650 --> 00:03:33.770
We should be able to
explain that again,
00:03:33.770 --> 00:03:39.040
based on the phenomena of
statistical-- the phenomenology
00:03:39.040 --> 00:03:44.830
of thermodynamics and the
rules of statistical mechanics.
00:03:44.830 --> 00:03:47.400
Now, a third thing that
is less often mentioned
00:03:47.400 --> 00:03:54.910
but is also important has
to do with heat capacity
00:03:54.910 --> 00:04:06.380
of the atomic gases such as
the air in this room, which
00:04:06.380 --> 00:04:09.260
is composed of, say,
oxygen and nitrogen that
00:04:09.260 --> 00:04:12.620
are diatomic gases.
00:04:12.620 --> 00:04:20.754
So probably, historically they
were answered and discussed
00:04:20.754 --> 00:04:25.570
and resolving the order
that I have drawn for you.
00:04:25.570 --> 00:04:27.480
But I will go backwards.
00:04:27.480 --> 00:04:30.590
so we will first
talk about this one,
00:04:30.590 --> 00:04:35.070
then about number two--
heat capacity of solids--
00:04:35.070 --> 00:04:39.500
and number three about
black body radiation.
00:04:39.500 --> 00:04:41.650
OK.
00:04:41.650 --> 00:04:45.520
Part of the reason is that
throughout the course,
00:04:45.520 --> 00:04:49.230
we have been using our
understanding of the gas
00:04:49.230 --> 00:04:55.420
as the sort of measure
of how well we understand
00:04:55.420 --> 00:04:57.870
thermal properties
of the matter.
00:04:57.870 --> 00:05:02.230
And so let's stick with
the gas and ask, what do I
00:05:02.230 --> 00:05:06.520
know about the heat capacity
of the gas in this room?
00:05:06.520 --> 00:05:11.497
So let's think
about heat capacity
00:05:11.497 --> 00:05:16.495
of dilute diatomic gas.
00:05:22.290 --> 00:05:28.440
It is a gas that is sufficiently
dilute that it is practically
00:05:28.440 --> 00:05:30.290
having ideal gas law.
00:05:30.290 --> 00:05:34.720
So PV is roughly
proportional to temperature.
00:05:34.720 --> 00:05:37.160
But rather than thinking
about its pressure,
00:05:37.160 --> 00:05:39.040
I want to make sure I
understand something
00:05:39.040 --> 00:05:42.810
about the heat capacity, another
quantity that I can measure.
00:05:42.810 --> 00:05:44.270
So what's going on here?
00:05:44.270 --> 00:05:46.990
I have, let's say, a box.
00:05:46.990 --> 00:05:49.640
And within this box,
we have a whole bunch
00:05:49.640 --> 00:05:51.910
of these diatomic molecules.
00:05:56.140 --> 00:06:01.910
Let's stick to the
canonical ensemble.
00:06:01.910 --> 00:06:05.660
So I tell you the volume
of this gas, the number
00:06:05.660 --> 00:06:07.690
of diatomic molecules,
and the temperature.
00:06:10.380 --> 00:06:16.260
And in this formalism, I
would calculate the partition
00:06:16.260 --> 00:06:16.780
function.
00:06:16.780 --> 00:06:18.800
Out of that, I should
be able to calculate
00:06:18.800 --> 00:06:22.220
the energy, heat
capacity, et cetera.
00:06:22.220 --> 00:06:24.470
So what do I have to do?
00:06:24.470 --> 00:06:30.980
I have to integrate over all
possible coordinates that
00:06:30.980 --> 00:06:33.340
occur in this system.
00:06:33.340 --> 00:06:37.060
To all intents and purposes,
the different molecules
00:06:37.060 --> 00:06:38.890
are identical.
00:06:38.890 --> 00:06:42.840
So I divide by the
phase space that
00:06:42.840 --> 00:06:45.930
is assigned to each one of them.
00:06:45.930 --> 00:06:51.210
And I said it is dilute
enough that for all intents
00:06:51.210 --> 00:06:55.880
and purposes, the pressure is
proportional to temperature.
00:06:55.880 --> 00:06:58.420
And that occur, I
know, when I can
00:06:58.420 --> 00:07:01.720
ignore the interactions
between particles.
00:07:01.720 --> 00:07:05.490
So if I can ignore the
interactions between particles,
00:07:05.490 --> 00:07:08.450
then the partition function
for the entire system
00:07:08.450 --> 00:07:10.820
would be the product of
the partition functions
00:07:10.820 --> 00:07:13.970
that I would write for
the individual molecules,
00:07:13.970 --> 00:07:16.270
or one of them raised
to the N power.
00:07:16.270 --> 00:07:20.230
So what's the Z1 that
I have to calculate?
00:07:20.230 --> 00:07:23.880
Z1 is obtained by
integrating over
00:07:23.880 --> 00:07:28.270
the coordinates and momenta
of a single diatomic particle.
00:07:28.270 --> 00:07:30.530
So I have a factor of d cubed p.
00:07:30.530 --> 00:07:33.600
I have a factor of d cubed q.
00:07:33.600 --> 00:07:38.480
But I have two particles, so
I have d cubed p1, d cubed q1,
00:07:38.480 --> 00:07:43.850
d cubed p2, d cubed
q 2, and I have
00:07:43.850 --> 00:07:48.170
six pairs of coordinate momenta.
00:07:48.170 --> 00:07:50.800
So I divide it by h cubed.
00:07:50.800 --> 00:07:57.820
I have e to the minus beta
times the energy of this system,
00:07:57.820 --> 00:08:04.230
which is p1 squared over
2m p2 squared over 2m.
00:08:04.230 --> 00:08:08.160
And some potential
of interaction
00:08:08.160 --> 00:08:12.220
that is responsible for bringing
and binding these things
00:08:12.220 --> 00:08:13.720
together.
00:08:13.720 --> 00:08:18.110
So there is some V that
is function of q1 and q2
00:08:18.110 --> 00:08:21.760
that binds the two particles
together and does not
00:08:21.760 --> 00:08:24.290
allow them to become separate.
00:08:24.290 --> 00:08:26.810
All right, so what
do we do here?
00:08:26.810 --> 00:08:30.275
We realize that immediately for
one of these particle,s there
00:08:30.275 --> 00:08:34.039
is a center of mass that
can go all over the place.
00:08:34.039 --> 00:08:36.750
So we change
coordinates to, let's
00:08:36.750 --> 00:08:42.049
say, Q, which is
q1 plus q2 over 2.
00:08:42.049 --> 00:08:45.910
And corresponding to the
center of mass position,
00:08:45.910 --> 00:08:51.660
there is also a center
of mass momentum, P,
00:08:51.660 --> 00:08:59.320
which is related to p1 minus p2.
00:08:59.320 --> 00:09:03.430
But when I make the change
of variables from these
00:09:03.430 --> 00:09:06.840
coordinates to these
coordinates, what I will get
00:09:06.840 --> 00:09:10.450
is that I will have
a simple integral
00:09:10.450 --> 00:09:12.380
over the relative coordinates.
00:09:12.380 --> 00:09:17.230
So I have d cubed Q d
cubed big P h cubed.
00:09:17.230 --> 00:09:20.580
And the only thing that
I have over there is
00:09:20.580 --> 00:09:27.640
e to the minus beta p squared
divided by 2 big M. Big M
00:09:27.640 --> 00:09:30.430
being the sum total
of the two masses.
00:09:30.430 --> 00:09:33.670
If the two masses are
identical, it would be 2M.
00:09:33.670 --> 00:09:36.316
Otherwise, it would
be M1 plus M2.
00:09:36.316 --> 00:09:42.040
And then I have an integration
over the relative coordinate.
00:09:42.040 --> 00:09:46.642
Let's call that q
relative momentum p
00:09:46.642 --> 00:09:52.330
h cubed e to the minus
beta p squared over 2 times
00:09:52.330 --> 00:09:56.000
the reduced mass here.
00:09:56.000 --> 00:09:59.280
And then, the
potential which only
00:09:59.280 --> 00:10:01.250
is a function of the
relative coordinate.
00:10:06.470 --> 00:10:12.070
Point is that what I have done
is I have separated out this 6
00:10:12.070 --> 00:10:15.270
degrees of freedom that
make up-- or actually,
00:10:15.270 --> 00:10:18.850
the 3 degrees of freedom and
their conjugate momenta that
00:10:18.850 --> 00:10:23.650
make up a single molecule into
some degrees of freedom that
00:10:23.650 --> 00:10:27.210
correspond to the center of
mass and some degrees of freedom
00:10:27.210 --> 00:10:29.910
that correspond to
the relative motion.
00:10:29.910 --> 00:10:31.870
Furthermore, for
the relative motion
00:10:31.870 --> 00:10:34.240
I expect that the
form of this potential
00:10:34.240 --> 00:10:43.980
as a function of the separation
has a form that is a minimum.
00:10:43.980 --> 00:10:48.270
Basically, the particles
at 0 temperature
00:10:48.270 --> 00:10:52.110
would be sitting
where this minimum is.
00:10:52.110 --> 00:10:55.900
So essentially, the shape
of this diatomic molecule
00:10:55.900 --> 00:11:00.580
would be something like this
if I find its minimum energy
00:11:00.580 --> 00:11:03.270
configuration.
00:11:03.270 --> 00:11:08.940
But then, I can allow
it to move with respect
00:11:08.940 --> 00:11:11.130
to, say, the minimum energy.
00:11:11.130 --> 00:11:13.590
Let's say it occurs
at some distance d.
00:11:16.350 --> 00:11:20.560
It can oscillate around
this minimum value.
00:11:20.560 --> 00:11:22.800
If it oscillates around
this minimum value,
00:11:22.800 --> 00:11:28.100
it basically will explore
the bottom of this potential.
00:11:28.100 --> 00:11:35.350
So I can basically think of
this center of mass contribution
00:11:35.350 --> 00:11:37.220
to the partition function.
00:11:37.220 --> 00:11:41.440
And this contribution
has a part that
00:11:41.440 --> 00:11:46.390
comes from these oscillations
around the center of mass.
00:11:46.390 --> 00:11:51.150
Let's call that u.
00:11:51.150 --> 00:11:53.430
Then, there is the
corresponding momentum.
00:11:53.430 --> 00:11:56.220
I don't know, let's call it pi.
00:11:56.220 --> 00:12:02.490
I divide by h and I
have e to the minus beta
00:12:02.490 --> 00:12:06.070
pi squared over 2 mu.
00:12:06.070 --> 00:12:10.000
And then I have minus beta.
00:12:10.000 --> 00:12:11.900
Well, to the lowest
order, I have
00:12:11.900 --> 00:12:14.650
v of d, which is a constant.
00:12:14.650 --> 00:12:22.170
And then I have some
frequency, some curvature
00:12:22.170 --> 00:12:23.890
at the bottom of
this potential that I
00:12:23.890 --> 00:12:26.660
choose to write as
mu omega squared
00:12:26.660 --> 00:12:30.160
over 2 multiplying by u squared.
00:12:30.160 --> 00:12:32.800
Essentially, what
I want to do is
00:12:32.800 --> 00:12:38.120
to say that really, there is a
vibrational degree of freedom
00:12:38.120 --> 00:12:41.480
and there is a harmonic
oscillator that describes that.
00:12:41.480 --> 00:12:44.117
The frequency of that is
related to the curvature
00:12:44.117 --> 00:12:45.950
that I have at the
bottom of this potential.
