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PROFESSOR: Particles
in quantum mechanics.
00:00:23.250 --> 00:00:27.470
In particular, the ones that are
identical and non-interacting.
00:00:37.140 --> 00:00:42.430
So basically, we were focusing
on a type of Hamiltonian
00:00:42.430 --> 00:00:45.780
for a system of N
particles, which
00:00:45.780 --> 00:00:53.190
could be written as the sum of
contributions that correspond
00:00:53.190 --> 00:00:57.460
respectively to particle
1, particle 2, particle N.
00:00:57.460 --> 00:01:03.090
So essentially, a sum of
terms that are all the same.
00:01:03.090 --> 00:01:06.960
And one-particle
terms is sufficient
00:01:06.960 --> 00:01:10.290
because we don't
have interactions.
00:01:10.290 --> 00:01:15.750
So if we look at one of
these H's-- so one of these
00:01:15.750 --> 00:01:19.673
one-particle Hamiltonians, we
said that we could find some
00:01:19.673 --> 00:01:21.950
kind of basis for it.
00:01:21.950 --> 00:01:25.270
In particular, typically we
were interested in particles
00:01:25.270 --> 00:01:26.140
in the box.
00:01:26.140 --> 00:01:31.430
We would label them
with some wave number k.
00:01:31.430 --> 00:01:37.130
And there was an associated
one-particle energy,
00:01:37.130 --> 00:01:39.900
which for the case of
one-particle in box
00:01:39.900 --> 00:01:44.270
was h bar squared
k squared over 2m.
00:01:44.270 --> 00:01:48.770
But in general, for the
one-particle system,
00:01:48.770 --> 00:01:54.330
we can think of a ladder of
possible values of energies.
00:01:54.330 --> 00:02:00.815
So there will be some
k1, k2, k3, et cetera.
00:02:05.590 --> 00:02:08.830
They may be distributed in any
particular way corresponding
00:02:08.830 --> 00:02:10.389
to different energies.
00:02:10.389 --> 00:02:12.760
Basically, you
would have a number
00:02:12.760 --> 00:02:15.600
of possible states
for one particle.
00:02:15.600 --> 00:02:27.730
So for the case of the particle
in the box, the wave functions
00:02:27.730 --> 00:02:32.335
and coordinate space that we
had x, k were of the form e
00:02:32.335 --> 00:02:37.020
to the i k dot x divided
by square root of V.
00:02:37.020 --> 00:02:45.220
We allowed the energies where h
bar squared k squared over 2m.
00:02:45.220 --> 00:02:50.520
And this discretization
was because the values of k
00:02:50.520 --> 00:02:56.165
were multiples of 2 pi over
l with integers in the three
00:02:56.165 --> 00:02:57.380
different directions.
00:02:57.380 --> 00:02:59.550
Assuming periodic
boundary conditions
00:02:59.550 --> 00:03:03.670
or appropriate discretization
for closed boundary conditions,
00:03:03.670 --> 00:03:07.010
or whatever you have.
00:03:07.010 --> 00:03:11.290
So that's the
one-particle state.
00:03:11.290 --> 00:03:15.350
If the Hamiltonian
is of this form,
00:03:15.350 --> 00:03:20.620
it is clear that we can
multiply a bunch of these states
00:03:20.620 --> 00:03:25.800
and form another
eigenstate for HN.
00:03:25.800 --> 00:03:27.805
Those we were calling
product states.
00:03:32.945 --> 00:03:38.180
So you basically pick a
bunch of these k's and you
00:03:38.180 --> 00:03:39.280
multiply them.
00:03:39.280 --> 00:03:43.685
So you have k1, k2, kN.
00:03:46.410 --> 00:03:49.050
And essentially, that
would correspond,
00:03:49.050 --> 00:03:52.460
let's say, in coordinate
representation
00:03:52.460 --> 00:03:57.910
to taking a bunch of x's and
the corresponding k's and having
00:03:57.910 --> 00:04:03.500
a wave function of the form e to
the i k alp-- sum over alpha k
00:04:03.500 --> 00:04:09.290
alpha x alpha, and then
divided by V to the N over 2.
00:04:13.210 --> 00:04:16.779
So in this procedure,
what did we do?
00:04:16.779 --> 00:04:22.340
We had a number of possibilities
for the one-particle state.
00:04:22.340 --> 00:04:26.030
And in order to, let's say,
make a two-particle state,
00:04:26.030 --> 00:04:31.450
we would pick two of
these k's and multiply
00:04:31.450 --> 00:04:33.900
the corresponding
wave functions.
00:04:33.900 --> 00:04:37.060
If you had three particles,
we could pick another one.
00:04:37.060 --> 00:04:39.710
If you had four particles,
we could potentially
00:04:39.710 --> 00:04:43.050
pick a second one
twice, et cetera.
00:04:43.050 --> 00:04:44.890
So in general,
basically we would
00:04:44.890 --> 00:04:48.900
put N of these crosses on
these one-particle states
00:04:48.900 --> 00:04:51.570
that we've selected.
00:04:51.570 --> 00:04:54.690
Problem was that
this was not allowed
00:04:54.690 --> 00:04:58.730
by quantum mechanics
for identical particles.
00:04:58.730 --> 00:05:01.960
Because if we took one
of these wave functions
00:05:01.960 --> 00:05:05.800
and exchanged two of
the labels, x1 and x2,
00:05:05.800 --> 00:05:10.280
we could potentially get
a different wave function.
00:05:10.280 --> 00:05:14.650
And in quantum mechanics, we
said that the wave function
00:05:14.650 --> 00:05:18.850
has to be either symmetric or
anti-symmetric with respect
00:05:18.850 --> 00:05:21.940
to exchange of a
pair of particles.
00:05:21.940 --> 00:05:27.040
And also, whatever it implied
for repeating this exchange
00:05:27.040 --> 00:05:31.340
many times to look at all
possible permutations.
00:05:31.340 --> 00:05:36.770
So what we saw was that
product states are good
00:05:36.770 --> 00:05:38.467
as long as you
are thinking about
00:05:38.467 --> 00:05:39.550
distinguishable particles.
00:05:44.950 --> 00:05:53.710
But if you have
identical particles,
00:05:53.710 --> 00:05:57.810
you had to appropriately
symmetrize or anti-symmetrize
00:05:57.810 --> 00:06:00.060
these states.
00:06:00.060 --> 00:06:05.320
So what we ended up was
a wabe of, for example,
00:06:05.320 --> 00:06:09.250
symmetrizing things for
the case of fermions.
00:06:09.250 --> 00:06:12.400
So we could take a bunch
of these k-values again.
00:06:15.360 --> 00:06:21.020
And a fermionic, or
anti-symmetrized version,
00:06:21.020 --> 00:06:25.980
was then constructed by
summing over all permutations.
00:06:25.980 --> 00:06:31.110
And for N particle, there would
be N factorial permutations.
00:06:31.110 --> 00:06:34.320
Basically, doing a
permutation of all of these
00:06:34.320 --> 00:06:40.200
indices k1, k2, et cetera,
that we had selected for this.
00:06:40.200 --> 00:06:43.670
And for the case
of fermions, we had
00:06:43.670 --> 00:06:49.720
to multiply each
permutation with a sign that
00:06:49.720 --> 00:06:56.130
was plus for even permutations,
minus for odd permutations.
00:06:56.130 --> 00:06:59.970
And this would give
us N factorial terms.
00:06:59.970 --> 00:07:03.780
And the appropriate
normalization
00:07:03.780 --> 00:07:08.490
was 1 over square
root of N factorial.
00:07:08.490 --> 00:07:12.030
So this was for the
case of fermions.
00:07:12.030 --> 00:07:13.445
And this was actually minus.
00:07:13.445 --> 00:07:15.410
I should have put
in a minus here.
00:07:15.410 --> 00:07:21.220
And this would have
been minus to the P.
00:07:21.220 --> 00:07:29.530
For the case of
bosons, we basically
00:07:29.530 --> 00:07:33.360
dispensed with this factor
of minus 1 to the P.
00:07:33.360 --> 00:07:35.145
So we had a Pk.
00:07:39.400 --> 00:07:44.480
Now, the corresponding
normalization that we had here
00:07:44.480 --> 00:07:46.420
was slightly different.
00:07:46.420 --> 00:07:50.020
The point was that
if we were doing
00:07:50.020 --> 00:07:55.570
this computation for
the case of fermions,
00:07:55.570 --> 00:07:58.165
we could not allow
a state where there
00:07:58.165 --> 00:08:02.030
is a double occupation of one
of the one-particle state.
00:08:02.030 --> 00:08:06.060
Under exchange of
the particles that
00:08:06.060 --> 00:08:08.470
would correspond
to these k-values,
00:08:08.470 --> 00:08:11.600
I would get the same state back,
but the exchange would give me
00:08:11.600 --> 00:08:14.250
a minus 1 and it
would give me 0.
00:08:14.250 --> 00:08:17.370
So the fermionic wave function
that I have constructed
00:08:17.370 --> 00:08:19.840
here, appropriately
anti-symmetrized,
00:08:19.840 --> 00:08:24.060
exists only as long as
there are no repeats.
00:08:24.060 --> 00:08:29.680
Whereas, for the case of bosons,
I could put 2 over same place.
00:08:29.680 --> 00:08:34.820
I could put 3 somewhere else,
any number that I liked,
00:08:34.820 --> 00:08:38.090
and there would be
no problem with this.
00:08:38.090 --> 00:08:41.780
Except that the normalization
would be more complicated.
00:08:41.780 --> 00:08:44.320
And we saw that
appropriate normalization
00:08:44.320 --> 00:08:47.135
was a product over
k nk factorial.
00:08:50.580 --> 00:08:55.380
So this was for fermions.
00:08:57.970 --> 00:08:58.820
This is for bosons.
00:09:03.380 --> 00:09:09.110
And the two I can
submerge into one formula
00:09:09.110 --> 00:09:16.890
by writing a symmetrized
or anti-symmetrized state,
00:09:16.890 --> 00:09:22.550
respectively indicated by eta,
where we have eta is minus 1
00:09:22.550 --> 00:09:27.750
for fermions and eta
is plus 1 for bosons,
00:09:27.750 --> 00:09:33.510
which is 1 over square
root of N factorial product
00:09:33.510 --> 00:09:37.100
over k nk factorials.
00:09:37.100 --> 00:09:42.360
And then the sum over all
N factorial permutations.
00:09:42.360 --> 00:09:49.360
This phase factor for fermions
and nothing for bosons
00:09:49.360 --> 00:09:52.120
of the appropriately
permuted set of k's.
00:09:57.630 --> 00:10:04.920
And in this way
of noting things,
00:10:04.920 --> 00:10:13.840
I have to assign values nk which
are either 0 or 1 for fermions.