00:12:49.530 --> 00:12:52.700
So this degree of freedom
corresponds to vibrations.
00:12:59.590 --> 00:13:05.490
But that's not the end of this
story because here I had three
00:13:05.490 --> 00:13:06.910
q's.
00:13:06.910 --> 00:13:10.000
One of them became the
amplitude of this oscillation.
00:13:10.000 --> 00:13:16.240
So basically, the relative
coordinate is a vector.
00:13:16.240 --> 00:13:19.730
One degree of freedom
corresponds to stretching,
00:13:19.730 --> 00:13:22.800
but there are two
other components of it.
00:13:22.800 --> 00:13:24.830
those two other
components correspond
00:13:24.830 --> 00:13:28.320
to essentially keeping
the length of this fixed
00:13:28.320 --> 00:13:31.340
but moving in the
other directions.
00:13:31.340 --> 00:13:32.470
What do they correspond to?
00:13:32.470 --> 00:13:34.790
They correspond to rotations.
00:13:34.790 --> 00:13:38.280
So then there is essentially
another partition function
00:13:38.280 --> 00:13:39.920
that I want right here.
00:13:39.920 --> 00:13:43.334
That corresponds to the
rotational degrees of freedom.
00:13:46.020 --> 00:13:49.640
Now, the rotational
degrees of freedom
00:13:49.640 --> 00:13:54.500
have a momentum contribution
because this p is also
00:13:54.500 --> 00:13:55.960
three components.
00:13:55.960 --> 00:13:58.850
One component went
in to the vibrations.
00:13:58.850 --> 00:14:01.360
There are two more
components that
00:14:01.360 --> 00:14:04.620
really combine to tell
you about the angular
00:14:04.620 --> 00:14:08.200
momentum and the energy that
is proportional to the square
00:14:08.200 --> 00:14:09.750
of the angular momentum.
00:14:09.750 --> 00:14:12.090
But there is no
restoring force for them.
00:14:12.090 --> 00:14:15.120
There is no corresponding
term that is like this.
00:14:15.120 --> 00:14:20.090
So maybe I will just write
that as an integral over angles
00:14:20.090 --> 00:14:22.270
that I can rotate this thing.
00:14:22.270 --> 00:14:24.860
An integral over
the two components
00:14:24.860 --> 00:14:28.610
of the angular momentum
divided by h squared.
00:14:28.610 --> 00:14:31.050
There's actually two angles.
00:14:31.050 --> 00:14:35.970
And the contribution is e
to the minus beta angular
00:14:35.970 --> 00:14:39.350
momentum squared over 2I.
00:14:39.350 --> 00:14:42.990
So I wrote the entire thing.
00:14:42.990 --> 00:14:49.550
So essentially,
all I have done is
00:14:49.550 --> 00:14:53.060
I have taken the
Hamiltonian that
00:14:53.060 --> 00:14:55.070
corresponds to
two particles that
00:14:55.070 --> 00:14:59.460
are bound together and
broken it into three pieces
00:14:59.460 --> 00:15:03.640
corresponding to the center
of mass, to the vibrations,
00:15:03.640 --> 00:15:04.515
and to the rotations.
00:15:08.300 --> 00:15:12.920
Now, the thing is
that if I now ask,
00:15:12.920 --> 00:15:18.500
what is the energy that I would
get for this one particle--
00:15:18.500 --> 00:15:24.660
I guess I'll call this Z1-- what
is the contribution of the one
00:15:24.660 --> 00:15:29.440
particular to the energy
of the entire system?
00:15:29.440 --> 00:15:34.380
I have minus the log Z1
with respect to beta.
00:15:34.380 --> 00:15:37.610
That's the usual formula
to calculate energies.
00:15:40.170 --> 00:15:44.360
So I go and look at
this entire thing.
00:15:44.360 --> 00:15:48.070
And where do the beta
dependencies come from?
00:15:48.070 --> 00:15:49.280
Well, let's see.
00:15:49.280 --> 00:16:00.220
So my Z1 has a part that comes
from this center of mass.
00:16:00.220 --> 00:16:05.720
It gives me a V. We expect that.
00:16:05.720 --> 00:16:09.570
And then from the
integration over the momenta,
00:16:09.570 --> 00:16:20.821
I will get something like 2 pi
m over beta h squared to the 3/2
00:16:20.821 --> 00:16:21.321
power.
00:16:25.090 --> 00:16:31.200
From the vibrations--
OK, what do I have?
00:16:31.200 --> 00:16:36.340
I have e to the minus beta V
of d, which is the constant.
00:16:36.340 --> 00:16:39.570
We really don't care.
00:16:39.570 --> 00:16:43.240
But there are these
two components
00:16:43.240 --> 00:16:52.775
that give me root 2
pi mu divided by beta.
00:16:57.960 --> 00:17:02.910
There is a
corresponding thing that
00:17:02.910 --> 00:17:06.990
comes from the variance that
goes with this object, which
00:17:06.990 --> 00:17:16.119
is square root of 2 pi divided
by beta mu omega squared.
00:17:16.119 --> 00:17:18.225
The entire thing
has a factor of 1/h.
00:17:22.483 --> 00:17:23.566
So this is the vibrations.
00:17:26.930 --> 00:17:29.570
And for the rotations,
what do I get?
00:17:29.570 --> 00:17:33.780
I will get a 4 pi
from integrating
00:17:33.780 --> 00:17:37.200
over all orientations.
00:17:37.200 --> 00:17:40.300
Divided by h squared.
00:17:40.300 --> 00:17:46.970
I have essentially the two
components of angular momentum.
00:17:46.970 --> 00:17:55.806
So I get essentially, the square
of 2 pi I divided by beta.
00:17:58.602 --> 00:18:03.240
So this is rotations.
00:18:03.240 --> 00:18:05.220
And this is center of mass.
00:18:12.010 --> 00:18:14.520
We can see that if
I take that formula,
00:18:14.520 --> 00:18:22.410
take its log divide by-- take
a derivative with respect
00:18:22.410 --> 00:18:23.710
to beta.
00:18:23.710 --> 00:18:26.360
First of all, I will
get this constant
00:18:26.360 --> 00:18:30.030
that is the energy of the
bond state at 0 temperature.
00:18:30.030 --> 00:18:32.630
But the more interesting
things are the things
00:18:32.630 --> 00:18:38.060
that I take from the derivatives
of the various factors of beta.
00:18:38.060 --> 00:18:41.470
Essentially, for each factor
of beta in the denominator,
00:18:41.470 --> 00:18:44.550
log Z will have a
minus log of beta.
00:18:44.550 --> 00:18:48.850
I take a derivative, I will
get a factor of 1 over beta.
00:18:48.850 --> 00:18:55.410
So from here, I will get 3/2
1 over beta, which is 3/2 kT.
00:18:55.410 --> 00:18:56.905
So this is the center of mass.
00:19:02.770 --> 00:19:09.180
From here, I have two
factors of beta to the 1/2.
00:19:09.180 --> 00:19:12.990
So they combine to give
me one factor of kT.
00:19:12.990 --> 00:19:13.960
This is for vibrations.
00:19:17.850 --> 00:19:21.360
And similarly, I have
two factors of beta
00:19:21.360 --> 00:19:28.790
to the 1/2, which correspond
to 1 kT for rotations.
00:19:35.660 --> 00:19:40.440
So then I say that the heat
capacity at constant volume
00:19:40.440 --> 00:19:46.710
is simply-- per particle
is related to d e1 by dT.
00:19:46.710 --> 00:19:54.960
And I see that that amounts
to kb times 3/2 plus 1 plus 1,
00:19:54.960 --> 00:19:59.160
or I should get 7/2 kb.
00:19:59.160 --> 00:20:05.350
Per particle, which says that
if you go and calculate the heat
00:20:05.350 --> 00:20:09.440
capacity of the
gas in this room,
00:20:09.440 --> 00:20:11.430
divide by the
number of molecules
00:20:11.430 --> 00:20:14.980
that we have-- doesn't
matter whether they
00:20:14.980 --> 00:20:16.410
are oxygens or nitrogen.
00:20:16.410 --> 00:20:18.890
They would basically give
the same contribution
00:20:18.890 --> 00:20:21.070
because you can see
that the masses and all
00:20:21.070 --> 00:20:25.020
the other properties
of the molecule
00:20:25.020 --> 00:20:27.890
do not appear in
the heat capacity.
00:20:27.890 --> 00:20:32.080
That as a function
of temperature,
00:20:32.080 --> 00:20:35.730
I should get a value of 7/2.
00:20:35.730 --> 00:20:38.890
So basically, C in units of kb.
00:20:38.890 --> 00:20:40.140
So I divide by kb.
00:20:40.140 --> 00:20:42.970
And my predictions is
that I should see 7/2.
00:20:49.030 --> 00:20:53.680
So you go and do a measurement
and what do you get?
00:20:53.680 --> 00:20:55.240
What you get is actually 5/2.
00:21:02.720 --> 00:21:07.000
So something is not quite right.
00:21:07.000 --> 00:21:11.700
We are not getting the
7/2 that we predicted.
00:21:11.700 --> 00:21:13.400
Except that I really
mentioned that you
00:21:13.400 --> 00:21:18.500
are getting this measurement
when you do measurements
00:21:18.500 --> 00:21:21.190
at room temperature,
you get this value.
00:21:21.190 --> 00:21:23.720
So when we measure
the heat capacity
00:21:23.720 --> 00:21:27.110
of the gas in this
room, we will get 5/2.
00:21:27.110 --> 00:21:29.890
But if we heat it
up, by the time
00:21:29.890 --> 00:21:35.420
we get to temperatures of a
few thousand degrees Kelvin.
00:21:35.420 --> 00:21:39.680
So if you heat the room
by a factor of 5 to 10,
00:21:39.680 --> 00:21:45.410
you will actually
get the value of 7/2.
00:21:45.410 --> 00:21:48.100
And if you cool
it, by the time you
00:21:48.100 --> 00:21:53.190
get to the order of
10 degrees or fewer,
00:21:53.190 --> 00:21:57.950
then you will find that
the heat capacity actually
00:21:57.950 --> 00:21:58.910
goes even further.
00:21:58.910 --> 00:22:00.970
It goes all the way to 3/2.
00:22:04.030 --> 00:22:06.910
And the 3/2 is
the thing that you
00:22:06.910 --> 00:22:09.320
would have predicted
for a gas that
00:22:09.320 --> 00:22:13.320
had monatomic particles,
no internal structure.
00:22:13.320 --> 00:22:16.340
Because then the only thing
that you would have gotten
00:22:16.340 --> 00:22:20.440
is the center of
mass contribution.
00:22:20.440 --> 00:22:25.320
So it seems like by going
to low temperatures,
00:22:25.320 --> 00:22:28.580
you somehow freeze
the degrees of freedom
00:22:28.580 --> 00:22:35.820
that correspond to vibrations
and rotations of the gas.
00:22:35.820 --> 00:22:38.086
And by going to really
high temperatures,
00:22:38.086 --> 00:22:41.750
you are able to liberate all
of these degrees of freedom
00:22:41.750 --> 00:22:43.550
and store energy in them.
00:22:43.550 --> 00:22:45.920
Heat capacity is the
measure of the ability
00:22:45.920 --> 00:22:50.905
to store heat and energy
into these molecules.