00:10:13.840 --> 00:10:19.370
Because as we said, multiple
occupations are not allowed.
00:10:19.370 --> 00:10:21.500
But there is no
restriction for bosons.
00:10:28.040 --> 00:10:33.830
Except of course, that
in this perspective,
00:10:33.830 --> 00:10:40.607
as I go along this k-axis,
I have 0, 1, 0, 2, 1, 3, 0,
00:10:40.607 --> 00:10:44.910
0 for the occupations.
00:10:44.910 --> 00:10:47.060
Of course, what I
need to construct
00:10:47.060 --> 00:10:50.040
whether I am dealing
with bosons or fermions
00:10:50.040 --> 00:10:54.660
is that the sum over k nk is
the total number of particles
00:10:54.660 --> 00:10:55.490
that I have.
00:11:01.800 --> 00:11:08.430
Now, the other thing to note
is that once I have given you
00:11:08.430 --> 00:11:13.700
a picture such as this in terms
of which one-particle states
00:11:13.700 --> 00:11:19.360
I want to look at, or which
set of occupation numbers
00:11:19.360 --> 00:11:23.300
I have nk, then
there is one and only
00:11:23.300 --> 00:11:27.400
one symmetrized or
anti-symmetrized state.
00:11:27.400 --> 00:11:30.730
So over here, I
could have permuted
00:11:30.730 --> 00:11:34.770
the k's in a number
of possible ways.
00:11:34.770 --> 00:11:36.740
But as a result
of symmetrization,
00:11:36.740 --> 00:11:39.940
anti-symmetrization,
various ways
00:11:39.940 --> 00:11:44.200
of permuting the
labels here ultimately
00:11:44.200 --> 00:11:48.210
come to the same set
of occupation numbers.
00:11:48.210 --> 00:11:54.900
So it is possible to actually
label the state rather than
00:11:54.900 --> 00:11:57.530
by the set of k's.
00:11:57.530 --> 00:11:59.600
By the set of nk's.
00:11:59.600 --> 00:12:04.085
It is kind of a more appropriate
way of representing the system.
00:12:16.850 --> 00:12:21.770
So that's essentially
the kinds of states
00:12:21.770 --> 00:12:24.320
that we are going to be using.
00:12:24.320 --> 00:12:27.200
Again, in talking about
identical particles,
00:12:27.200 --> 00:12:31.650
which could be either
bosons or fermions.
00:12:31.650 --> 00:12:35.970
Let's take a step back,
remind you of something
00:12:35.970 --> 00:12:41.020
that we did before that
had only one particle.
00:12:41.020 --> 00:12:43.750
Because I will soon
go to many particles.
00:12:43.750 --> 00:12:46.940
But before that,
let's remind you
00:12:46.940 --> 00:12:52.510
what the one particle
in a box looked like.
00:12:52.510 --> 00:12:57.180
So indeed, in this case,
the single-particle states
00:12:57.180 --> 00:13:00.030
were the ones that
I told you before,
00:13:00.030 --> 00:13:03.360
2 pi over l some
set of integers.
00:13:03.360 --> 00:13:08.260
Epsilon k, that was h bar
squared k squared over 2m.
00:13:08.260 --> 00:13:13.260
If I want to calculate
the partition function
00:13:13.260 --> 00:13:19.330
for one particle in the box,
I have to do a trace of e
00:13:19.330 --> 00:13:24.185
to the minus beta
h for one particle.
00:13:24.185 --> 00:13:26.820
The trace I can very
easily calculate
00:13:26.820 --> 00:13:30.340
in the basis in which
this is diagonal.
00:13:30.340 --> 00:13:32.330
That's the basis
that is parameterized
00:13:32.330 --> 00:13:34.320
by these k-values.
00:13:34.320 --> 00:13:40.280
So I do a sum over k of
e to the minus beta h
00:13:40.280 --> 00:13:42.305
bar squared k squared over 2m.
00:13:45.040 --> 00:13:47.940
And then in the limit
of very large box,
00:13:47.940 --> 00:13:50.880
we saw that the sum
over k I can replace
00:13:50.880 --> 00:13:55.700
with V integral
over k 2 pi cubed.
00:13:55.700 --> 00:13:59.040
This was the density
of states in k.
00:13:59.040 --> 00:14:05.910
e to the minus beta h bar
squared k squared over 2m.
00:14:05.910 --> 00:14:09.820
And this was three
Gaussian integrals
00:14:09.820 --> 00:14:15.100
that gave us the usual formula
of V over lambda cubed, where
00:14:15.100 --> 00:14:18.830
lambda was this thermal
[INAUDIBLE] wavelength
00:14:18.830 --> 00:14:20.665
h root 2 pi mk.
00:14:26.970 --> 00:14:31.560
But we said that the essence
of statistical mechanics
00:14:31.560 --> 00:14:34.120
is to tell you
about probabilities
00:14:34.120 --> 00:14:37.680
of various micro-states, various
positions of the particle
00:14:37.680 --> 00:14:41.870
in the box, which in
the quantum perspective
00:14:41.870 --> 00:14:45.210
is probability becomes
a density matrix.
00:14:45.210 --> 00:14:48.560
And we evaluated
this density matrix
00:14:48.560 --> 00:14:49.986
in the coordinate
representation.
00:14:52.810 --> 00:14:57.030
And in the coordinate
representation,
00:14:57.030 --> 00:15:00.600
essentially what
we had to do was
00:15:00.600 --> 00:15:07.360
to go into the basis in
which rho is diagonal.
00:15:07.360 --> 00:15:11.750
So we had x prime k.
00:15:11.750 --> 00:15:18.616
In the k basis, the density
matrix is just this formula.
00:15:18.616 --> 00:15:23.030
It's the Boltzmann weight
appropriately normalized by Z1.
00:15:25.540 --> 00:15:26.880
And then we go kx.
00:15:31.860 --> 00:15:35.300
And basically,
again replacing this
00:15:35.300 --> 00:15:42.360
with V integral d cubed k 2 pi
cubed e to the minus beta h bar
00:15:42.360 --> 00:15:45.510
squared k squared over 2m.
00:15:45.510 --> 00:15:51.430
These two factors of xk and x
prime k gave us a factor of e
00:15:51.430 --> 00:16:05.130
to the kx, xk I have as
ik dot x prime minus x.
00:16:05.130 --> 00:16:07.070
Completing the square.
00:16:07.070 --> 00:16:11.140
Actually, I had to divide by Z1.
00:16:11.140 --> 00:16:14.140
There is a factor of 1 over
V from the normalization
00:16:14.140 --> 00:16:16.400
of these things.
00:16:16.400 --> 00:16:21.380
The two V's here cancel,
but Z1 is proportional to V.
00:16:21.380 --> 00:16:23.510
The lambda cubes
cancel and so what
00:16:23.510 --> 00:16:28.740
we have is 1 over
V e to the minus x
00:16:28.740 --> 00:16:33.730
minus x prime squared
pi over lambda squared.
00:16:37.730 --> 00:16:41.340
So basically, what
you have here is
00:16:41.340 --> 00:16:47.040
that we have a box
of volume V. There
00:16:47.040 --> 00:16:50.135
is a particle inside
at some location x.
00:16:53.860 --> 00:16:56.450
And the probability
to find it at location
00:16:56.450 --> 00:17:00.000
x is the diagonal
element of this entity.
00:17:00.000 --> 00:17:03.400
It's just 1 over
V. But this entity
00:17:03.400 --> 00:17:08.369
has off-diagonal elements
reflecting the fact
00:17:08.369 --> 00:17:12.510
that the best that you can do
to localize something in quantum
00:17:12.510 --> 00:17:15.990
mechanics is to make some
kind of a wave packet.
00:17:24.890 --> 00:17:27.170
OK.
00:17:27.170 --> 00:17:31.740
So this we did last time.
00:17:31.740 --> 00:17:37.216
What we want to do now is
to go from one particle
00:17:37.216 --> 00:17:40.570
to the case of N particles.
00:17:40.570 --> 00:17:44.020
So rather than
having 1x prime, I
00:17:44.020 --> 00:17:49.420
will have a whole bunch of x
primes labeled 1 through N.
00:17:49.420 --> 00:17:55.660
And I want to calculate the N
particle density matrix that
00:17:55.660 --> 00:18:03.440
connects me from set of points
x to another set of points
00:18:03.440 --> 00:18:04.080
x prime.
00:18:04.080 --> 00:18:06.880
So if you like in
the previous picture,
00:18:06.880 --> 00:18:11.160
this would have been
x1 and x1 prime,
00:18:11.160 --> 00:18:19.160
and then I now have x2 and x2
prime, x3 and x3 prime, xN and
00:18:19.160 --> 00:18:20.500
xN prime.
00:18:20.500 --> 00:18:23.520
I have a bunch of
different coordinates
00:18:23.520 --> 00:18:27.140
and I'd like to calculate that.
00:18:27.140 --> 00:18:28.100
OK.
00:18:28.100 --> 00:18:37.620
Once more, we know that rho
is diagonal in the basis that
00:18:37.620 --> 00:18:40.710
is represented by
these occupations
00:18:40.710 --> 00:18:44.430
of one-particle states.
00:18:44.430 --> 00:18:49.480
And so what I can
do is I can sum over
00:18:49.480 --> 00:18:52.360
a whole bunch of plane waves.
00:18:52.360 --> 00:18:58.240
And I have to pick N factors
of k out of this list
00:18:58.240 --> 00:19:02.580
in order to make one of these
symmetrized or anti-symmetrized
00:19:02.580 --> 00:19:04.740
wave functions.
00:19:04.740 --> 00:19:08.230
But then I have to
remember, as I said,
00:19:08.230 --> 00:19:14.170
that I should not over-count
distinct set of k-values
00:19:14.170 --> 00:19:17.690
because permutations
of these list of k's
00:19:17.690 --> 00:19:20.750
that I have over here,
because of symmetrization
00:19:20.750 --> 00:19:24.140
or anti-symmetrization,
will give me the same state.
00:19:24.140 --> 00:19:26.670
So I have to be
careful about that.
00:19:26.670 --> 00:19:31.650
Then, I go from x prime to k.
00:19:35.770 --> 00:19:39.510
Now, the density matrix
in the k-basis I know.
00:19:39.510 --> 00:19:43.110
It is simply e to the minus
beta, the energy which
00:19:43.110 --> 00:19:48.610
is sum over alpha h bar squared
k alpha squared over 2m.
00:19:48.610 --> 00:19:52.690
So I sum over the
list of k alphas
00:19:52.690 --> 00:19:54.510
that appear in this series.
00:19:54.510 --> 00:19:56.350
There will be n of them.
00:19:56.350 --> 00:19:58.360
I have to
appropriately normalize
00:19:58.360 --> 00:20:03.100
that by the N-particle partition
function, which we have yet
00:20:03.100 --> 00:20:04.800
to calculate.