00:22:50.905 --> 00:22:52.990
So what is happening?
00:22:58.370 --> 00:23:03.020
Well, by 1905,
Planck Had already
00:23:03.020 --> 00:23:09.010
proposed that there is some
underlying quantization
00:23:09.010 --> 00:23:13.770
for heat that you have
in the black body case.
00:23:13.770 --> 00:23:18.250
And in 1905, Einstein
said, well, maybe we
00:23:18.250 --> 00:23:22.000
should think about the
vibrational degrees
00:23:22.000 --> 00:23:26.360
of the molecule also as
being similarly quantized.
00:23:26.360 --> 00:23:32.450
So quantize vibrations.
00:23:36.950 --> 00:23:39.880
It's totally a
phenomenological statement.
00:23:39.880 --> 00:23:42.290
We have to justify it later.
00:23:42.290 --> 00:23:52.120
But the statement is that for
the case where classically we
00:23:52.120 --> 00:23:53.720
had a harmonic oscillator.
00:23:53.720 --> 00:23:55.180
And let's say in
this case we would
00:23:55.180 --> 00:23:58.000
have said that
its energy depends
00:23:58.000 --> 00:24:03.590
on its momentum and its
position or displacement--
00:24:03.590 --> 00:24:11.510
I guess I called it u--
through a formula such as this.
00:24:14.270 --> 00:24:18.240
Certainly, you can pick
lots of values of u and p
00:24:18.240 --> 00:24:21.430
that are compatible with
any value of the energy
00:24:21.430 --> 00:24:24.010
that you choose.
00:24:24.010 --> 00:24:27.720
But to get the
black body spectrum
00:24:27.720 --> 00:24:33.740
to work, Planck had proposed
that really what you should do
00:24:33.740 --> 00:24:37.760
is rather than thinking of
this harmonic oscillator
00:24:37.760 --> 00:24:42.180
as being able to take all
possible values, that somehow
00:24:42.180 --> 00:24:45.710
the values of energy that
it can take are quantized.
00:24:45.710 --> 00:24:48.180
And furthermore, he
had proposed that they
00:24:48.180 --> 00:24:52.050
are proportional to
the frequency involved.
00:24:52.050 --> 00:24:53.930
And how did he guess that?
00:24:53.930 --> 00:24:55.830
Ultimately, it was
related to what
00:24:55.830 --> 00:24:58.260
I said about black
body radiation.
00:24:58.260 --> 00:25:01.880
That as you heat
up the body, you
00:25:01.880 --> 00:25:04.720
will find that there's
a light that comes out
00:25:04.720 --> 00:25:07.190
and the frequency of
that light is somehow
00:25:07.190 --> 00:25:10.140
related to temperature
and nothing else.
00:25:10.140 --> 00:25:11.950
And based on that,
he had proposed
00:25:11.950 --> 00:25:16.715
that frequencies should come
up in certain packages that
00:25:16.715 --> 00:25:20.200
are proportional
to-- the energies
00:25:20.200 --> 00:25:23.470
of the particular frequencies
should come in packages that
00:25:23.470 --> 00:25:26.210
are proportional
to that frequency.
00:25:26.210 --> 00:25:30.100
So there is an integer
here n that tells you
00:25:30.100 --> 00:25:33.930
about the number
of these packets.
00:25:33.930 --> 00:25:39.220
And not that it really matters
for what we are doing now,
00:25:39.220 --> 00:25:42.780
but just to be consistent
with what we currently know
00:25:42.780 --> 00:25:46.830
with quantum mechanics,
let me add the 0 point
00:25:46.830 --> 00:25:48.590
energy of the harmonic
oscillator here.
00:25:51.830 --> 00:25:55.950
So then, to calculate
the contribution
00:25:55.950 --> 00:26:00.270
of a system in which energy
is in quantized packages,
00:26:00.270 --> 00:26:02.240
you would say, OK,
I will calculate
00:26:02.240 --> 00:26:06.720
a Z1 for these
vibrational levels,
00:26:06.720 --> 00:26:10.740
assuming this
quantization of energy.
00:26:10.740 --> 00:26:14.460
And so that says that
the possible states
00:26:14.460 --> 00:26:18.510
of my harmonic
oscillator have energies
00:26:18.510 --> 00:26:24.210
that are in these units
h bar omega n plus 1/2.
00:26:24.210 --> 00:26:28.960
And if I still continue to
believe statistical mechanics,
00:26:28.960 --> 00:26:33.620
I would say that
at a temperature t,
00:26:33.620 --> 00:26:36.860
the probability that I
will be in a state that
00:26:36.860 --> 00:26:40.550
is characterized by
integer n is e to the minus
00:26:40.550 --> 00:26:44.570
beta times the energy that
corresponds to that integer n.
00:26:44.570 --> 00:26:50.590
And then I can go and sum
over all possible energies
00:26:50.590 --> 00:26:52.925
and that would be
the normalization
00:26:52.925 --> 00:26:57.510
of the probability that
I'm in one of these states.
00:26:57.510 --> 00:27:04.195
So this is e to the minus
beta h bar omega over 2
00:27:04.195 --> 00:27:09.020
from the ground
state contribution.
00:27:09.020 --> 00:27:12.490
The rest of it is simply
a geometric series.
00:27:12.490 --> 00:27:18.820
Geometric series, we can sum
very easily to get 1 minus e
00:27:18.820 --> 00:27:21.780
to the minus beta h bar omega.
00:27:21.780 --> 00:27:28.820
And the interesting thing--
or a few interesting things
00:27:28.820 --> 00:27:31.570
about this expression
is that if I
00:27:31.570 --> 00:27:35.261
evaluate this in the
limit of low temperatures.
00:27:35.261 --> 00:27:37.510
Well, actually, let's go
first to the high temperature
00:27:37.510 --> 00:27:39.060
where beta goes to 0.
00:27:39.060 --> 00:27:42.940
So t goes to become
large, beta goes to 0.
00:27:42.940 --> 00:27:48.080
Numerator goes to 1, denominator
I can expand the exponential.
00:27:48.080 --> 00:27:52.000
And to lowest order, I will
get 1 over beta h bar omega.
00:27:56.320 --> 00:28:03.960
Now, compare this result
with the classical result
00:28:03.960 --> 00:28:07.330
that we have over here
for the vibration.
00:28:07.330 --> 00:28:10.440
Contribution of a harmonic
oscillator to the partition
00:28:10.440 --> 00:28:11.680
function.
00:28:11.680 --> 00:28:16.880
You can see that
the mu's cancel out.
00:28:16.880 --> 00:28:21.280
I will get 1 over beta.
00:28:21.280 --> 00:28:24.950
I will get h divided by 2 pi.
00:28:24.950 --> 00:28:29.090
So if I call h divided
by 2 pi to be h bar,
00:28:29.090 --> 00:28:36.370
then I will get
exactly this limit.
00:28:36.370 --> 00:28:43.250
So somehow this constant
that we had introduced
00:28:43.250 --> 00:28:46.420
that had dimensions
of action made
00:28:46.420 --> 00:28:50.960
to make our calculations
of partition function
00:28:50.960 --> 00:28:54.870
to be dimensionless
will be related
00:28:54.870 --> 00:28:58.220
to this h bar that
quantizes the energy
00:28:58.220 --> 00:29:01.480
levels through the
usual formula of h being
00:29:01.480 --> 00:29:05.640
h bar-- h bar being h over 2 pi.
00:29:05.640 --> 00:29:10.920
So basically, this
quantization of energy
00:29:10.920 --> 00:29:14.310
clearly does not affect
the high temperature limit.
00:29:14.310 --> 00:29:16.400
This oscillator at
high temperature
00:29:16.400 --> 00:29:21.920
behaves exactly like what we
had calculated classically.
00:29:21.920 --> 00:29:24.670
Yes?
00:29:24.670 --> 00:29:26.670
AUDIENCE: Is it h
equals h bar over 2 pi?
00:29:26.670 --> 00:29:30.850
Or is it the other way, based
on your definitions above?
00:29:30.850 --> 00:29:31.834
PROFESSOR: Thank you.
00:29:38.071 --> 00:29:38.570
Good.
00:29:44.170 --> 00:29:45.710
All right?
00:29:45.710 --> 00:29:48.740
So this is, I guess, the
corresponding formula.
00:29:52.150 --> 00:29:54.830
Now, when you go to low
temperature, what do you get?
00:29:54.830 --> 00:30:02.130
You essentially get the first
few terms in the series.
00:30:02.130 --> 00:30:05.710
Because at the
lowest temperature
00:30:05.710 --> 00:30:10.460
you get the term that
corresponds to n equals to 0,
00:30:10.460 --> 00:30:12.850
and then you will
get corrections
00:30:12.850 --> 00:30:13.910
from subsequent terms.
00:30:21.090 --> 00:30:28.350
Now, what this does is that
it affects the heat capacity
00:30:28.350 --> 00:30:29.020
profoundly.
00:30:29.020 --> 00:30:31.180
So let's see how that happens.
00:30:31.180 --> 00:30:36.330
So the contribution of 1
degrees of freedom to the energy
00:30:36.330 --> 00:30:44.120
in this quantized fashion
d log Z by d beta.
00:30:44.120 --> 00:30:48.700
So if I just take the
log of this expression,
00:30:48.700 --> 00:30:52.920
log of this expression will get
this factor of minus beta h bar
00:30:52.920 --> 00:30:55.530
omega over 2 from the numerator.
00:30:55.530 --> 00:30:58.590
The derivative of that will
give you this ground state
00:30:58.590 --> 00:31:01.880
energy, which is always there.
00:31:01.880 --> 00:31:05.570
And then you'll have
to take the derivative
00:31:05.570 --> 00:31:08.370
of the log of what
is coming out here.
00:31:08.370 --> 00:31:11.730
Taking a derivative
with respect to beta,
00:31:11.730 --> 00:31:16.880
we'll always pick out a
factor of h bar omega.
00:31:16.880 --> 00:31:19.980
Indeed, it will pick out
a factor of h bar omega
00:31:19.980 --> 00:31:23.160
e to the minus beta h bar omega.
00:31:23.160 --> 00:31:26.880
And then in the denominator,
because I took the log,
00:31:26.880 --> 00:31:28.595
I will get this expression back.
00:31:36.720 --> 00:31:40.740
So again, in this
expression, if I
00:31:40.740 --> 00:31:50.870
take the limit where beta
goes to 0, what do I get?
00:31:50.870 --> 00:31:53.370
I will get this h
bar omega over 2.
00:31:53.370 --> 00:31:54.990
It's always there.
00:31:54.990 --> 00:31:59.750
Expanding these results here,
I will have a beta h bar omega.
00:31:59.750 --> 00:32:03.610
It will cancel this and it
will give me a 1 over beta.
00:32:03.610 --> 00:32:07.830
I will get this kT
that I had before.
00:32:07.830 --> 00:32:12.440
Indeed, if I am correct
to the right order,
00:32:12.440 --> 00:32:16.810
I will just simply
get 1 over beta.
00:32:16.810 --> 00:32:24.470
Whereas, if I go to
large beta, what I get
00:32:24.470 --> 00:32:28.710
is this h bar omega
over 2 plus a correction
00:32:28.710 --> 00:32:34.446
from here, which is h bar omega
e to the minus beta h bar.
00:32:39.640 --> 00:32:47.790
And that will be reflected
in the heat capacity, which
00:32:47.790 --> 00:32:49.876
is dE by dT.
00:32:52.690 --> 00:32:56.760
This h bar omega over 2 does
not continue to heat capacity,
00:32:56.760 --> 00:32:59.210
not surprisingly.
00:32:59.210 --> 00:33:03.620
From here, I have to take
derivatives with temperatures.
00:33:03.620 --> 00:33:07.930
They appear in the combination
h bar omega over kT.
00:33:07.930 --> 00:33:10.000
So what happens is
I will get something
00:33:10.000 --> 00:33:12.990
that is of the order
of h bar omega.
00:33:12.990 --> 00:33:15.500
And then from here, I
will get another h bar
00:33:15.500 --> 00:33:20.290
omega divided by kb T squared.
00:33:20.290 --> 00:33:25.900
I will write it in this fashion
and put the kb out here.
00:33:25.900 --> 00:33:29.510
And then the rest
of these objects
00:33:29.510 --> 00:33:32.270
will give me a
contribution that is minus
00:33:32.270 --> 00:33:37.800
h bar omega over kT
divided by 1 minus e
00:33:37.800 --> 00:33:42.760
to the minus h bar
omega over kT squared.
00:33:47.140 --> 00:33:49.260
The important thing
is the following--
00:33:52.110 --> 00:33:54.460
If I plot the heat
capacity that I
00:33:54.460 --> 00:33:57.760
get from one of
these oscillators--
00:33:57.760 --> 00:34:00.480
and the natural
units of all heat
00:34:00.480 --> 00:34:03.770
capacities are kb, essentially.
00:34:03.770 --> 00:34:08.860
Energy divided by temperature,
as kb has that units.
00:34:08.860 --> 00:34:12.969
At high temperatures,
what I can see
00:34:12.969 --> 00:34:17.360
is that the energy is
proportional to kT.
00:34:17.360 --> 00:34:20.500
So heat capacity of the
vibrational degree of freedom
00:34:20.500 --> 00:34:25.870
will be in these
units going to 1.
00:34:25.870 --> 00:34:30.920
At low temperatures, however, it
becomes this exponentially hard
00:34:30.920 --> 00:34:35.210
problem to create excitations.
00:34:35.210 --> 00:34:38.280
Because of that, you
will get a contribution
00:34:38.280 --> 00:34:43.130
that as T goes to 0 will
exponentially go to 0.
00:34:43.130 --> 00:34:47.160
So the shape of the heat
capacity that you would get
00:34:47.160 --> 00:34:49.098
will be something like this.
00:34:53.662 --> 00:35:01.450
The natural way to
draw this figure
00:35:01.450 --> 00:35:05.500
is actually what I
made the vertical axis
00:35:05.500 --> 00:35:06.650
to be dimensionless.
00:35:06.650 --> 00:35:08.970
So it goes between 0 and 1.
00:35:08.970 --> 00:35:11.310
I can make the
horizontal axis to be
00:35:11.310 --> 00:35:16.180
dimensionless by introducing
a theta of vibrations,
00:35:16.180 --> 00:35:21.340
so that all of the exponential
terms are of the form e
00:35:21.340 --> 00:35:25.970
to the minus T over this
theta of vibrations, which
00:35:25.970 --> 00:35:29.750
means that this
theta of vibration
00:35:29.750 --> 00:35:33.130
is h bar omega over kb.
00:35:33.130 --> 00:35:36.260
That is, you tell me what the
frequency of your oscillator
00:35:36.260 --> 00:35:37.260
is.
00:35:37.260 --> 00:35:40.270
I can calculate the
corresponding temperature,
00:35:40.270 --> 00:35:41.410
theta.
00:35:41.410 --> 00:35:45.490
And then the heat capacity
of a harmonic oscillator
00:35:45.490 --> 00:35:49.140
is this universal
function there, presumably
00:35:49.140 --> 00:35:53.160
at some value that
is of the order of 1.
00:35:53.160 --> 00:35:55.980
It switches from
being of the order
00:35:55.980 --> 00:35:58.810
of 1 to going
exponentially to 0.
00:35:58.810 --> 00:36:03.060
So basically, the dependence
down here to leading order
00:36:03.060 --> 00:36:05.404
is e to the minus T
over theta vibration.
00:36:10.250 --> 00:36:10.750
OK.
00:36:14.020 --> 00:36:17.750
So you say, OK,
Planck has given us
00:36:17.750 --> 00:36:20.720
some estimate of what
this h bar is based
00:36:20.720 --> 00:36:25.250
on looking at the spectrum
of black body radiation.
00:36:25.250 --> 00:36:29.450
We can, more or, less
estimate the typical energies
00:36:29.450 --> 00:36:31.760
of interactions of molecules.
00:36:31.760 --> 00:36:35.120
And from that, we
can estimate what
00:36:35.120 --> 00:36:37.960
this frequency of vibration is.
00:36:37.960 --> 00:36:40.720
So we should be able to
get an order of magnitude
00:36:40.720 --> 00:36:43.380
estimate of what
this theta y is.
00:36:43.380 --> 00:36:47.350
And what you find is that
theta y is of the order of 10
00:36:47.350 --> 00:36:50.014
to the 3 degrees Kelvin.
00:36:50.014 --> 00:36:52.910
It depends, of course, on
what gas you are looking at,
00:36:52.910 --> 00:36:53.830
et cetera.
00:36:53.830 --> 00:36:59.140
But as an order of magnitude,
it is something like that.
00:36:59.140 --> 00:37:03.540
So we can now transport this
curve that we have over here
00:37:03.540 --> 00:37:06.680
and more or less get this
first part of the curve
00:37:06.680 --> 00:37:08.280
that we have over here.
00:37:08.280 --> 00:37:11.910
So essentially, in this
picture what we have
00:37:11.910 --> 00:37:14.760
is that there is no vibrations.
00:37:14.760 --> 00:37:18.212
The vibrations have
been frozen out.
00:37:18.212 --> 00:37:19.960
And here you have vibrations.
00:37:26.480 --> 00:37:28.220
Of course, in all
of the cases, you
00:37:28.220 --> 00:37:30.640
have the kinetic energy
of the center of mass.
00:37:36.450 --> 00:37:42.100
And presumably since we are
getting the right answer
00:37:42.100 --> 00:37:45.790
at very high temperatures now,
we also have the rotations.
00:37:51.150 --> 00:37:52.910
And it makes sense
that essentially
00:37:52.910 --> 00:37:59.420
what happened as we go
to very low temperatures
00:37:59.420 --> 00:38:02.130
is that the rotations
are also frozen out.
00:38:08.590 --> 00:38:11.750
Now, that's part of
the story-- actually,
00:38:11.750 --> 00:38:15.770
you would think that among all
of the examples that I gave
00:38:15.770 --> 00:38:20.010
you, this last one should be
the simplest thing because it's
00:38:20.010 --> 00:38:21.810
really a two-body problem.
00:38:21.810 --> 00:38:23.730
Whereas, solids you
have many things.
00:38:23.730 --> 00:38:25.470
Radiation, you
have to think about
00:38:25.470 --> 00:38:27.930
the electromagnetic
waves, et cetera.
00:38:27.930 --> 00:38:29.840
That somehow,
historically, this would
00:38:29.840 --> 00:38:33.080
be the one that
is resolved first.
00:38:33.080 --> 00:38:36.800
And indeed, as I said
in 1905, Einstein
00:38:36.800 --> 00:38:38.950
figured out
something about this.
00:38:38.950 --> 00:38:43.780
But this part dealing with the
rotational degrees of freedom
00:38:43.780 --> 00:38:46.170
and quantizing
them appropriately
00:38:46.170 --> 00:38:50.420
had to really wait until you
had developed quantum mechanics
00:38:50.420 --> 00:38:53.300
beyond the statement
that harmonic oscillators
00:38:53.300 --> 00:38:54.830
are quantized in energy.
00:38:54.830 --> 00:38:57.440
You had to know something more.
00:38:57.440 --> 00:39:02.000
So since in retrospect we
do know something more,
00:39:02.000 --> 00:39:04.410
let's finish and
give that answer
00:39:04.410 --> 00:39:09.180
before going on
to something else.
00:39:09.180 --> 00:39:10.970
OK?
00:39:10.970 --> 00:39:15.820
So the next part of the
story of the diatomic gas
00:39:15.820 --> 00:39:19.630
is quantizing rotations.
00:39:24.630 --> 00:39:32.150
So currently what I have is that
there is an energy classically
00:39:32.150 --> 00:39:38.960
for rotations that is
simply the kinetic energy
00:39:38.960 --> 00:39:41.680
of rotational
degrees of freedom.
00:39:41.680 --> 00:39:43.780
So there is an
angular momentum L,
00:39:43.780 --> 00:39:46.150
and then there's
L squared over 2I.
00:39:46.150 --> 00:39:49.880
It looks pretty much
like P squared over 2M,
00:39:49.880 --> 00:39:54.360
except that the
degrees of freedom
00:39:54.360 --> 00:39:57.950
for translation and
motion are positions.
00:39:57.950 --> 00:40:00.030
They can be all over the place.
00:40:00.030 --> 00:40:01.750
Whereas, the degrees
of freedom that you
00:40:01.750 --> 00:40:04.000
have to think in
terms of rotations
00:40:04.000 --> 00:40:06.830
are angles that go
between 0, 2 pi,
00:40:06.830 --> 00:40:11.190
or on the surface of
a sphere, et cetera.
00:40:11.190 --> 00:40:16.810
So once we figure out how
to do quantum mechanics,
00:40:16.810 --> 00:40:22.110
we find that the
allowed values of this
00:40:22.110 --> 00:40:29.270
are of the form h bar squared
over 2I l, l plus 1, where
00:40:29.270 --> 00:40:36.290
l now is the number that gives
you the discrete values that
00:40:36.290 --> 00:40:41.070
are possible for the square
of the angular momentum.
00:40:41.070 --> 00:40:45.690
So you say OK, let's calculate
a Z for the rotational degrees
00:40:45.690 --> 00:40:50.110
of freedom assuming this
kind of quantization.
00:40:50.110 --> 00:40:54.160
So what I have to do, like I
did for the harmonic oscillator,
00:40:54.160 --> 00:40:58.890
is I sum over all possible
values of l that are allowed.
00:40:58.890 --> 00:41:04.220
The energy e to the minus
beta h bar squared over 2I
00:41:04.220 --> 00:41:05.340
l, l plus 1.
00:41:07.860 --> 00:41:10.500
Except that there
is one other thing,
00:41:10.500 --> 00:41:13.910
which is that these
different values of l
00:41:13.910 --> 00:41:19.370
have degeneracy
that is 2l plus 1.
00:41:19.370 --> 00:41:26.290
And so you have to multiply by
the corresponding degeneracy.
00:41:30.820 --> 00:41:34.280
So what am I doing over here?
00:41:34.280 --> 00:41:41.110
I have to do a sum over
different values of l,
00:41:41.110 --> 00:41:45.410
contributions that are
really the probability that I
00:41:45.410 --> 00:41:51.329
am in these different values of
the index l-- 0, 1, 2, 3, 4, 5,
00:41:51.329 --> 00:41:51.828
6.
00:41:54.790 --> 00:41:58.970
And I have to add all
of these contributions.