00:20:04.800 --> 00:20:09.780
And then I go back from k to x.
00:20:17.245 --> 00:20:20.230
Now, let's do this.
00:20:20.230 --> 00:20:23.390
The first thing that
I mentioned last time
00:20:23.390 --> 00:20:26.430
is that I would,
in principle, like
00:20:26.430 --> 00:20:30.580
to sum over k1 going
over the entire list,
00:20:30.580 --> 00:20:34.570
k2 going the entire list, k3
going over the entire list.
00:20:34.570 --> 00:20:39.170
That is, I would like to make
the sum over k's unrestricted.
00:20:42.100 --> 00:20:45.170
But then I have to
take into account
00:20:45.170 --> 00:20:48.260
the over-counting that I have.
00:20:48.260 --> 00:20:51.490
If I am looking at the case
where all of the k's are
00:20:51.490 --> 00:20:57.250
distinct-- they don't show
any double occupancy-- then
00:20:57.250 --> 00:21:00.160
I have over-counted by the
number of permutations.
00:21:00.160 --> 00:21:03.890
Because any permutation would
have given me the same number.
00:21:03.890 --> 00:21:07.000
So I have to divide by
the number of permutations
00:21:07.000 --> 00:21:12.710
to avoid the over-counting
due to symmetrization here.
00:21:12.710 --> 00:21:14.680
Now, when I have
something like this,
00:21:14.680 --> 00:21:22.580
which is a multiple occupancy,
I have overdone this division.
00:21:22.580 --> 00:21:28.730
I have to multiply
by this factor,
00:21:28.730 --> 00:21:36.850
and that's the correct number
of over-countings that I have.
00:21:36.850 --> 00:21:40.880
And as I said, this was a good
thing because the quantity
00:21:40.880 --> 00:21:43.980
that I had the hardest
time for, and comes
00:21:43.980 --> 00:21:47.020
in the normalizations
that occurs here,
00:21:47.020 --> 00:21:49.750
is this factor of 1
over nk factorial.
00:21:53.140 --> 00:21:56.680
Naturally, again,
all of these things
00:21:56.680 --> 00:21:58.670
do depend on the symmetry.
00:21:58.670 --> 00:22:03.240
So I better make sure
I indicate the index.
00:22:03.240 --> 00:22:06.400
Whether I'm calculating this
density matrix for fermions
00:22:06.400 --> 00:22:09.070
or bosons, it is important.
00:22:09.070 --> 00:22:16.000
In either case-- well,
what I need to do
00:22:16.000 --> 00:22:23.960
is to do a summation over P
here for this one and P prime
00:22:23.960 --> 00:22:26.150
here or P prime here and P here.
00:22:26.150 --> 00:22:28.440
It doesn't matter, there's
two sets of permutations
00:22:28.440 --> 00:22:30.580
that I have to do.
00:22:30.580 --> 00:22:36.240
In each case, I have to
take care of this eta P,
00:22:36.240 --> 00:22:36.815
eta P prime.
00:22:42.390 --> 00:22:45.450
And then the normalization.
00:22:45.450 --> 00:22:50.040
So I divide by twice, or the
square of the square root.
00:22:50.040 --> 00:22:56.350
I get the N factorial
product over k nk factorial.
00:22:56.350 --> 00:23:00.190
And very nicely, the
over-counting factor
00:23:00.190 --> 00:23:03.580
here cancels the
normalization factor
00:23:03.580 --> 00:23:06.660
that I would have had here.
00:23:06.660 --> 00:23:08.820
So we got that.
00:23:08.820 --> 00:23:11.280
Now, what do we have?
00:23:11.280 --> 00:23:23.500
We have P prime permutation
of these objects going to x,
00:23:23.500 --> 00:23:27.330
and then we have
here P permutation
00:23:27.330 --> 00:23:31.346
of these k numbers going to x.
00:23:31.346 --> 00:23:36.270
I guess the first
one I got wrong.
00:23:36.270 --> 00:23:37.860
I start with x prime.
00:23:40.460 --> 00:23:44.630
Go through P prime to k.
00:23:44.630 --> 00:23:50.810
And again, symmetries are
already taken into account.
00:23:50.810 --> 00:23:52.730
I don't need to write that.
00:23:52.730 --> 00:23:57.490
And I have the factor
of e to the minus beta h
00:23:57.490 --> 00:24:01.670
bar squared sum over
alpha k alpha squared
00:24:01.670 --> 00:24:04.540
over 2m divided by ZN.
00:24:09.940 --> 00:24:14.880
OK, so let's bring all of the
denominator factors out front.
00:24:14.880 --> 00:24:17.170
I have a ZN.
00:24:17.170 --> 00:24:20.970
I have an N factorial squared.
00:24:20.970 --> 00:24:24.620
Two factors of N factorial.
00:24:24.620 --> 00:24:31.620
I have a sum over two sets of
permutations P and P prime.
00:24:34.350 --> 00:24:44.480
The product of the associated
phase factor of their parities,
00:24:44.480 --> 00:24:48.830
and then I have this
integration over k's.
00:24:48.830 --> 00:24:50.820
Now, unrestricted.
00:24:50.820 --> 00:24:55.730
Since it is unrestricted, I
can integrate independently
00:24:55.730 --> 00:25:01.320
over each one of the k's, or
sum over each one of them.
00:25:01.320 --> 00:25:04.000
When I sum, the sum
becomes the integral
00:25:04.000 --> 00:25:12.700
over d cubed k alpha divided
by 2 pi-- yeah, 2 pi cubed V.
00:25:12.700 --> 00:25:16.820
Basically, the density in
replacing the sum over k
00:25:16.820 --> 00:25:20.350
alpha with the
corresponding integration.
00:25:20.350 --> 00:25:25.430
So basically, this
set of factors
00:25:25.430 --> 00:25:28.550
is what happened to that.
00:25:34.240 --> 00:25:37.210
OK, what do we have here?
00:25:37.210 --> 00:25:47.680
We have e to the i x--
well, let's be careful here.
00:25:47.680 --> 00:26:02.130
I have e to the i x prime alpha
acting on k of p prime alpha
00:26:02.130 --> 00:26:07.850
because I permuted
the k-label that
00:26:07.850 --> 00:26:15.580
went with, say, the alpha
component here with p prime.
00:26:15.580 --> 00:26:18.350
From here, I would
have minus because it's
00:26:18.350 --> 00:26:20.050
the complex conjugate.
00:26:20.050 --> 00:26:30.746
I have x alpha k p alpha,
because I permuted this by k.
00:26:34.080 --> 00:26:40.010
I have one of these factors for
each V. With each one of them,
00:26:40.010 --> 00:26:43.900
there is a normalization
of square root of V.
00:26:43.900 --> 00:26:48.930
So the two of them together will
give me V. But that's only one
00:26:48.930 --> 00:26:53.210
of the N-particle So
there are N of them.
00:26:53.210 --> 00:26:57.265
So if I want, I can
extend this product
00:26:57.265 --> 00:26:59.660
to also encompass this term.
00:27:02.520 --> 00:27:07.420
And then having done so, I can
also write here e to the minus
00:27:07.420 --> 00:27:10.000
beta h bar squared
k alpha squared
00:27:10.000 --> 00:27:12.680
over 2m within the product.
00:27:18.682 --> 00:27:23.040
AUDIENCE: [INAUDIBLE] after
this-- is it quantity xk minus
00:27:23.040 --> 00:27:25.540
[INAUDIBLE].
00:27:25.540 --> 00:27:27.200
PROFESSOR: I forgot an a here.
00:27:27.200 --> 00:27:28.740
What else did I miss out?
00:27:31.433 --> 00:27:32.724
AUDIENCE: [INAUDIBLE] quantity.
00:27:35.592 --> 00:27:37.210
PROFESSOR: So I forgot the i.
00:27:39.890 --> 00:27:41.340
OK, good?
00:27:41.340 --> 00:27:44.848
So the V's cancel out.
00:27:44.848 --> 00:27:48.200
All right, so that's fine.
00:27:48.200 --> 00:27:49.380
What do we have?
00:27:49.380 --> 00:27:57.530
We have 1 over ZN N
factorial squared.
00:27:57.530 --> 00:28:02.070
Two sets of permutations
summed over, p and p prime.
00:28:02.070 --> 00:28:07.170
Corresponding parities
eta p eta of p prime.
00:28:07.170 --> 00:28:13.300
And then, I have a product
of these integrations
00:28:13.300 --> 00:28:17.980
that I have to do that are
three-dimensional Gaussians
00:28:17.980 --> 00:28:20.900
for each k alpha.
00:28:20.900 --> 00:28:22.700
What do I get?
00:28:22.700 --> 00:28:24.890
Well, first of all,
if I didn't have this,
00:28:24.890 --> 00:28:27.730
if I just was doing
the integration of e
00:28:27.730 --> 00:28:32.100
to the minus beta h bar
squared k squared over 2m,
00:28:32.100 --> 00:28:34.120
I did that already.
00:28:34.120 --> 00:28:36.720
I get a 1 over lambda cubed.
00:28:36.720 --> 00:28:39.760
So basically, from each
one of them I will get a 1
00:28:39.760 --> 00:28:40.645
over lambda cubed.
00:28:43.830 --> 00:28:47.470
But the integration is
shifted by this amount.
00:28:47.470 --> 00:28:50.690
Actually, I already did the
shifted integration here also
00:28:50.690 --> 00:28:53.060
for one particle.
00:28:53.060 --> 00:29:03.400
So I get the corresponding
factor of e to the minus-- ah.
00:29:03.400 --> 00:29:07.590
I have to be a little
bit careful over here
00:29:07.590 --> 00:29:14.450
because what I am integrating
is over k alpha squared.
00:29:14.450 --> 00:29:18.910
Whereas, in the way that
I have the list over here,
00:29:18.910 --> 00:29:22.830
I have x prime
alpha and x alpha,
00:29:22.830 --> 00:29:27.450
but a different k
playing around with each.
00:29:27.450 --> 00:29:28.360
What should I do?
00:29:28.360 --> 00:29:31.700
I really want this
integration over k
00:29:31.700 --> 00:29:36.620
alpha to look like
what I have over here.
00:29:36.620 --> 00:29:39.680
Well, as I sum over
all possibilities
00:29:39.680 --> 00:29:44.680
in each one of these terms, I
am bound to encounter k alpha.
00:29:44.680 --> 00:29:49.690
Essentially, I have permuted
all of the k's that I originally
00:29:49.690 --> 00:29:50.810
had.
00:29:50.810 --> 00:29:55.340
So the k alpha has now been
sent to some other location.
00:29:55.340 --> 00:30:00.170
But as I sum over all possible
alpha, I will hit that.
00:30:00.170 --> 00:30:03.540
When I hit that, I will
find that the thing that
00:30:03.540 --> 00:30:15.030
was multiplying k alpha is the
inverse permutation of alpha.
00:30:15.030 --> 00:30:18.880
And the thing that was
multiplying k alpha here
00:30:18.880 --> 00:30:23.390
is the inverse permutation of p.
00:30:29.140 --> 00:30:34.140
So then I can do the
integration over k alpha easily.
00:30:34.140 --> 00:30:35.570
And so what do I have?
00:30:35.570 --> 00:30:39.510
I have x prime of
p prime inverse
00:30:39.510 --> 00:30:43.780
alpha-- the inverse
permutation-- minus x
00:30:43.780 --> 00:30:49.180
of p inverse alpha squared
pi over lambda squared.
00:31:04.610 --> 00:31:09.190
Now, this is still
inconvenient because I
00:31:09.190 --> 00:31:14.860
am summing over two N
factorial sets of permutations.
00:31:14.860 --> 00:31:19.070
And I expect that since
the sum only involves
00:31:19.070 --> 00:31:24.610
comparison of things that
are occurring N times,
00:31:24.610 --> 00:31:29.690
as I go over the list of N
factorial permutation squared,
00:31:29.690 --> 00:31:33.240
I will get the same
thing appearing twice.
00:31:33.240 --> 00:31:37.590
So it is very much like when we
are doing an integration over x
00:31:37.590 --> 00:31:39.610
and x prime, but
the function only
00:31:39.610 --> 00:31:41.370
depends on x minus x prime.
00:31:41.370 --> 00:31:43.470
We get a factor of volume.
00:31:43.470 --> 00:31:47.250
Here, it is easy to
see that one of these
00:31:47.250 --> 00:31:54.470
sums I can very easily do
because it is just repetition
00:31:54.470 --> 00:31:57.390
of all of the results
that I have previously.
00:31:57.390 --> 00:32:01.400
And there will be N
factorial such terms.
00:32:01.400 --> 00:32:07.810
So doing that, I can get rid
of one of the N factorials.
00:32:07.810 --> 00:32:11.780
And I will have only
one permutation left,
00:32:11.780 --> 00:32:19.150
Q. And what will appear here
would be the parity of this Q
00:32:19.150 --> 00:32:21.520
that is the combination,
or if you like,
00:32:21.520 --> 00:32:25.600
the relative of these
two permutations.
00:32:25.600 --> 00:32:32.680
And I have an
exponential of minus sum
00:32:32.680 --> 00:32:41.190
over alpha x alpha minus
x prime Q alpha squared
00:32:41.190 --> 00:32:44.650
pi over lambda squared.
00:32:44.650 --> 00:32:50.173
And I think I forgot a factor
of lambda to the 3 [INAUDIBLE].
00:32:55.103 --> 00:32:57.075
This factor of lambda 3.
00:33:03.510 --> 00:33:05.970
So this is actually
the final result.
00:33:14.480 --> 00:33:21.090
And let's see what
that precisely
00:33:21.090 --> 00:33:22.510
means for two particles.
00:33:27.030 --> 00:33:28.560
So let's look at two particles.
00:33:32.810 --> 00:33:40.020
So for two particle,s I will
have on one side coordinates
00:33:40.020 --> 00:33:42.885
of 1 prime and 2 prime.
00:33:45.440 --> 00:33:49.525
On the right-hand side, I
have coordinates 1 and 2.
00:33:54.210 --> 00:33:57.702
And let's see what this
density matrix tells us.
00:33:57.702 --> 00:34:04.470
It tells us that to go from x1
prime x2 prime, a two particle
00:34:04.470 --> 00:34:11.540
density matrix connecting
to x1 x2 on the other side,
00:34:11.540 --> 00:34:15.670
I have 1 over the two-particle
partition function
00:34:15.670 --> 00:34:18.560
that i haven't yet calculated.
00:34:18.560 --> 00:34:20.920
Lambda to the sixth.
00:34:20.920 --> 00:34:24.699
N factorial in this case is 2.
00:34:24.699 --> 00:34:32.320
And then for two things,
there are two permutations.
00:34:32.320 --> 00:34:37.710
So the identity
maps 1 to 1, 2 to 2.
00:34:37.710 --> 00:34:39.560
And therefore, what
I will get here
00:34:39.560 --> 00:34:47.144
would be exponential of minus
x1' minus x1 prime squared
00:34:47.144 --> 00:34:55.070
pi over lambda squared minus
x2 minus x2 prime squared pi
00:34:55.070 --> 00:34:56.060
over lambda squared.
00:35:02.450 --> 00:35:07.610
So that's Q being
identity and identity
00:35:07.610 --> 00:35:10.470
has essentially 0 parity.
00:35:10.470 --> 00:35:14.120
It's an even permutation.
00:35:14.120 --> 00:35:17.410
The next thing is when
I exchange 1 and 2.
00:35:17.410 --> 00:35:19.650
That would have odd parity.
00:35:19.650 --> 00:35:23.570
So I would get minus 1 for
fermions, plus for bosons.
00:35:23.570 --> 00:35:28.460
And what I would get here
is exponential of minus x1'
00:35:28.460 --> 00:35:33.990
minus x2 prime squared pi
over lambda squared minus x2
00:35:33.990 --> 00:35:37.650
minus x1 prime squared
pi over lambda squared.
00:35:43.950 --> 00:35:49.020
So essentially, one of
the terms-- the first term
00:35:49.020 --> 00:35:53.320
is just the square of what I
had before for one particle.
00:35:53.320 --> 00:35:57.560
I take the one-particle result,
going from 1 to 1 prime,
00:35:57.560 --> 00:36:02.310
going from 2 to 2 prime
and multiply them together.
00:36:02.310 --> 00:36:07.700
But then you say, I can't tell
apart 2 prime and 1 prime.
00:36:07.700 --> 00:36:09.860
Maybe the thing that
you are calling 1 prime
00:36:09.860 --> 00:36:13.330
is really 2 prime
and vice versa.
00:36:13.330 --> 00:36:16.280
So I have to allow
for the possibility
00:36:16.280 --> 00:36:18.430
that rather than
x1 prime here, I
00:36:18.430 --> 00:36:23.080
should put x2 prime and
the other way around.
00:36:23.080 --> 00:36:27.560
And this also corresponds
to a permutation
00:36:27.560 --> 00:36:28.630
that is an exchange.
00:36:28.630 --> 00:36:33.840
It's an odd parity and will
give you something like that.
00:36:33.840 --> 00:36:38.780
Say OK, I have no
idea what that means.
00:36:38.780 --> 00:36:43.720
I'll tell you, OK, you were
happy when I put x prime and x
00:36:43.720 --> 00:36:45.640
here because that
was the probability
00:36:45.640 --> 00:36:48.790
to find the particle somewhere.
00:36:48.790 --> 00:36:50.800
So let me look at
the diagonal term
00:36:50.800 --> 00:36:52.650
here, which is a probability.
00:36:57.720 --> 00:36:59.520
This should give
me the probability
00:36:59.520 --> 00:37:02.120
to find one particle
at position x1,
00:37:02.120 --> 00:37:03.650
one particle at position x2.
00:37:06.990 --> 00:37:09.130
Because the particles
were non-interacting,
00:37:09.130 --> 00:37:10.890
one particle-- it
could be anywhere.
00:37:10.890 --> 00:37:15.010
I had the 1 over V.
Is it 1 over V squared
00:37:15.010 --> 00:37:17.240
or something like that?
00:37:17.240 --> 00:37:19.350
Well, we find that
there is factor
00:37:19.350 --> 00:37:21.520
out front that we
haven't yet evaluated.
00:37:21.520 --> 00:37:24.460
It turns out that this
factor will give me a 1
00:37:24.460 --> 00:37:27.070
over V squared.
00:37:27.070 --> 00:37:31.650
And if I set x1
prime to x1, x2 prime
00:37:31.650 --> 00:37:36.800
to x2, which is what I've done
here, this factor becomes 1.
00:37:36.800 --> 00:37:41.510
But then the other factor will
give me eta e to the minus 2
00:37:41.510 --> 00:37:46.590
pi over lambda squared
x1 minus x2 squared.
00:37:53.070 --> 00:37:58.960
So the physical probability
to find one particle-- or more
00:37:58.960 --> 00:38:05.240
correctly, a wave packet
here and a wave package there
00:38:05.240 --> 00:38:07.580
is not 1 over V squared.
00:38:07.580 --> 00:38:10.796
It's some function of the
separation between these two
00:38:10.796 --> 00:38:11.295
particles.
00:38:15.030 --> 00:38:18.410
So that separation
is contained here.
00:38:18.410 --> 00:38:23.290
If I really call that
separation to be r,
00:38:23.290 --> 00:38:27.210
this is an additional
weight that depends on r.
00:38:27.210 --> 00:38:30.650
This is r squared.
00:38:30.650 --> 00:38:38.410
So you can think of this
as an interaction, which
00:38:38.410 --> 00:38:41.140
is because solely of
quantum statistics.
00:38:44.880 --> 00:38:47.310
And what is this interaction?
00:38:47.310 --> 00:38:56.600
This interaction V of r
would be minus kT log of 1
00:38:56.600 --> 00:39:02.337
plus eta e to the minus 2 pi
r squared over lambda squared.
00:39:07.110 --> 00:39:11.670
I will plot out for you
what this V or r looks
00:39:11.670 --> 00:39:16.730
like as a function of how far
apart the centers of these two
00:39:16.730 --> 00:39:19.390
wave packets are.
00:39:19.390 --> 00:39:23.240
You can see that the
result depends on eta.
00:39:23.240 --> 00:39:27.850
If eta is minus 1, which is
for the case of fermions,
00:39:27.850 --> 00:39:30.330
this is 1 minus something.
00:39:30.330 --> 00:39:32.600
It's something that
is less than 1.
00:39:32.600 --> 00:39:34.960
[INAUDIBLE] would be negative.
00:39:34.960 --> 00:39:39.580
So the whole potential would
be positive, or repulsive.
00:39:39.580 --> 00:39:44.210
At large distances, indeed it
would be exponentially going
00:39:44.210 --> 00:39:48.840
to 0 because I can expand
the log at large distances.
00:39:48.840 --> 00:39:51.650
So here I have a term
that is minus 2 pi
00:39:51.650 --> 00:39:55.620
r squared over lambda squared.
00:39:55.620 --> 00:40:00.440
As I go towards r equals
to 0, actually things
00:40:00.440 --> 00:40:05.610
become very bad because r
goes to 0 I will get 1 minus 1
00:40:05.610 --> 00:40:07.310
and the log will diverge.
00:40:07.310 --> 00:40:14.290
So basically, there is, if you
like, an effective potential
00:40:14.290 --> 00:40:15.950
that says you
can't put these two
00:40:15.950 --> 00:40:19.460
fermions on top of each other.