00:42:02.570 --> 00:42:05.660
Now, the first
thing that I will do
00:42:05.660 --> 00:42:09.320
is I ask whether the limit
of high temperatures that I
00:42:09.320 --> 00:42:13.860
had calculated before is
correctly reproduced or not.
00:42:13.860 --> 00:42:17.090
So I have to go to the limit
where temperature is high
00:42:17.090 --> 00:42:19.340
or beta goes to 0.
00:42:19.340 --> 00:42:22.310
If beta goes to 0,
you can see that going
00:42:22.310 --> 00:42:25.150
from one l to another
l, it is multiply
00:42:25.150 --> 00:42:28.050
this exponent by a small number.
00:42:28.050 --> 00:42:29.290
So what does that mean?
00:42:29.290 --> 00:42:32.730
It means that the values from
one point to another point
00:42:32.730 --> 00:42:37.350
of what am I summing over is
not really that different.
00:42:37.350 --> 00:42:41.020
And I can think of a
continuous curve that
00:42:41.020 --> 00:42:44.090
goes through all
of these points.
00:42:44.090 --> 00:42:46.450
So if I do that, then
I can essentially
00:42:46.450 --> 00:42:48.240
replace the sum
with an integral.
00:43:00.970 --> 00:43:04.580
In fact, you can systematically
calculate corrections
00:43:04.580 --> 00:43:08.330
to replacing the sum with
an integral mathematically
00:43:08.330 --> 00:43:13.590
and you have a problem set
that shows you how to do that.
00:43:13.590 --> 00:43:18.580
But now what I can do is I
can call this combination l, l
00:43:18.580 --> 00:43:21.330
plus 1 x.
00:43:21.330 --> 00:43:26.940
And then dx will
simply be 2l plus 1 dl.
00:43:29.550 --> 00:43:35.630
So essentially, the
degeneracy works out precisely
00:43:35.630 --> 00:43:39.110
so that when I go to the
continuum limit, whatever
00:43:39.110 --> 00:43:43.120
quantization I had for
these angular momenta
00:43:43.120 --> 00:43:45.110
corresponds to the
weight or measure
00:43:45.110 --> 00:43:48.770
that I would have in stepping
around the l-directions.
00:43:48.770 --> 00:43:53.320
And then, this is something
that I can easily do.
00:43:53.320 --> 00:43:57.280
It's just an integral dx
e to the minus alpha x.
00:43:57.280 --> 00:44:01.830
The answer is going to be 1 over
alpha, or the answer to this
00:44:01.830 --> 00:44:06.785
is simply 2I beta h bar squared.
00:44:14.680 --> 00:44:18.130
So this is the classical
limit of the expression
00:44:18.130 --> 00:44:20.260
that we had over here.
00:44:20.260 --> 00:44:23.465
Let's go and see what we had
when we did things classically.
00:44:27.570 --> 00:44:29.820
So when we did
things classically,
00:44:29.820 --> 00:44:35.890
I had two factors of
h and 2 pi and 4 pi.
00:44:35.890 --> 00:44:40.100
So I can write the whole
thing as h bar squared and 2.
00:44:40.100 --> 00:44:43.902
I have I, and then I have beta.
00:44:43.902 --> 00:44:48.360
And you can see that
this is exactly what we
00:44:48.360 --> 00:44:49.610
have over there.
00:44:49.610 --> 00:44:55.920
So once more, properly
accounting for phase space,
00:44:55.920 --> 00:45:01.150
measure, productive p
q's being dimension--
00:45:01.150 --> 00:45:05.400
made dimensionless
by this quantity h
00:45:05.400 --> 00:45:10.080
is equivalent to the
high temperature limit
00:45:10.080 --> 00:45:11.990
that you would get
in quantum mechanics
00:45:11.990 --> 00:45:13.230
where things are discretized.
00:45:13.230 --> 00:45:15.030
Yes.
00:45:15.030 --> 00:45:17.832
AUDIENCE: When you're
talking about the quantum
00:45:17.832 --> 00:45:20.686
interpretations, then h
bar is the precise value
00:45:20.686 --> 00:45:23.150
of Planck's constant, which
can be an experimental measure.
00:45:23.150 --> 00:45:23.858
PROFESSOR: Right.
00:45:23.858 --> 00:45:25.260
AUDIENCE: But when
you're talking
00:45:25.260 --> 00:45:29.082
about the classical derivations,
h is just some factor
00:45:29.082 --> 00:45:30.540
that we mention of
curve dimension.
00:45:30.540 --> 00:45:32.026
PROFESSOR: That's correct.
00:45:32.026 --> 00:45:35.802
AUDIENCE: So if you're
comparing the limits
00:45:35.802 --> 00:45:40.710
of large temperatures,
how can you
00:45:40.710 --> 00:45:45.263
be sure to establish the
h bar in two places means
00:45:45.263 --> 00:45:46.555
the same thing?
00:45:46.555 --> 00:45:47.930
PROFESSOR: So far,
I haven't told
00:45:47.930 --> 00:45:50.390
you anything to justify that.
00:45:50.390 --> 00:45:52.760
So when we were doing
things classically,
00:45:52.760 --> 00:45:55.870
we said that just to make
things dimensionless,
00:45:55.870 --> 00:46:00.380
let's introduce this
quantity that we call h.
00:46:00.380 --> 00:46:03.800
Now, I have shown you
two examples where
00:46:03.800 --> 00:46:07.710
if you do things quantum
mechanically properly
00:46:07.710 --> 00:46:10.930
and take the limit of
going to high temperatures,
00:46:10.930 --> 00:46:14.060
you will see that the
h that you would get--
00:46:14.060 --> 00:46:16.660
because the quantum
mechanical partition functions
00:46:16.660 --> 00:46:19.500
are dimensionless
quantities, right?
00:46:19.500 --> 00:46:21.650
So these are
dimensionless quantities.
00:46:21.650 --> 00:46:24.140
They have to be made
dimensionless by something.
00:46:24.140 --> 00:46:27.530
They're made dimensionless
by Boltzmann's constant.
00:46:27.530 --> 00:46:30.880
By a Planck's constant, h bar.
00:46:30.880 --> 00:46:33.270
And we can see
that as long as we
00:46:33.270 --> 00:46:36.330
are consistent with this
measure of phase space,
00:46:36.330 --> 00:46:41.980
the same constant shows up both
for the case of the vibrations,
00:46:41.980 --> 00:46:46.610
for the case of the rotations.
00:46:46.610 --> 00:46:49.100
And very soon, we will
see that it will also
00:46:49.100 --> 00:46:51.780
arise in the case of
the center of mass.
00:46:51.780 --> 00:47:00.010
And so there is certainly
something in the transcriptions
00:47:00.010 --> 00:47:04.130
that we ultimately will make
between quantum mechanics
00:47:04.130 --> 00:47:08.080
and classical mechanics
that must account for this.
00:47:08.080 --> 00:47:12.640
And somehow in the limit where
quantum mechanics is dealing
00:47:12.640 --> 00:47:16.510
with large energies,
it is indistinguishable
00:47:16.510 --> 00:47:19.160
from classical mechanics.
00:47:19.160 --> 00:47:23.840
And quantum partition
functions are--
00:47:23.840 --> 00:47:26.990
all of the countings that we do
in quantum mechanics are kind
00:47:26.990 --> 00:47:30.880
of unambiguous because we are
dealing with discrete levels.
00:47:30.880 --> 00:47:34.960
So if you remember the
original part of the difficulty
00:47:34.960 --> 00:47:39.150
was that we could define
things like entropy only
00:47:39.150 --> 00:47:41.820
properly when we
had discrete levels.
00:47:41.820 --> 00:47:44.380
If we had a continuum
probability distribution
00:47:44.380 --> 00:47:46.650
and if we made a
change of variable,
00:47:46.650 --> 00:47:49.240
then the entropy was changed.
00:47:49.240 --> 00:47:52.290
But in quantum mechanics,
we don't have that problem.
00:47:52.290 --> 00:47:57.080
We have discretized values
for the different states.
00:47:57.080 --> 00:48:00.910
Probabilities will be-- once
we deal with them appropriately
00:48:00.910 --> 00:48:02.640
be discretized.
00:48:02.640 --> 00:48:06.190
And all of the things
here are dimensionless.
00:48:06.190 --> 00:48:11.490
And somehow they reproduce the
correct classical dynamics.
00:48:11.490 --> 00:48:13.830
Quantum mechanics goes
to classical mechanics
00:48:13.830 --> 00:48:16.090
in the appropriate
high-energy limit.
00:48:16.090 --> 00:48:22.440
And what we find is that what
happens is that this shows up.
00:48:22.440 --> 00:48:28.720
If you like, another
way of achieving--
00:48:28.720 --> 00:48:32.630
why is there this
correspondence?
00:48:32.630 --> 00:48:36.920
In classical
statistical mechanics,
00:48:36.920 --> 00:48:43.390
I emphasize that I should really
write h in units of p and q.
00:48:43.390 --> 00:48:47.870
And it was only when I
calculated partition functions
00:48:47.870 --> 00:48:51.880
in coordinates p and q that were
canonically conjugate that I
00:48:51.880 --> 00:48:54.350
was getting results
that were meaningful.
00:48:58.260 --> 00:49:01.600
One way of constructing
quantum mechanics
00:49:01.600 --> 00:49:04.720
is that you take the
Hamiltonian and you change these
00:49:04.720 --> 00:49:06.200
into operators.
00:49:06.200 --> 00:49:14.950
And you have to impose these
kinds of commutation relations.
00:49:14.950 --> 00:49:19.480
So you can see that somehow
the same prescription in terms
00:49:19.480 --> 00:49:23.780
of phase space appears both
in statistical mechanics,
00:49:23.780 --> 00:49:26.340
in calculating measures
of partition function,
00:49:26.340 --> 00:49:27.950
in quantum mechanics.
00:49:27.950 --> 00:49:31.710
And not surprisingly, you have
introduced in quantum mechanics
00:49:31.710 --> 00:49:35.260
some unit for phase space p, q.
00:49:35.260 --> 00:49:37.810
It shows up in
classical mechanics
00:49:37.810 --> 00:49:39.020
as the quantity [INAUDIBLE].
00:49:43.220 --> 00:49:46.200
But there is, indeed,
a little bit more work
00:49:46.200 --> 00:49:49.100
than I have shown you
here that one can do.
00:49:49.100 --> 00:49:53.650
Once we have developed
the appropriate formalism
00:49:53.650 --> 00:49:56.900
for quantum statistical
mechanics, which
00:49:56.900 --> 00:50:01.230
is this [INAUDIBLE] performed
and appropriate quantities
00:50:01.230 --> 00:50:04.500
defined for partition
functions, et cetera,
00:50:04.500 --> 00:50:06.780
in quantum statistical
mechanics that we
00:50:06.780 --> 00:50:08.620
will do in a couple of lectures.
00:50:08.620 --> 00:50:11.260
Then if you take the
limit h bar goes to 0,
00:50:11.260 --> 00:50:15.100
you should get the classical
integration over phase space
00:50:15.100 --> 00:50:18.412
with this factor
of h showing up.
00:50:23.290 --> 00:50:25.400
But right now, we
are just giving you
00:50:25.400 --> 00:50:29.910
some heuristic response.
00:50:29.910 --> 00:50:32.610
If I go, however, in
the other limit, where
00:50:32.610 --> 00:50:40.370
beta is much larger
than 1, what do I get?