00:40:19.460 --> 00:40:21.860
So there is a
statistical potential.
00:40:21.860 --> 00:40:26.670
So this is for eta
minus 1, or fermions.
00:40:30.430 --> 00:40:34.490
For the case of
bosons, eta plus 1.
00:40:34.490 --> 00:40:36.910
It is log of 1 plus
something, so it's
00:40:36.910 --> 00:40:39.385
a positive number
inside the log.
00:40:39.385 --> 00:40:43.520
The potential will
be attractive.
00:40:43.520 --> 00:40:49.660
And it will actually saturate
to a value of kT log 2
00:40:49.660 --> 00:40:51.360
when r goes to 0.
00:40:51.360 --> 00:40:57.850
So this is again, eta of plus
1 for the case of bosons.
00:41:38.320 --> 00:41:42.840
So the one thing that this
formula does not have yet
00:41:42.840 --> 00:41:46.480
is the value for this
partition function ZN.
00:41:46.480 --> 00:41:50.590
It gives you the qualitative
behavior in either case.
00:41:50.590 --> 00:41:53.390
And let's calculate what ZN is.
00:41:53.390 --> 00:41:56.320
Well, basically, that
would come from noting
00:41:56.320 --> 00:42:00.000
that the trace of
rho has to be 1.
00:42:00.000 --> 00:42:12.050
So ZN is trace of e
to the minus beta H.
00:42:12.050 --> 00:42:22.340
And essentially, I can take
this ZN to the other side
00:42:22.340 --> 00:42:35.110
and evaluate this as x
e to the minus beta H x.
00:42:35.110 --> 00:42:44.770
That is, I can calculate
the diagonal elements
00:42:44.770 --> 00:42:46.600
of this matrix that
I have calculated--
00:42:46.600 --> 00:42:48.830
that I have over there.
00:42:48.830 --> 00:42:52.260
So there is an
overall factor of 1
00:42:52.260 --> 00:42:59.430
over lambda cubed to the power
of N. I have N factorial.
00:43:02.250 --> 00:43:12.800
And then I have a sum over
permutations Q eta of Q.
00:43:12.800 --> 00:43:15.580
The diagonal element
is obtained by putting
00:43:15.580 --> 00:43:18.610
x prime to be the same as x.
00:43:18.610 --> 00:43:30.290
So I have exponential of minus
x-- sum over alpha x alpha
00:43:30.290 --> 00:43:33.130
minus x of Q alpha.
00:43:33.130 --> 00:43:36.620
I set x prime to
be the same as x.
00:43:36.620 --> 00:43:37.560
Squared.
00:43:37.560 --> 00:43:41.530
And then there's an overall
pi over lambda squared.
00:43:44.200 --> 00:43:50.030
And if I am taking the trace,
it means that I have to do
00:43:50.030 --> 00:43:52.580
integration over all x's.
00:44:03.640 --> 00:44:08.560
So I'm evaluating this trace
in coordinate basis, which
00:44:08.560 --> 00:44:11.040
means that I should
put x and x prime to be
00:44:11.040 --> 00:44:13.560
the same for the
trace, and then I
00:44:13.560 --> 00:44:19.970
have to sum or integrate over
all possible values of x.
00:44:19.970 --> 00:44:21.190
So let's do this.
00:44:21.190 --> 00:44:31.070
I have 1 over N factorial lambda
cubed raised to the power of N.
00:44:31.070 --> 00:44:31.900
OK.
00:44:31.900 --> 00:44:36.290
Now I have to make
a choice because I
00:44:36.290 --> 00:44:40.660
have a whole bunch of terms
because of these permutations.
00:44:43.180 --> 00:44:46.520
Let's do them one by one.
00:44:46.520 --> 00:44:51.120
Let's first do the case
where Q is identity.
00:44:51.120 --> 00:44:55.530
That is, I map
everybody to themselves.
00:44:55.530 --> 00:45:00.580
Actually, let me write down
the integrations first.
00:45:00.580 --> 00:45:06.330
I will do the integrations
over all pairs
00:45:06.330 --> 00:45:10.780
of coordinates of
these Gaussians.
00:45:10.780 --> 00:45:17.730
These Gaussians I will evaluate
for different permutations.
00:45:17.730 --> 00:45:22.160
Let's look at the case
where Q is identity.
00:45:22.160 --> 00:45:27.540
When Q is identity, essentially
I will put all of the x prime
00:45:27.540 --> 00:45:29.040
to be the same as x.
00:45:29.040 --> 00:45:33.600
It is like what I did here
for two particles and I got 1.
00:45:33.600 --> 00:45:36.960
I do the same thing for
more than one particle.
00:45:36.960 --> 00:45:38.210
I will still get 1.
00:45:42.740 --> 00:45:46.420
Then, I will do the same
thing that I did over here.
00:45:46.420 --> 00:45:52.370
Here, the next term that I
did was to exchange 1 and 2.
00:45:52.370 --> 00:45:56.070
So this became x1 minus x2.
00:45:56.070 --> 00:45:58.330
I'll do the same thing here.
00:45:58.330 --> 00:46:01.570
I look at the case
where Q corresponds
00:46:01.570 --> 00:46:05.340
to exchange of
particles 1 and 2.
00:46:05.340 --> 00:46:07.690
And then that will
give me a factor which
00:46:07.690 --> 00:46:15.180
is e to the minus pi over lambda
squared x1 minus x2 squared.
00:46:15.180 --> 00:46:19.110
There are two of these making
together 2 pi over lambda
00:46:19.110 --> 00:46:21.340
squared, which I hope
I had there, too.
00:46:25.420 --> 00:46:29.030
But then there was a whole bunch
of other terms that I can do.
00:46:29.030 --> 00:46:35.360
I can exchange,
let's say, 7 and 9.
00:46:35.360 --> 00:46:38.720
And then I will get
here 2 pi over lambda
00:46:38.720 --> 00:46:43.340
squared x7 minus x9 squared.
00:46:43.340 --> 00:46:46.230
And there's a whole
bunch of such exchanges
00:46:46.230 --> 00:46:50.220
that I can make in
which I just switch
00:46:50.220 --> 00:46:55.950
between two particles
in this whole story.
00:46:55.950 --> 00:47:00.840
And clearly, the number of
exchanges that I can make
00:47:00.840 --> 00:47:05.060
is the number of pairs,
N N minus 1 over 2.
00:47:08.830 --> 00:47:12.750
Once I am done with
all of the exchanges,
00:47:12.750 --> 00:47:14.780
then I have to go to
the next thing that
00:47:14.780 --> 00:47:18.120
doesn't have an analog
here for two particles.
00:47:18.120 --> 00:47:20.850
But if I take three
particles, I can permute them
00:47:20.850 --> 00:47:22.980
like a triangle.
00:47:22.980 --> 00:47:25.770
So presumably there
would be next set
00:47:25.770 --> 00:47:32.190
of terms, which is a permutation
that is like 1, 2, 3, 2, 3, 1.
00:47:32.190 --> 00:47:36.830
There's a bunch of things that
involve two permutations, four
00:47:36.830 --> 00:47:40.190
permutations, and so forth.
00:47:40.190 --> 00:47:42.600
So there is a whole
list of things
00:47:42.600 --> 00:47:47.710
that would go to here where
these two-particle exchanges
00:47:47.710 --> 00:47:49.190
are the simplest class.
00:47:52.970 --> 00:47:57.590
Now, as we shall see,
there is a systematic way
00:47:57.590 --> 00:48:02.540
of looking at things where
the two-particle exchanges are
00:48:02.540 --> 00:48:05.570
the first correction
due to quantum effects.
00:48:05.570 --> 00:48:09.300
Three-particle exchanges would
be higher-order corrections.
00:48:09.300 --> 00:48:14.880
And we can systematically
do them in order.
00:48:14.880 --> 00:48:19.685
So let's see what happens if
we compare the case where there
00:48:19.685 --> 00:48:24.210
is no exchange and the case
where there is one exchange.
00:48:24.210 --> 00:48:26.540
When there is no
exchange, I am essentially
00:48:26.540 --> 00:48:31.500
integrating over each
position over the volume.
00:48:31.500 --> 00:48:38.880
So what I would get is V
raised to the power of N.
00:48:38.880 --> 00:48:41.880
The next term?
00:48:41.880 --> 00:48:45.420
Well, I have to do
the integrations.
00:48:45.420 --> 00:48:51.120
The integrations over x3, x4,
x5, all the way to x to the N,
00:48:51.120 --> 00:48:52.270
there is no factors.
00:48:52.270 --> 00:48:56.790
So they will give me factors
of V. And there are N minus 2
00:48:56.790 --> 00:48:57.290
of them.
00:49:01.060 --> 00:49:04.600
And then I have to do the
integration over x1 and x2
00:49:04.600 --> 00:49:08.840
of this factor, but
it's only a function
00:49:08.840 --> 00:49:10.690
of the relative coordinate.
00:49:10.690 --> 00:49:12.650
So there is one
other integration
00:49:12.650 --> 00:49:17.690
that I can trivially do, which
is the center of mass gives me
00:49:17.690 --> 00:49:20.760
a factor of V.
And then I am left
00:49:20.760 --> 00:49:24.973
with the integral over the
relative coordinate of e
00:49:24.973 --> 00:49:30.170
to the minus 2 pi r squared
over lambda squared.
00:49:30.170 --> 00:49:34.070
And I forgot-- it's
very important.
00:49:34.070 --> 00:49:40.200
This will carry a factor of eta
because any exchange is odd.
00:49:40.200 --> 00:49:42.520
And so there will be
a factor of eta here.
00:49:45.950 --> 00:49:52.310
And I said that I would
get the same expression
00:49:52.310 --> 00:49:55.510
for any of my N N minus
1 over 2 exchanges.
00:49:59.230 --> 00:50:05.480
So the result of all of
these exchange calculations
00:50:05.480 --> 00:50:08.480
would be the same thing.
00:50:08.480 --> 00:50:10.580
And then there would
be the contribution
00:50:10.580 --> 00:50:12.520
from three-body
exchange and so forth.
00:50:19.460 --> 00:50:21.950
So let's re-organize this.
00:50:21.950 --> 00:50:27.630
I can pull out the factor
of V to the N outside.
00:50:27.630 --> 00:50:32.620
So I would have V over lambda
cubed to the power of N.
00:50:32.620 --> 00:50:35.710
So the first term is 1.
00:50:35.710 --> 00:50:38.830
The next term has
the parity factor
00:50:38.830 --> 00:50:42.460
that distinguishes
bosons and fermions, goes
00:50:42.460 --> 00:50:47.920
with a multiplicity of pairs
which is N N minus 1 over 2.