00:50:40.370 --> 00:50:43.210
Basically, then
all of the weight
00:50:43.210 --> 00:50:48.560
is going to be in the
lowest energy level, 0, 1.
00:50:48.560 --> 00:50:52.400
And then the rest of them
will be exponentially small.
00:50:52.400 --> 00:50:55.840
I cannot replace the
sum with an integral,
00:50:55.840 --> 00:51:00.380
so basically I will get a
contribution that starts with 1
00:51:00.380 --> 00:51:03.140
for l equals to 0.
00:51:03.140 --> 00:51:10.000
And then I will get
3e to the minus beta h
00:51:10.000 --> 00:51:13.630
bar squared divided by 2I.
00:51:13.630 --> 00:51:17.770
l being 1, this will
give me 1 times 2.
00:51:17.770 --> 00:51:20.760
So I will have a 2 here.
00:51:20.760 --> 00:51:22.220
And then, higher-order terms.
00:51:28.360 --> 00:51:35.120
So once you have the
partition function,
00:51:35.120 --> 00:51:36.730
you go through
the same procedure
00:51:36.730 --> 00:51:39.230
as we described before.
00:51:39.230 --> 00:51:45.440
You calculate the energy,
which is d log Z by d beta.
00:51:48.460 --> 00:51:51.230
What do you get?
00:51:51.230 --> 00:51:53.300
Again, in the high
temperature limit
00:51:53.300 --> 00:51:55.610
you will get the same
answer as before.
00:51:55.610 --> 00:51:59.390
So you will get beta goes to 0.
00:51:59.390 --> 00:52:01.320
You will get kT.
00:52:04.100 --> 00:52:06.770
If you go to the low
temperature limit-- well,
00:52:06.770 --> 00:52:08.440
let's be more precise.
00:52:08.440 --> 00:52:10.390
What do I mean by
low temperatures?
00:52:10.390 --> 00:52:13.740
Beta larger than what?
00:52:13.740 --> 00:52:17.050
Clearly, the unit that
is appearing everywhere
00:52:17.050 --> 00:52:26.690
is this beta h bar squared
over 2I, which has units of 1
00:52:26.690 --> 00:52:30.200
over temperature from beta.
00:52:30.200 --> 00:52:34.220
So I can introduce a
theta for rotations
00:52:34.220 --> 00:52:37.920
to make this demonstrate
that this is dimensionless.
00:52:37.920 --> 00:52:41.120
So the theta that
goes with rotations
00:52:41.120 --> 00:52:48.650
is h bar squared over 2I kb.
00:52:48.650 --> 00:52:53.630
And so what I mean by going
to the low temperatures
00:52:53.630 --> 00:52:55.610
is that I go for
temperatures that
00:52:55.610 --> 00:53:00.910
are much less than the
theta of these rotations.
00:53:00.910 --> 00:53:06.780
And then what happens is
that essentially this state
00:53:06.780 --> 00:53:10.590
will occur with exponentially
small probability
00:53:10.590 --> 00:53:14.320
and will contribute to
the energy and amount that
00:53:14.320 --> 00:53:19.490
is of the order of h
bar squared 2I times 2.
00:53:19.490 --> 00:53:22.780
That's the energy of
the l equals to 1 state.
00:53:22.780 --> 00:53:26.570
There are three of them, and
they occur with probability e
00:53:26.570 --> 00:53:31.400
to the minus theta
rotation divided
00:53:31.400 --> 00:53:34.118
by T times a factor of 2.
00:53:37.870 --> 00:53:40.870
All of those factors is
not particularly important.
00:53:40.870 --> 00:53:45.100
Really, the only thing
that is important
00:53:45.100 --> 00:53:50.600
is that if I look now at the
rotational heat capacity, which
00:53:50.600 --> 00:53:55.160
again should properly
have units of kb,
00:53:55.160 --> 00:53:56.770
as a function of temperature.
00:53:56.770 --> 00:53:59.320
Well, temperatures
I have to make
00:53:59.320 --> 00:54:04.380
dimensionless by dividing by
this rotational heat capacity.
00:54:04.380 --> 00:54:06.280
I say that at
high, temperature I
00:54:06.280 --> 00:54:08.810
get the classical result back.
00:54:08.810 --> 00:54:15.160
So basically, I will get
to 1 at high temperatures.
00:54:15.160 --> 00:54:19.070
At low temperatures, again
I have this situation
00:54:19.070 --> 00:54:22.830
that there is a gap in
the allowed energies.
00:54:22.830 --> 00:54:27.160
So there is the lowest
energy, which is 0.
00:54:27.160 --> 00:54:31.490
The next one, the first type of
rotational mode that is allowed
00:54:31.490 --> 00:54:34.100
has a finite energy
that is larger than that
00:54:34.100 --> 00:54:37.960
by an amount that is of the
order of h bar squared over I.
00:54:37.960 --> 00:54:40.120
And if I am at these
temperatures that
00:54:40.120 --> 00:54:42.840
are less than this
theta of rotation,
00:54:42.840 --> 00:54:45.870
I simply don't
have enough energy
00:54:45.870 --> 00:54:50.660
from thermal fluctuations
to get to that level.
00:54:50.660 --> 00:54:54.820
So the occupation of that level
will be exponentially small.
00:54:54.820 --> 00:54:57.940
And so I will have a curve that
will, in fact, look something
00:54:57.940 --> 00:54:58.912
like this.
00:55:04.270 --> 00:55:09.270
So again, you basically
go over at a temperature
00:55:09.270 --> 00:55:12.500
of the order of 1
from heat capacity
00:55:12.500 --> 00:55:17.000
that is order of 1 to heat
capacity that is exponentially
00:55:17.000 --> 00:55:24.740
small when you get to
temperatures that are lower
00:55:24.740 --> 00:55:28.740
than this rotational
temperature.
00:55:28.740 --> 00:55:30.680
AUDIENCE: Is that
over-shooting, or is that--
00:55:30.680 --> 00:55:31.780
PROFESSOR: Yes.
00:55:31.780 --> 00:55:35.260
So you have a
problem set where you
00:55:35.260 --> 00:55:37.700
calculate the next correction.
00:55:37.700 --> 00:55:41.100
So there is the summation
replacing the sum
00:55:41.100 --> 00:55:43.290
with an integral.
00:55:43.290 --> 00:55:45.850
This gives you this
to the first order,
00:55:45.850 --> 00:55:48.040
and then there's a correction.
00:55:48.040 --> 00:55:51.440
And you will show that
the correction is such
00:55:51.440 --> 00:55:54.790
that there is actually
the approach to one
00:55:54.790 --> 00:55:59.250
for the case of the rotational
heat capacity is from above.
00:55:59.250 --> 00:56:01.940
Whereas, for the
vibrational heat capacity,
00:56:01.940 --> 00:56:05.000
it is from below.
00:56:05.000 --> 00:56:07.308
So there is, indeed,
a small bump.
00:56:12.676 --> 00:56:15.610
OK?
00:56:15.610 --> 00:56:24.050
So you can ask, well, I
know the typical size of one
00:56:24.050 --> 00:56:26.610
of these oxygen molecules.
00:56:26.610 --> 00:56:28.020
I know the mass.
00:56:28.020 --> 00:56:33.120
I can figure out what the
moment of inertia I is.
00:56:33.120 --> 00:56:37.100
I put it over here and I figure
out what the theta of rotation
00:56:37.100 --> 00:56:37.840
is.
00:56:37.840 --> 00:56:41.880
And you find that, again, as a
matter of order of magnitudes,
00:56:41.880 --> 00:56:46.220
theta of rotations is of
the order of 10 degrees
00:56:46.220 --> 00:56:56.120
K. So this kind of accounts for
why when you go to sufficiently
00:56:56.120 --> 00:56:58.820
low temperatures for the
heat capacity of the gas
00:56:58.820 --> 00:57:01.770
in this room, we
see that essentially
00:57:01.770 --> 00:57:05.220
the rotational degrees of
freedom are also frozen out.
00:57:13.558 --> 00:57:14.058
OK.
00:57:18.990 --> 00:57:25.370
So now let's go to the
second item that we have,
00:57:25.370 --> 00:57:27.780
which is the heat
capacity of the solid.
00:57:27.780 --> 00:57:28.855
So what do I mean?
00:57:31.920 --> 00:57:35.610
So this is item 2,
heat capacity of solid.
00:57:43.710 --> 00:57:51.020
And you measure heat
capacities for some solid
00:57:51.020 --> 00:57:53.880
as a function of temperature.
00:57:53.880 --> 00:57:57.430
And what you find is
that the heat capacity
00:57:57.430 --> 00:58:00.970
has a behavior such as this.
00:58:04.950 --> 00:58:07.810
So it seems to
vanish as you to go
00:58:07.810 --> 00:58:10.320
to lower and lower temperatures.
00:58:10.320 --> 00:58:11.610
So what's going on here?
00:58:15.690 --> 00:58:21.060
Again, Einstein looked at
this and said, well, it's
00:58:21.060 --> 00:58:25.270
another case of the story of
vibrations and some things
00:58:25.270 --> 00:58:28.370
that we have looked at here.
00:58:28.370 --> 00:58:32.490
And in fact, I really don't
have to do any calculation.
00:58:32.490 --> 00:58:34.340
I'll do the following.
00:58:34.340 --> 00:58:37.620
Let's imagine that this is
what we have for the solid.
00:58:37.620 --> 00:58:41.290
It's some regular arrangement
of atoms or molecules.
00:58:46.630 --> 00:58:50.910
And presumably, this
is the situation
00:58:50.910 --> 00:58:52.450
that I have at 0 temperature.
00:58:52.450 --> 00:58:55.770
Everybody is
sitting nicely where
00:58:55.770 --> 00:58:58.500
they should be to
minimize the energy.
00:58:58.500 --> 00:59:02.420
If I go to finite temperature,
then these atoms and molecules
00:59:02.420 --> 00:59:03.650
start to vibrate.
00:59:08.320 --> 00:59:13.480
And he said, well,
basically, I can
00:59:13.480 --> 00:59:17.912
estimate the frequencies
of vibrations.
00:59:17.912 --> 00:59:26.840
And what I will do is I
will say that each atom is
00:59:26.840 --> 00:59:31.430
in a cage by its neighbors.
00:59:35.740 --> 00:59:43.200
That is, this particular atom
here, if it wants to move,
00:59:43.200 --> 00:59:51.610
it find that its distance to
the neighbors has been changed.
00:59:51.610 --> 00:59:53.370
And if I imagine
that there are kind
00:59:53.370 --> 00:59:57.750
of springs that are
connecting this atom only
00:59:57.750 --> 01:00:01.630
to its neighbors,
moving around there
01:00:01.630 --> 01:00:05.610
will be some kind of
a restoring force.
01:00:05.610 --> 01:00:09.160
So it's like it is
sitting in some kind
01:00:09.160 --> 01:00:11.690
of a harmonic potential.
01:00:11.690 --> 01:00:15.820
And if it tries to
move, it will experience
01:00:15.820 --> 01:00:17.270
this restoring force.
01:00:17.270 --> 01:00:21.730
And so it will have some
kind of a frequency.
01:00:21.730 --> 01:00:30.520
So each atom vibrates
at some frequency.
01:00:35.499 --> 01:00:40.370
Let's call it omega E.