00:50:47.920 --> 00:50:51.230
Since I already pulled
out a factor of V to the N
00:50:51.230 --> 00:50:54.440
and I really had V to
the N minus 1 here,
00:50:54.440 --> 00:51:00.210
I better put a factor
of 1 over V here.
00:51:00.210 --> 00:51:02.420
And then I just am
left with having
00:51:02.420 --> 00:51:06.420
to evaluate these
Gaussian integrals.
00:51:06.420 --> 00:51:09.192
Each Gaussian
integral will give me
00:51:09.192 --> 00:51:11.750
2 pi times the
variance, which is
00:51:11.750 --> 00:51:16.620
lambda squared divided by 2 pi.
00:51:16.620 --> 00:51:21.040
And then there's
actually a factor of 2.
00:51:21.040 --> 00:51:25.260
And there are three of
them, so I will have 3/2.
00:51:25.260 --> 00:51:30.870
So what I get here is lambda
cubed divided by 2 to the 3/2.
00:51:39.580 --> 00:51:43.690
Now, you can see that
any time I go further
00:51:43.690 --> 00:51:47.240
in this series of
exchanges, I will
00:51:47.240 --> 00:51:50.440
have more of these
Gaussian factors.
00:51:50.440 --> 00:51:52.920
And whenever I have
a Gaussian factor,
00:51:52.920 --> 00:51:54.990
I have an additional
integration to do
00:51:54.990 --> 00:51:57.770
that has an x minus
something squared in it.
00:51:57.770 --> 00:52:01.990
I will lose a factor of V. I
don't have that factor of V.
00:52:01.990 --> 00:52:07.470
And so subsequent terms will
be even smaller in powers of V.
00:52:07.470 --> 00:52:12.240
And presumably, compensated by
corresponding factors of lambda
00:52:12.240 --> 00:52:13.550
squared-- lambda cubed.
00:52:27.610 --> 00:52:33.630
Now, first thing to note is
that in the very, very high
00:52:33.630 --> 00:52:37.540
temperature limit,
lambda goes to 0.
00:52:37.540 --> 00:52:40.680
So I can forget even
this correction.
00:52:40.680 --> 00:52:42.170
What do I get?
00:52:42.170 --> 00:52:47.770
I get 1 over N factorial V over
lambda cubed to the power of N.
00:52:47.770 --> 00:52:52.000
Remember that many, many
lectures back we introduced
00:52:52.000 --> 00:52:55.610
by hand the factor of 1 over
N factorial for measuring
00:52:55.610 --> 00:52:58.780
phase spaces of
identical particles.
00:52:58.780 --> 00:53:01.130
And I promise to you
that we would get it
00:53:01.130 --> 00:53:03.880
when we did identical
particles in quantum mechanics,
00:53:03.880 --> 00:53:05.940
so here it is.
00:53:05.940 --> 00:53:09.390
So automatically, we
did the calculation,
00:53:09.390 --> 00:53:12.030
keeping track of
identity of particles
00:53:12.030 --> 00:53:14.300
at the level of quantum states.
00:53:14.300 --> 00:53:17.400
Went through the calculation and
in the high temperature limit,
00:53:17.400 --> 00:53:22.410
we get this 1 over N
factorial emerging.
00:53:22.410 --> 00:53:26.770
Secondly, we see that the
corrections to ideal gas
00:53:26.770 --> 00:53:34.470
behavior emerge as a series in
powers of lambda cubed over V.
00:53:34.470 --> 00:53:40.120
And for example, if I were to
take the log of the partition
00:53:40.120 --> 00:53:45.550
function, I would
get log of what
00:53:45.550 --> 00:53:50.250
I would have had classically,
which is this V over lambda
00:53:50.250 --> 00:53:53.770
cubed to the power of N
divided by N factorial.
00:53:57.110 --> 00:54:01.750
And then the log
of this expression.
00:54:01.750 --> 00:54:05.070
And this I'm going to
replace with N squared.
00:54:05.070 --> 00:54:07.720
There is not that
much difference.
00:54:07.720 --> 00:54:13.630
And since I'm regarding this
as a correction, log of 1
00:54:13.630 --> 00:54:18.620
plus something, I will
replace with the something
00:54:18.620 --> 00:54:27.670
eta N squared 2V lambda cubed
2 to the 3/2 plus higher order.
00:54:30.580 --> 00:54:32.170
What does this mean?
00:54:32.170 --> 00:54:34.130
Once we have the
partition function,
00:54:34.130 --> 00:54:36.740
we can calculate pressure.
00:54:36.740 --> 00:54:42.885
Reminding you that beta
p was the log Z by dV.
00:54:45.615 --> 00:54:49.850
The first part is
the ideal gas that we
00:54:49.850 --> 00:54:52.040
had looked at classically.
00:54:52.040 --> 00:54:57.480
So once I go to the appropriate
large-end limit of this,
00:54:57.480 --> 00:55:03.330
what this gives me is
the density n over V.
00:55:03.330 --> 00:55:06.150
And then when I look
at the derivative here,
00:55:06.150 --> 00:55:11.400
the derivative of 1/V will give
me a minus 1 over V squared.
00:55:11.400 --> 00:55:15.630
So I will get minus eta.
00:55:15.630 --> 00:55:18.500
N over V, the whole
thing squared.
00:55:18.500 --> 00:55:25.180
So I will have n squared
lambda cubed 2 to the 5/2,
00:55:25.180 --> 00:55:26.600
and so forth.
00:55:30.730 --> 00:55:35.600
So I see that the
pressure of this ideal gas
00:55:35.600 --> 00:55:40.140
with no interactions
is already different
00:55:40.140 --> 00:55:43.525
from the classical result
that we had calculated
00:55:43.525 --> 00:55:47.800
by a factor that actually
reflects the statistics.
00:55:47.800 --> 00:55:52.660
For fermions eta of minus 1,
you get an additional pressure
00:55:52.660 --> 00:55:56.170
because of the kind of repulsion
that we have over here.
00:55:56.170 --> 00:55:59.640
Whereas, for bosons
you get an attraction.
00:55:59.640 --> 00:56:02.740
You can see that also the
thing that determines this--
00:56:02.740 --> 00:56:06.420
so basically, this
corresponds to a second Virial
00:56:06.420 --> 00:56:16.830
coefficient, which is minus
eta lambda cubed 2 to the 5/2,
00:56:16.830 --> 00:56:23.820
is the volume of
these wave packets.
00:56:23.820 --> 00:56:27.310
So essentially,
the corrections are
00:56:27.310 --> 00:56:32.720
of the order of n
lambda cubed that
00:56:32.720 --> 00:56:37.750
is within one of these wave
packets how many particles
00:56:37.750 --> 00:56:39.970
you will encounter.
00:56:39.970 --> 00:56:43.780
As you go to high temperature,
the wave packets shrink.
00:56:43.780 --> 00:56:48.550
As you go to low temperature,
the wave packets expand.
00:56:48.550 --> 00:56:52.010
If you like, the interactions
become more important
00:56:52.010 --> 00:56:55.147
and you get corrections
to ideal gas wave.
00:56:58.471 --> 00:57:01.393
AUDIENCE: You assume that
we can use perturbation,
00:57:01.393 --> 00:57:05.289
but the higher terms actually
had a factor [INAUDIBLE].
00:57:09.185 --> 00:57:13.110
And you can't really use
perturbation in that.
00:57:13.110 --> 00:57:14.570
PROFESSOR: OK.
00:57:14.570 --> 00:57:19.370
So what you are worried
about is the story here,
00:57:19.370 --> 00:57:27.410
that I took log of 1
plus something here
00:57:27.410 --> 00:57:31.700
and I'm interested in the
limit of n going to infinity,
00:57:31.700 --> 00:57:36.530
that finite density n over
V. So already in that limit,
00:57:36.530 --> 00:57:38.800
you would say that
this factor really
00:57:38.800 --> 00:57:42.080
is overwhelmingly
larger than that.
00:57:42.080 --> 00:57:46.220
And as you say, the next
factor will be even larger.
00:57:46.220 --> 00:57:50.490
So what is the justification
in all of this?
00:57:50.490 --> 00:57:54.990
We have already encountered
this same problem
00:57:54.990 --> 00:57:57.850
when we were doing
these perturbations
00:57:57.850 --> 00:57:59.710
due to interactions.
00:57:59.710 --> 00:58:08.932
And the answer is that what
you really want to ensure
00:58:08.932 --> 00:58:20.710
is that not log Z,
but Z has a form
00:58:20.710 --> 00:58:23.245
that is e to the N something.
00:58:25.985 --> 00:58:28.375
And that something
will have corrections,
00:58:28.375 --> 00:58:34.330
potentially that are powers
of N, the density, which
00:58:34.330 --> 00:58:39.490
is N over V. And if you try to
force it into a perturbation
00:58:39.490 --> 00:58:44.830
series such as this, naturally
things like this happen.
00:58:44.830 --> 00:58:46.820
What does that really mean?
00:58:46.820 --> 00:58:51.630
That really means that the
correct thing that you should
00:58:51.630 --> 00:58:55.860
be expanding is,
indeed, log Z. If you
00:58:55.860 --> 00:58:59.350
were to do the kind of
hand-waving that I did here
00:58:59.350 --> 00:59:03.950
and do the expansion for Z, if
you also try to do it over here
00:59:03.950 --> 00:59:06.360
you will generate
terms that look
00:59:06.360 --> 00:59:08.780
kind of at the wrong order.
00:59:08.780 --> 00:59:12.480
But higher order terms
that you would get
00:59:12.480 --> 00:59:17.120
would naturally conspire so
that when you evaluate log Z,
00:59:17.120 --> 00:59:19.870
they come out right.
00:59:19.870 --> 00:59:22.420
You have to do this correctly.
00:59:22.420 --> 00:59:24.120
And once you have
done it correctly,
00:59:24.120 --> 00:59:26.120
then you can rely
on the calculation
00:59:26.120 --> 00:59:28.430
that you did before
as an example.
00:59:28.430 --> 00:59:33.300
And we did it correctly when
we were doing these cluster
00:59:33.300 --> 00:59:38.440
expansions and the
corresponding calculation we
00:59:38.440 --> 00:59:41.500
did for Q. We saw how the
different diagrams were
00:59:41.500 --> 00:59:44.400
appearing in both
Q and the log Q,
00:59:44.400 --> 00:59:47.910
and how they could be
summed over in log Q.
00:59:47.910 --> 00:59:51.485
But indeed, this
mathematically looks awkward
00:59:51.485 --> 00:59:55.250
and I kind of jumped a
step in writing log of 1
00:59:55.250 --> 01:00:00.492
plus something that is huge
as if it was a small number.
01:00:14.550 --> 01:00:17.300
All right.
01:00:17.300 --> 01:00:20.870
So we have a problem.
01:00:20.870 --> 01:00:23.410
We want to calculate
the simplest
01:00:23.410 --> 01:00:26.940
system, which is the ideal gas.
01:00:26.940 --> 01:00:30.680
So classically, we did
all of our calculations
01:00:30.680 --> 01:00:32.210
first for the ideal gas.
01:00:32.210 --> 01:00:34.030
We had exact results.
01:00:34.030 --> 01:00:36.220
Then, let's say we
had interactions.
01:00:36.220 --> 01:00:39.100
We did perturbations around
that and all of that.
01:00:39.100 --> 01:00:43.690
And we saw that having to do
things for interacting systems
01:00:43.690 --> 01:00:46.540
is very difficult.
01:00:46.540 --> 01:00:49.960
Now, when we start to do
calculations for the quantum
01:00:49.960 --> 01:00:53.810
problem, at least in the way
that I set it up for you,
01:00:53.810 --> 01:00:57.250
it seems that quantum
problems are inherently
01:00:57.250 --> 01:01:00.640
interacting problems.
01:01:00.640 --> 01:01:03.850
I showed you that even at
the level of two particles,
01:01:03.850 --> 01:01:08.820
it is like having an interaction
between bosons and fermions.
01:01:08.820 --> 01:01:11.070
For three particles,
it becomes even worse
01:01:11.070 --> 01:01:14.890
because it's not only the
two-particle interaction.
01:01:14.890 --> 01:01:16.920
Because of the
three-particle exchanges,
01:01:16.920 --> 01:01:19.720
you would get an additional
three-particle interaction,
01:01:19.720 --> 01:01:24.020
four-particle interaction,
all of these things emerge.
01:01:24.020 --> 01:01:27.550
So really, if you
want to look at this
01:01:27.550 --> 01:01:32.870
from the perspective of
a partition function,
01:01:32.870 --> 01:01:36.520
we already see that
the exchange term
01:01:36.520 --> 01:01:38.800
involved having to
do a calculation that
01:01:38.800 --> 01:01:42.500
is equivalent to calculating
the second Virial
01:01:42.500 --> 01:01:46.780
coefficient for an
interacting system.
01:01:46.780 --> 01:01:49.570
The next one, for the
third Virial coefficient,
01:01:49.570 --> 01:01:52.550
I would need to look at
the three-body exchanges,
01:01:52.550 --> 01:01:56.570
kind of like the point
clusters, four-point clusters,
01:01:56.570 --> 01:01:59.510
all kinds of other
things are there.
01:01:59.510 --> 01:02:00.410
So is there any hope?
01:02:05.150 --> 01:02:11.300
And the answer is that it is
all a matter of perspective.
01:02:11.300 --> 01:02:22.560
And somehow it is true
that these particles
01:02:22.560 --> 01:02:26.890
in quantum mechanics
because of the statistics
01:02:26.890 --> 01:02:31.310
are subject to all kinds of
complicated interactions.
01:02:31.310 --> 01:02:34.250
But also, the
underlying Hamiltonian
01:02:34.250 --> 01:02:37.000
is simple and non-interacting.
01:02:37.000 --> 01:02:39.870
We can enumerate all
of the wave functions.
01:02:39.870 --> 01:02:41.460
Everything is simple.
01:02:41.460 --> 01:02:44.770
So by looking at things
in the right basis,
01:02:44.770 --> 01:02:49.470
we should be able to calculate
everything that we need.
01:02:49.470 --> 01:02:55.290
So here, I was kind of looking
at calculating the partition
01:02:55.290 --> 01:03:00.590
function in the coordinate
basis, which is the worst case
01:03:00.590 --> 01:03:06.390
scenario because the Hamiltonian
is diagonal in the momentum
01:03:06.390 --> 01:03:07.860
basis.
01:03:07.860 --> 01:03:16.690
So let's calculate ZN trace
of e to the minus beta H
01:03:16.690 --> 01:03:19.630
in the basis in
which H is diagonal.
01:03:23.860 --> 01:03:28.240
So what are the eigenvalues
and eigenfunctions?
01:03:28.240 --> 01:03:32.890
Well, the eigenfunctions are
the symmetrized/anti-symmetrized
01:03:32.890 --> 01:03:34.000
quantities.
01:03:34.000 --> 01:03:37.170
The eigenvalues are simply
e to the minus beta H
01:03:37.170 --> 01:03:40.890
bar squared k alpha
squared over 2m.
01:03:40.890 --> 01:03:43.820
So this is basically
the thing that I
01:03:43.820 --> 01:03:47.690
could write as the set
of k's appropriately
01:03:47.690 --> 01:03:52.370
symmetrized or anti-symmetrized
e to the minus beta
01:03:52.370 --> 01:04:00.060
sum over alpha H bar squared
k alpha squared over 2m k eta.
01:04:06.230 --> 01:04:11.710
Actually, I'm going to-- rather
than go through this procedure
01:04:11.710 --> 01:04:15.530
that we have up there
in which I wrote
01:04:15.530 --> 01:04:22.860
these, what I need to do
here is a sum over all k
01:04:22.860 --> 01:04:25.070
in order to evaluate the trace.
01:04:25.070 --> 01:04:31.200
So this is inherently a
sum over all sets of k's.
01:04:31.200 --> 01:04:35.180
But this sum is
restricted, just like what
01:04:35.180 --> 01:04:38.790
I had indicated for you before.
01:04:38.790 --> 01:04:42.190
Rather than trying
to do it that way,
01:04:42.190 --> 01:04:48.180
I note that these
k's I could also
01:04:48.180 --> 01:04:52.860
write in terms of these
occupation numbers.
01:04:52.860 --> 01:04:57.200
So equivalently,
my basis would be
01:04:57.200 --> 01:05:04.370
the set of occupation
numbers times the energy.
01:05:04.370 --> 01:05:13.400
The energy is then e to the
minus beta sum over k epsilon k
01:05:13.400 --> 01:05:20.830
nk, where epsilon
k is this beta H
01:05:20.830 --> 01:05:23.650
bar squared k alpha
squared over 2m.
01:05:23.650 --> 01:05:28.760
But I could do this in
principle for any epsilon k
01:05:28.760 --> 01:05:30.800
that I have over here.
01:05:30.800 --> 01:05:35.230
So the result that I am writing
for you is more general.
01:05:35.230 --> 01:05:37.120
Then I sandwich it
again, since I'm
01:05:37.120 --> 01:05:40.365
calculating the trace,
with the same state.
01:05:43.360 --> 01:05:46.900
Now, the states have
this restriction
01:05:46.900 --> 01:05:47.970
that I have over there.
01:05:47.970 --> 01:05:54.070
That is, for the case of
fermions, my nk can be 0 or 1.
01:05:54.070 --> 01:05:58.440
But there is no restriction
for nk on the bosons.
01:05:58.440 --> 01:06:03.360
Except, of course, that there
is this overall restriction
01:06:03.360 --> 01:06:11.670
that the sum over
k nk has to be N
01:06:11.670 --> 01:06:14.865
because I am looking
at N-particle states.
01:06:22.810 --> 01:06:29.610
Actually, I can remove
this because in this basis,
01:06:29.610 --> 01:06:32.560
e to the minus
beta H is diagonal.
01:06:32.560 --> 01:06:39.330
So I can, basically,
remove these entities.
01:06:39.330 --> 01:06:41.715
And I'm just summing a
bunch of exponentials.
01:06:47.860 --> 01:06:55.160
So that is good because I
should be able to do for each nk
01:06:55.160 --> 01:06:57.580
a sum of e to something nk.
01:07:00.410 --> 01:07:03.150
Well, the problem is this
that I can't sum over
01:07:03.150 --> 01:07:04.553
each nk independently.
01:07:08.990 --> 01:07:14.660
Essentially in the picture
that I have over here,
01:07:14.660 --> 01:07:16.730
I have some n1 here.
01:07:16.730 --> 01:07:18.870
I have some n2 here.
01:07:18.870 --> 01:07:21.660
Some n3 here, which are
the occupation numbers
01:07:21.660 --> 01:07:24.260
of these things.
01:07:24.260 --> 01:07:26.290
And for that
partition function, I
01:07:26.290 --> 01:07:29.760
have to do the sum of
these exponentials e
01:07:29.760 --> 01:07:33.760
to the minus epsilon 1 n1,
e to the minus epsilon 2 n2.
01:07:33.760 --> 01:07:40.140
But the sum of all of these
n's is kind of maxed out by N.
01:07:40.140 --> 01:07:46.092
I cannot independently sum
over them going over the entire
01:07:46.092 --> 01:07:47.740
range.
01:07:47.740 --> 01:07:52.745
But we've seen previously
how those constraints can
01:07:52.745 --> 01:07:55.400
be removed in
statistical mechanics.
01:07:55.400 --> 01:07:56.790
So our usual trick.
01:07:56.790 --> 01:08:01.910
We go to the ensemble in
which n can take any value.
01:08:01.910 --> 01:08:06.040
So we go to the grand
canonical prescription.
01:08:12.880 --> 01:08:19.479
We remove this constraint on n
by evaluating a grand partition
01:08:19.479 --> 01:08:30.800
function Q, which is a sum over
all N of e to the beta mu N ZN.
01:08:30.800 --> 01:08:34.240
So we do, essentially,
a Laplace transform.
01:08:34.240 --> 01:08:38.300
We exchange our n with
the chemical potential mu.
01:08:41.850 --> 01:08:47.620
Then, this constraint no
longer we need to worry about.
01:08:47.620 --> 01:08:55.390
So now I can sum over all
of the nk's without worrying
01:08:55.390 --> 01:09:00.140
about any constraint, provided
that I multiply with e
01:09:00.140 --> 01:09:03.560
to the beta mu n, which
is a sum over k nk.
01:09:07.770 --> 01:09:10.660
And then, the factor
that I have here,
01:09:10.660 --> 01:09:15.771
which is e to the minus beta
sum over k epsilon of k nk.
01:09:22.180 --> 01:09:28.319
So essentially for each
k, I can independently
01:09:28.319 --> 01:09:38.359
sum over its nk of e to the
beta mu minus epsilon of k nk.
01:09:45.189 --> 01:09:49.069
Now, the symmetry issues remain.
01:09:49.069 --> 01:09:53.180
This answer still
depends on whether or not
01:09:53.180 --> 01:09:57.870
I am calculating things
for bosons or fermions
01:09:57.870 --> 01:10:03.566
because these sums are
differently constrained
01:10:03.566 --> 01:10:06.200
whether I'm dealing
with fermions.
01:10:06.200 --> 01:10:09.420
In which case, nk
is only 0 or 1.
01:10:09.420 --> 01:10:12.075
Or bosons, in which case
there is no constraint.
01:10:17.190 --> 01:10:20.020
So what do I get?
01:10:20.020 --> 01:10:28.540
For the case of fermions,
I have a Q minus,
01:10:28.540 --> 01:10:33.020
which is product over all k.