01:00:40.370 --> 01:00:44.210
Now, in principle,
in this picture
01:00:44.210 --> 01:00:47.050
if this cage is not
exactly symmetric,
01:00:47.050 --> 01:00:49.600
you may imagine that
oscillations in the three
01:00:49.600 --> 01:00:52.820
different directions could
give you different frequencies.
01:00:52.820 --> 01:00:56.140
But let's ignore that and let's
imagine that the frequencies is
01:00:56.140 --> 01:00:58.790
the same in all of these.
01:00:58.790 --> 01:01:00.190
So what have we done?
01:01:00.190 --> 01:01:04.220
We have reduced the problem
of the excitation energy
01:01:04.220 --> 01:01:08.340
that you can put in the
atoms of the solid to be
01:01:08.340 --> 01:01:21.330
the same now as 3N harmonic
oscillators of frequency omega.
01:01:24.150 --> 01:01:25.100
Why 3N?
01:01:25.100 --> 01:01:28.620
Because each atom
essentially sees restoring
01:01:28.620 --> 01:01:30.380
force in three directions.
01:01:30.380 --> 01:01:32.790
And forgetting about
boundary effects,
01:01:32.790 --> 01:01:35.240
it's basically
three per particle.
01:01:35.240 --> 01:01:39.270
So you would have said that
the heat capacity that I would
01:01:39.270 --> 01:01:47.200
calculate per particle
in units of kb
01:01:47.200 --> 01:01:52.770
should essentially be exactly
what we have over here,
01:01:52.770 --> 01:01:55.910
except that I multiply by 3
because each particular has
01:01:55.910 --> 01:01:57.830
3 possible degrees of freedom.
01:01:57.830 --> 01:02:01.620
So all I need to do is
to take that green curve
01:02:01.620 --> 01:02:04.540
and multiply it
by a factor of 3.
01:02:04.540 --> 01:02:11.080
And indeed, the limiting value
that you get over here is 3.
01:02:11.080 --> 01:02:14.830
Except that if I just
take that green curve
01:02:14.830 --> 01:02:18.420
and superpose it on
this, what I will get
01:02:18.420 --> 01:02:19.665
is something like this.
01:02:24.980 --> 01:02:31.545
So this is 3 times
harmonic oscillator.
01:02:39.530 --> 01:02:44.240
What do I mean by that, is
I try to sort of do my best
01:02:44.240 --> 01:02:47.110
to match the
temperature at which you
01:02:47.110 --> 01:02:49.210
go from one to the other.
01:02:49.210 --> 01:02:53.550
But then what I find is that
as we had established before,
01:02:53.550 --> 01:02:57.300
the green curve goes
to 0 exponentially.
01:02:57.300 --> 01:03:00.770
So there is going to be
some theta associated
01:03:00.770 --> 01:03:02.950
with this frequency.
01:03:02.950 --> 01:03:06.400
Let's call it theta
Einstein divided by T.
01:03:06.400 --> 01:03:10.440
And so the prediction
of this model
01:03:10.440 --> 01:03:16.140
is that the heat capacities
should vanish very rapidly
01:03:16.140 --> 01:03:18.255
as this form of exponential.
01:03:18.255 --> 01:03:22.050
Whereas, what is actually
observed in the experiment
01:03:22.050 --> 01:03:25.182
is that it is going
to 0 proportional
01:03:25.182 --> 01:03:33.200
to T cubed, which is a
much slower type of decay.
01:03:33.200 --> 01:03:35.304
OK?
01:03:35.304 --> 01:03:38.650
AUDIENCE: That's
negative [INAUDIBLE]?
01:03:38.650 --> 01:03:43.810
PROFESSOR: As T goes to 0,
the heat capacity goes to 0.
01:03:43.810 --> 01:03:45.030
T to the third power.
01:03:48.920 --> 01:03:51.910
So it's the limit-- did I
make a mistake somewhere else?
01:03:56.710 --> 01:03:57.670
All right.
01:03:57.670 --> 01:04:02.080
So what's happening here?
01:04:02.080 --> 01:04:05.370
OK, so what's happening
is the following.
01:04:08.590 --> 01:04:12.010
In some average
sense, it is correct
01:04:12.010 --> 01:04:18.430
that if you try to oscillate
some atom in the crystal,
01:04:18.430 --> 01:04:22.660
it's going to have some
characteristic restoring force.
01:04:22.660 --> 01:04:24.850
The characteristic
restoring force
01:04:24.850 --> 01:04:28.660
will give you some
corresponding typical scale
01:04:28.660 --> 01:04:30.460
for the frequencies
of the vibrations.
01:04:32.845 --> 01:04:33.345
Yes?
01:04:36.076 --> 01:04:38.490
AUDIENCE: Is this the
historical progression?
01:04:38.490 --> 01:04:39.115
PROFESSOR: Yes.
01:04:42.710 --> 01:04:44.970
AUDIENCE: I mean, it seems
interesting that they
01:04:44.970 --> 01:04:48.570
would know that-- like this
cage hypothesis is very good,
01:04:48.570 --> 01:04:52.707
considering where a
quantum [INAUDIBLE] exists.
01:04:56.980 --> 01:04:59.170
I don't understand how
that's the logic based--
01:04:59.170 --> 01:05:01.690
if what we know is the
top board over there,
01:05:01.690 --> 01:05:08.657
the logical progression is that
you would have-- I don't know.
01:05:08.657 --> 01:05:09.240
PROFESSOR: No.
01:05:12.660 --> 01:05:18.120
At that time, the proposal
was that essentially
01:05:18.120 --> 01:05:22.000
if you have oscillator
of frequency omega,
01:05:22.000 --> 01:05:25.200
its energy is quantized
in multiples of omega.
01:05:25.200 --> 01:05:27.800
So that's really the only
aspect of quantum mechanics.
01:05:27.800 --> 01:05:31.310
So I actually jumped the
historical development
01:05:31.310 --> 01:05:35.030
where I gave you the
rotational degrees of freedom.
01:05:35.030 --> 01:05:42.761
So as I said, historically this
was resolved last in this part
01:05:42.761 --> 01:05:44.885
because they didn't know
what to do with rotations.
01:05:48.910 --> 01:05:52.040
But now I'm saying that
you know about rotations,
01:05:52.040 --> 01:05:54.860
you know that the heat
capacity goes to 0.
01:05:54.860 --> 01:05:58.910
You say, well,
solid is composed.
01:05:58.910 --> 01:06:01.450
The way that you put
heat into the system,
01:06:01.450 --> 01:06:05.240
enhance its heat capacity, is
because there is kinetic energy
01:06:05.240 --> 01:06:07.620
that you put in the
atoms of the solid.
01:06:07.620 --> 01:06:10.090
And as you try to
put kinetic energy,
01:06:10.090 --> 01:06:14.080
there is this cage model
and there's restoring force.
01:06:14.080 --> 01:06:16.610
The thing that is
wrong about this model
01:06:16.610 --> 01:06:24.420
is that, basically, if you ask
how easy it is to give energy
01:06:24.420 --> 01:06:28.530
to the system, if rather
than having one frequency
01:06:28.530 --> 01:06:33.150
you have multiple frequencies,
then at low temperatures
01:06:33.150 --> 01:06:36.840
you would put energy
in the lower frequency.
01:06:36.840 --> 01:06:44.800
Because the typical scale we
saw for connecting temperature
01:06:44.800 --> 01:06:47.850
and frequency, they are kind
of proportional to each other.
01:06:47.850 --> 01:06:50.310
So if you want to go
to low temperature,
01:06:50.310 --> 01:06:55.290
you are bound to excite things
that have lower frequency.
01:06:55.290 --> 01:06:58.025
So the thing is that
it is true that there
01:06:58.025 --> 01:07:00.310
is a typical frequency.
01:07:00.310 --> 01:07:03.990
But the typical frequency
becomes less and less important
01:07:03.990 --> 01:07:06.010
as you go to low temperature.
01:07:06.010 --> 01:07:08.840
The issue is, what are
the lowest frequencies
01:07:08.840 --> 01:07:11.160
of excitation?
01:07:11.160 --> 01:07:13.510
And basically, the
correct picture
01:07:13.510 --> 01:07:17.890
of excitations of the solid
is that you bang on something
01:07:17.890 --> 01:07:22.180
and you generate
these sound waves.
01:07:22.180 --> 01:07:32.550
So what you have is that
oscillations or vibrations
01:07:32.550 --> 01:07:50.240
of solid are characterized
by wavelength and wave number
01:07:50.240 --> 01:08:01.290
k, 2 pi over lambda.
01:08:01.290 --> 01:08:19.450
So if I really take a
better model of the solid
01:08:19.450 --> 01:08:27.069
in which I have springs that
connect all of these things
01:08:27.069 --> 01:08:33.290
together and ask, what are
the normal modes of vibration?
01:08:33.290 --> 01:08:36.970
I find that the normal
modes can be characterized
01:08:36.970 --> 01:08:39.660
by some wave number k.
01:08:39.660 --> 01:08:43.399
As I said, it's the
inverse of the wavelength.
01:08:43.399 --> 01:08:47.819
And frequency depends
on wave number.
01:08:47.819 --> 01:08:56.069
In a manner that when you go
to 0k, frequency goes to 0.
01:08:56.069 --> 01:08:57.340
And why is that?
01:08:57.340 --> 01:09:00.819
Essentially, what I'm
saying is that if you
01:09:00.819 --> 01:09:05.420
look at particles
that are along a line
01:09:05.420 --> 01:09:07.649
and may be connected by springs.
01:09:07.649 --> 01:09:11.859
So a kind of one-dimensional
version of a solid.
01:09:11.859 --> 01:09:14.920
Then, the normal modes
are characterized
01:09:14.920 --> 01:09:18.124
by distortions that have
some particular wavelength.
01:09:23.160 --> 01:09:26.350
And in the limit
where the wavelength
01:09:26.350 --> 01:09:30.439
goes to 0, essentially--
01:09:30.439 --> 01:09:32.580
Sorry, in the limit
where the wavelength goes
01:09:32.580 --> 01:09:36.729
to infinity or k
goes to 0, it looks
01:09:36.729 --> 01:09:38.990
like I am taking
all of the particles
01:09:38.990 --> 01:09:43.120
and translating them together.
01:09:43.120 --> 01:09:47.910
And if I take the entire
solid here and translate it,
01:09:47.910 --> 01:09:50.124
there is no restoring force.
01:09:50.124 --> 01:09:55.370
So omega has to go to
0 as your k goes to 0,
01:09:55.370 --> 01:09:57.810
or wavelength goes to infinity.
01:09:57.810 --> 01:10:01.920
And there is a symmetry
between k and minus k,
01:10:01.920 --> 01:10:04.240
in fact, that forces
the restoring force
01:10:04.240 --> 01:10:06.560
to be proportional to k squared.
01:10:06.560 --> 01:10:08.570
And when you take the
square root of that,
01:10:08.570 --> 01:10:10.170
you get the frequency.
01:10:10.170 --> 01:10:15.160
You always get a linear
behavior as k goes to 0.
01:10:15.160 --> 01:10:19.810
So essentially,
that's the observation
01:10:19.810 --> 01:10:22.620
that whatever you
do with your solid,
01:10:22.620 --> 01:10:27.140
no matter how complicated,
you have sound modes.
01:10:27.140 --> 01:10:29.520
And sound modes are
things that happen
01:10:29.520 --> 01:10:32.360
in the limit where you
have long wavelengths
01:10:32.360 --> 01:10:37.930
and there is a relationship
between omega and k
01:10:37.930 --> 01:10:39.785
through some kind of
velocity of sound.
01:10:43.400 --> 01:10:49.370
Now, to be precise
there are really
01:10:49.370 --> 01:10:51.910
three types of sound waves.
01:10:51.910 --> 01:10:55.360
If I choose the
direction k along which
01:10:55.360 --> 01:10:58.650
I want to create an
oscillation, the distortions
01:10:58.650 --> 01:11:01.760
can be either along that
direction or perpendicular
01:11:01.760 --> 01:11:02.790
to that.
01:11:02.790 --> 01:11:07.830
They can either be
longitudinal or transfers.
01:11:07.830 --> 01:11:12.340
So there could be one
or two other branches.
01:11:12.340 --> 01:11:15.470
So there could, in principle,
be different straight lines
01:11:15.470 --> 01:11:17.560
as k goes to 0.
01:11:17.560 --> 01:11:19.920
And the other
thing is that there
01:11:19.920 --> 01:11:23.470
is a shortest wavelength
that you can think about.
01:11:23.470 --> 01:11:28.670
So if these particles
are a distance a apart,
01:11:28.670 --> 01:11:31.840
there is no sense in
going to wave numbers that
01:11:31.840 --> 01:11:34.390
are larger than pi over a.
01:11:34.390 --> 01:11:38.400
So you have some
limit to these curves.
01:11:38.400 --> 01:11:41.010
And indeed, when you
approach the boundary,
01:11:41.010 --> 01:11:43.475
this linear dependence
can shift and change
01:11:43.475 --> 01:11:46.270
in all kinds of possible ways.
01:11:46.270 --> 01:11:51.480
And calculating the
frequency inside one
01:11:51.480 --> 01:12:00.526
of these units that is
called a Brillouin zone
01:12:00.526 --> 01:12:05.980
is a nice thing
to do for the case
01:12:05.980 --> 01:12:09.180
of using methods of solid state.
01:12:09.180 --> 01:12:11.660
And you've probably seen that.
01:12:11.660 --> 01:12:18.390
And there is a whole
spectrum of frequencies
01:12:18.390 --> 01:12:21.570
as a function of
wave number that
01:12:21.570 --> 01:12:24.280
correctly characterize a solid.
01:12:24.280 --> 01:12:27.370
So it may be that somewhere
in the middle of this spectrum
01:12:27.370 --> 01:12:32.300
is a typical frequency
omega E. But the point is
01:12:32.300 --> 01:12:35.900
that as you go to lower
and lower temperatures,
01:12:35.900 --> 01:12:42.830
because of these factors of e
to the minus beta h bar omega,
01:12:42.830 --> 01:12:45.970
you can see that as you go to
lower and lower temperature,
01:12:45.970 --> 01:12:50.370
the only things that get
excited are omegas that are also
01:12:50.370 --> 01:12:54.340
going to 0
proportionately to kT.
01:12:54.340 --> 01:13:01.810
So I can draw a line here that
corresponds to frequencies that
01:13:01.810 --> 01:13:04.372
are of the order
of kT over h bar.
01:13:08.630 --> 01:13:10.680
All of the harmonic
oscillators that
01:13:10.680 --> 01:13:15.190
have these larger frequencies
that occur at short wavelengths
01:13:15.190 --> 01:13:16.440
are unimportant.
01:13:16.440 --> 01:13:19.960
They're kind of frozen,
just like the vibrations
01:13:19.960 --> 01:13:22.410
of the oxygen molecules
in this room are frozen.
01:13:22.410 --> 01:13:24.710
You cannot put energy in them.
01:13:24.710 --> 01:13:27.610
They don't contribute
to heat capacity.
01:13:27.610 --> 01:13:30.590
But all of these
long wavelength modes
01:13:30.590 --> 01:13:35.080
down here have
frequencies that go to 0.
01:13:35.080 --> 01:13:38.500
Their excitation
possibility is large.
01:13:38.500 --> 01:13:40.500
And it, indeed,
these long wavelength
01:13:40.500 --> 01:13:44.885
modes that are easy to excite
and continue to heat capacity.
01:13:48.090 --> 01:13:51.130
I'll do maybe the precise
calculation next time,
01:13:51.130 --> 01:13:53.610
but even within this
picture we can figure out
01:13:53.610 --> 01:13:56.190
why the answer should be
proportional to T cubed.
01:13:58.770 --> 01:14:02.460
So what I need to do,
rather than counting
01:14:02.460 --> 01:14:06.790
all harmonic oscillators--
the factor of 3n--
01:14:06.790 --> 01:14:10.380
I have to count how many
oscillators have frequencies
01:14:10.380 --> 01:14:13.600
that are less than
this kT over h bar.
01:14:16.140 --> 01:14:28.260
So I claim that number of modes
with frequency less than kT
01:14:28.260 --> 01:14:40.970
over h bar goes like
kT over h bar cubed V.
01:14:40.970 --> 01:14:43.657
Essentially, what
I have to do is
01:14:43.657 --> 01:14:50.710
to do a summation over all k
that is less than some k max.
01:14:50.710 --> 01:14:55.080
This k max is set
by this condition
01:14:55.080 --> 01:14:58.450
that Vk max is of the
order of kT over h bar.
01:14:58.450 --> 01:15:04.620
So this k max is of the
order of kT over h bar V.
01:15:04.620 --> 01:15:09.770
So actually, to be more
precise I have to put a V here.
01:15:09.770 --> 01:15:14.700
So I have to count
all of the modes.
01:15:14.700 --> 01:15:18.090
Now, this separation
between these modes--
01:15:18.090 --> 01:15:21.230
if you have a box of
size l is 2 pi over l.
01:15:21.230 --> 01:15:24.870
So maybe we will
discuss that later on.
01:15:24.870 --> 01:15:27.770
But the summations
over k you will always
01:15:27.770 --> 01:15:31.730
replace with
integrations over k times
01:15:31.730 --> 01:15:37.900
the density of state, which
is V divided by 2 pi cubed.
01:15:37.900 --> 01:15:41.880
So this has to go
between 0 and k max.
01:15:41.880 --> 01:15:46.330
And so this is
proportional to V k max
01:15:46.330 --> 01:15:50.685
cubed, which is what
I wrote over there.
01:15:53.300 --> 01:15:56.000
So as I go to lower
and lower temperature,
01:15:56.000 --> 01:15:59.930
there are fewer and
fewer oscillators.
01:15:59.930 --> 01:16:04.620
The number of those
oscillators grows like T cubed.
01:16:04.620 --> 01:16:07.300
Each one of those
oscillators is fully excited
01:16:07.300 --> 01:16:11.550
as energy kT contributes
1 unit to heat capacity.
01:16:11.550 --> 01:16:15.480
Since the number of oscillators
goes to 0 as T cubed,
01:16:15.480 --> 01:16:18.030
the heat capacity that
they contribute also
01:16:18.030 --> 01:16:22.450
goes to 0 as T cubed.
01:16:22.450 --> 01:16:27.860
So you don't really need
to know-- this is actually
01:16:27.860 --> 01:16:29.570
an interesting thing to ponder.
01:16:29.570 --> 01:16:31.420
So rather than doing
the calculations,
01:16:31.420 --> 01:16:34.480
maybe just think about this.
01:16:34.480 --> 01:16:40.980
That somehow the solid could
be arbitrarily complicated.
01:16:40.980 --> 01:16:44.460
So it could be
composed of molecules
01:16:44.460 --> 01:16:46.270
that have some particular shape.
01:16:46.270 --> 01:16:49.460
They are forming
some strange lattice
01:16:49.460 --> 01:16:50.970
of some form, et cetera.
01:16:53.920 --> 01:16:58.180
And given the complicated
nature of the molecules,
01:16:58.180 --> 01:17:02.080
the spectrum that you have
for potential frequencies
01:17:02.080 --> 01:17:05.220
that a solid can
take, because of all
01:17:05.220 --> 01:17:07.680
of the different
vibrations, et cetera,
01:17:07.680 --> 01:17:09.560
could be arbitrary complicated.
01:17:09.560 --> 01:17:13.430
You can have kinds of
oscillations such as the ones
01:17:13.430 --> 01:17:16.040
that I have indicated.
01:17:16.040 --> 01:17:19.230
However, if you go
to low temperature,
01:17:19.230 --> 01:17:22.730
you are only interested
in vibrations
01:17:22.730 --> 01:17:25.470
that are very low in frequency.
01:17:25.470 --> 01:17:28.090
Vibrations that are
very low in frequencies
01:17:28.090 --> 01:17:30.340
must correspond
to the formations
01:17:30.340 --> 01:17:32.920
that are very long wavelength.
01:17:32.920 --> 01:17:34.340
And when you are
looking at things
01:17:34.340 --> 01:17:36.250
that are long wavelength,
this is, again,
01:17:36.250 --> 01:17:38.950
another thing that has
statistical in character.
01:17:38.950 --> 01:17:41.350
That is, rather you
are here looking
01:17:41.350 --> 01:17:47.510
at things that span thousands
of atoms or molecules.
01:17:47.510 --> 01:17:50.510
However, as you go to lower
and lower temperature,
01:17:50.510 --> 01:17:53.240
more and more atoms
and molecules.
01:17:53.240 --> 01:17:57.280
And so again, some kind of
averaging is taking place.
01:17:57.280 --> 01:18:00.840
All of the details,
et cetera, wash out.
01:18:00.840 --> 01:18:04.380
You really see some
global characteristic.
01:18:04.380 --> 01:18:06.370
The global characteristic
that you see
01:18:06.370 --> 01:18:07.650
is set by this symmetry.
01:18:07.650 --> 01:18:12.580
Just the fact that when I
go to exactly k equals to 0,
01:18:12.580 --> 01:18:13.800
I am translating.
01:18:13.800 --> 01:18:18.190
I have 0 frequency.
01:18:18.190 --> 01:18:21.560
So when I'm doing something
that is long wavelength,
01:18:21.560 --> 01:18:24.710
the frequency should
somehow be proportional
01:18:24.710 --> 01:18:25.520
to that wavelength.
01:18:25.520 --> 01:18:29.120
So that's just a statement
of continuity if you like.
01:18:29.120 --> 01:18:33.300
Once I have made that
statement, then it's
01:18:33.300 --> 01:18:37.170
just a calculation of how
many modes are possible.
01:18:37.170 --> 01:18:40.350
The number of modes will
be proportional to T cubed.
01:18:40.350 --> 01:18:43.885
And I will get this T
cubed law irrespective
01:18:43.885 --> 01:18:46.520
of how complicated the solid is.
01:18:46.520 --> 01:18:50.570
All of the solids will have
the same T cubed behavior.
01:18:50.570 --> 01:18:55.120
The place where they come
from the classical behavior
01:18:55.120 --> 01:18:57.880
to this quantum
behavior will depend
01:18:57.880 --> 01:19:01.110
on the details of
the solid, et cetera.
01:19:01.110 --> 01:19:04.005
But the low temperature
law, this T cubed law,
01:19:04.005 --> 01:19:07.700
is something that is universal.
01:19:07.700 --> 01:19:10.975
OK, so next time around,
we will do this calculation
01:19:10.975 --> 01:19:15.210
in more detail, and then
see also its connection
01:19:15.210 --> 01:19:17.590
to the blackbody radius.