01:10:33.020 --> 01:10:39.580
And for each k, the nk
takes either 0 or 1.
01:10:39.580 --> 01:10:45.500
So if it takes 0, I will
write e to the 0, which is 1.
01:10:45.500 --> 01:10:46.465
Or it takes 1.
01:10:46.465 --> 01:10:50.562
It is e to the beta
mu minus epsilon of k.
01:10:57.110 --> 01:11:04.640
For the case of bosons,
I have a Q plus.
01:11:04.640 --> 01:11:10.370
Q plus is, again, a product
over Q. In this case,
01:11:10.370 --> 01:11:15.240
nk going from 0 to infinity, I
am summing a geometric series
01:11:15.240 --> 01:11:20.320
that starts as 1, and
then the subsequent terms
01:11:20.320 --> 01:11:26.950
are smaller by a factor of
beta mu minus epsilon of k.
01:11:26.950 --> 01:11:31.260
Actually, for future
reference note
01:11:31.260 --> 01:11:35.350
that I would be able to
do this geometric sum
01:11:35.350 --> 01:11:39.830
provided that this combination
beta mu minus epsilon of k
01:11:39.830 --> 01:11:41.130
is negative.
01:11:41.130 --> 01:11:44.570
So that the subsequent terms
in this series are decaying.
01:11:51.890 --> 01:11:54.680
Typically, we would be
interested in things
01:11:54.680 --> 01:11:58.660
like partition functions,
grand partition functions.
01:11:58.660 --> 01:12:05.620
So we have something like log of
Q, which would be a sum over k.
01:12:09.760 --> 01:12:15.600
And I would have either
the log of this quantity
01:12:15.600 --> 01:12:20.330
or the log of this
quantity with a minus sign.
01:12:20.330 --> 01:12:23.180
I can combine the
two results together
01:12:23.180 --> 01:12:28.910
by putting a factor of minus
eta because in taking the log,
01:12:28.910 --> 01:12:32.794
over here for the bosons I
would pick a factor of minus 1
01:12:32.794 --> 01:12:34.460
because the thing is
in the denominator.
01:12:37.100 --> 01:12:42.310
And then I would
write the log of 1.
01:12:42.310 --> 01:12:45.030
And then I have in both
cases, a factor which
01:12:45.030 --> 01:12:49.810
is e to the beta mu
minus epsilon of k.
01:12:49.810 --> 01:12:52.050
But occurring with
different signs
01:12:52.050 --> 01:12:54.850
for the bosons and
fermions, which again I
01:12:54.850 --> 01:12:58.890
can combine into a single
expression by putting a minus
01:12:58.890 --> 01:13:01.920
eta here.
01:13:01.920 --> 01:13:10.890
So this is a general
result for any Hamiltonian
01:13:10.890 --> 01:13:14.210
that has the characteristic
that we wrote over here.
01:13:14.210 --> 01:13:16.960
So this does not have to
be particles in a box.
01:13:16.960 --> 01:13:20.350
It could be particles in
a harmonic oscillator.
01:13:20.350 --> 01:13:23.750
These could be energy levels
of a harmonic oscillator.
01:13:23.750 --> 01:13:27.470
All you need to do is to
make the appropriate sum
01:13:27.470 --> 01:13:30.520
over the one-particle
levels harmonic oscillator,
01:13:30.520 --> 01:13:34.010
or whatever else you have,
of these factors that
01:13:34.010 --> 01:13:37.650
depend on the
individual energy levels
01:13:37.650 --> 01:13:39.120
of the one-particle system.
01:13:50.720 --> 01:13:52.790
Now, one of the things
that we will encounter
01:13:52.790 --> 01:13:56.870
having made this transition
from canonical, where we knew
01:13:56.870 --> 01:14:01.070
how many particles we had, to
grand canonical, where we only
01:14:01.070 --> 01:14:04.510
know the chemical potential,
is that we would ultimately
01:14:04.510 --> 01:14:09.360
want to express things in terms
of the number of particles.
01:14:09.360 --> 01:14:16.070
So it makes sense to
calculate how many particles
01:14:16.070 --> 01:14:21.060
you have given that you have
fixed the chemical potential.
01:14:21.060 --> 01:14:42.985
So for that we
note the following.
01:14:45.560 --> 01:14:55.410
That essentially, we were able
to do this calculation for Q
01:14:55.410 --> 01:14:58.580
because it was a
product of contributions
01:14:58.580 --> 01:15:04.560
that we had for the individual
one-particle states.
01:15:04.560 --> 01:15:10.510
So clearly, as far as this
normalization is concerned,
01:15:10.510 --> 01:15:15.680
the individual one-particle
states are independent.
01:15:15.680 --> 01:15:20.752
And indeed, what we can say
is that in this ensemble,
01:15:20.752 --> 01:15:27.610
there is a classical probability
for a set of occupation numbers
01:15:27.610 --> 01:15:31.550
of one particle
states, which is simply
01:15:31.550 --> 01:15:37.960
a product over the different
one-particle states of e
01:15:37.960 --> 01:15:49.830
to the beta mu minus epsilon
k nk appropriately normalized.
01:15:49.830 --> 01:15:54.010
And again, the
restriction on n's
01:15:54.010 --> 01:15:57.200
being 0 or 1 for
fermions or anything
01:15:57.200 --> 01:16:01.100
for bosons would be
implicit in either case.
01:16:01.100 --> 01:16:05.830
But in either case, essentially
the occupation numbers
01:16:05.830 --> 01:16:08.690
are independently taken
from distributions
01:16:08.690 --> 01:16:11.810
that I've discussed [INAUDIBLE].
01:16:11.810 --> 01:16:13.860
So you can, in
fact, independently
01:16:13.860 --> 01:16:17.350
calculate the average
occupation number
01:16:17.350 --> 01:16:22.460
that you have for each one of
these single-particle states.
01:16:22.460 --> 01:16:29.790
And it's clear that you could
get that by, for example,
01:16:29.790 --> 01:16:34.150
bringing down a
factor of nk here.
01:16:34.150 --> 01:16:37.190
And you can bring
down a factor of nk
01:16:37.190 --> 01:16:44.170
by taking a derivative of Q
with respect to beta epsilon k
01:16:44.170 --> 01:16:48.420
with a minus sign
and normalizing it,
01:16:48.420 --> 01:16:50.632
so you would have log.
01:16:50.632 --> 01:16:53.350
So you would have an
expression such as this.
01:16:57.250 --> 01:16:59.890
So you basically would
need to calculate,
01:16:59.890 --> 01:17:01.960
since you are taking
derivative with respect
01:17:01.960 --> 01:17:08.740
to epsilon k the corresponding
log for which epsilon
01:17:08.740 --> 01:17:10.500
k appears.
01:17:10.500 --> 01:17:12.890
Actually, for the
case of fermions,
01:17:12.890 --> 01:17:14.890
really there are
two possibilities.
01:17:14.890 --> 01:17:18.360
n is either 0 or 1.
01:17:18.360 --> 01:17:21.260
So you would say that the
expectation value would
01:17:21.260 --> 01:17:27.440
be when it is 1, you have e to
the beta epsilon of k minus mu.
01:17:27.440 --> 01:17:28.370
Oops.
01:17:28.370 --> 01:17:31.550
e to the beta mu
minus epsilon of k.
01:17:34.330 --> 01:17:36.910
The two possibilities
are 1 plus e
01:17:36.910 --> 01:17:40.450
to the beta mu
minus epsilon of k.
01:17:40.450 --> 01:17:44.630
So when I look at
some particular state,
01:17:44.630 --> 01:17:46.030
it is either empty.
01:17:46.030 --> 01:17:48.830
In which case, contributes 0.
01:17:48.830 --> 01:17:51.140
Or, it is occupied.
01:17:51.140 --> 01:17:53.362
In which case, it
contributes this weight,
01:17:53.362 --> 01:17:55.070
which has to be
appropriately normalized.
01:17:57.650 --> 01:18:01.460
If I do the same thing
for the case of bosons,
01:18:01.460 --> 01:18:04.400
it is a bit more
complicated because I
01:18:04.400 --> 01:18:07.490
have to look at this series
rather than geometric 1
01:18:07.490 --> 01:18:10.440
plus x plus x squared
plus x cubed is
01:18:10.440 --> 01:18:14.970
1 plus x plus 2x squared plus
3x cubed, which can be obtained
01:18:14.970 --> 01:18:19.430
by taking the derivative
of the appropriate log.
01:18:19.430 --> 01:18:22.750
Or you can fall back
on your calculations
01:18:22.750 --> 01:18:27.160
of geometric series
and convince yourself
01:18:27.160 --> 01:18:29.880
that it is essentially the same
thing with a factor of minus
01:18:29.880 --> 01:18:30.380
here.
01:18:34.850 --> 01:18:40.798
So this is fermions
and this is bosons.
01:18:45.480 --> 01:18:52.000
And indeed, I can put the
two expressions together
01:18:52.000 --> 01:18:57.390
by dividing through this
factor in both of them
01:18:57.390 --> 01:19:03.380
and write it as 1
over Z inverse e
01:19:03.380 --> 01:19:08.700
to the beta epsilon
of k minus eta,
01:19:08.700 --> 01:19:11.560
where for convenience
I have introduced
01:19:11.560 --> 01:19:16.117
Z to be the contribution
e to the beta.
01:19:30.100 --> 01:19:36.530
So for this system of
non-interacting particles
01:19:36.530 --> 01:19:40.090
that are identical,
we have expressions
01:19:40.090 --> 01:19:45.890
for log of the grand partition
function, the grand potential.
01:19:45.890 --> 01:19:48.220
And for the average
number of particles,
01:19:48.220 --> 01:19:51.260
which is an appropriate
derivative of this,
01:19:51.260 --> 01:19:54.410
expressed in terms of the
single-particle energy
01:19:54.410 --> 01:19:58.440
levels and the
chemical potential.
01:19:58.440 --> 01:20:01.710
So next time, what
we will do is we
01:20:01.710 --> 01:20:03.880
will start this
with this expression
01:20:03.880 --> 01:20:06.576
for the case of the
particles in a box
01:20:06.576 --> 01:20:11.350
to get the pressure of
the ideal quantum gas
01:20:11.350 --> 01:20:13.510
as a function of mu.
01:20:13.510 --> 01:20:15.090
But we want to
write the pressure
01:20:15.090 --> 01:20:19.260
as a function of density, so
we will invert this expression
01:20:19.260 --> 01:20:21.610
to get density as a
function of-- chemical
01:20:21.610 --> 01:20:24.620
potential as a function of
density [INAUDIBLE] here.
01:20:24.620 --> 01:20:27.580
And therefore, get the
expression for pressure
01:20:27.580 --> 01:20:30.620
as a function of density.