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PROFESSOR: So last
lecture, we laid out
00:00:26.000 --> 00:00:28.700
the foundations of
quantum stat mech.
00:00:36.565 --> 00:00:39.700
And to remind you,
in quantum mechanics
00:00:39.700 --> 00:00:45.830
we said we could very
formally describe a state
00:00:45.830 --> 00:00:49.950
through a vector
in Hilbert space
00:00:49.950 --> 00:00:53.080
that has complex components
in terms of some basis
00:00:53.080 --> 00:00:55.570
that we haven't specified.
00:00:55.570 --> 00:00:58.850
And quantum mechanics
describes, essentially
00:00:58.850 --> 00:01:01.170
what happens to one
of these states,
00:01:01.170 --> 00:01:05.000
how to interpret it, et cetera.
00:01:05.000 --> 00:01:08.700
But this was, in our
language, a pure state.
00:01:08.700 --> 00:01:13.480
We were interested in cases
where we have an ensemble
00:01:13.480 --> 00:01:16.620
and there are many members
of the ensemble that
00:01:16.620 --> 00:01:19.790
correspond to the same
macroscopic state.
00:01:19.790 --> 00:01:24.050
And we can distinguish
them macroscopically.
00:01:24.050 --> 00:01:27.510
If we think about each
one of those states
00:01:27.510 --> 00:01:32.070
being one particular
member, psi alpha,
00:01:32.070 --> 00:01:35.760
some vector in this
Hilbert space occurring
00:01:35.760 --> 00:01:41.060
with some probability
P alpha, then that's
00:01:41.060 --> 00:01:44.210
kind of a more
probabilistic representation
00:01:44.210 --> 00:01:49.470
of what's going on here
for the mech state.
00:01:49.470 --> 00:01:52.460
And the question was, well,
how do we manipulate things
00:01:52.460 --> 00:01:58.020
when we have a collection of
states in this mixed form?
00:01:58.020 --> 00:02:01.060
And we saw that in
quantum mechanics,
00:02:01.060 --> 00:02:03.255
most of the things
that are observable
00:02:03.255 --> 00:02:06.420
are expressed in
terms of matrices.
00:02:06.420 --> 00:02:14.820
And one way to convert a
state vector into a matrix
00:02:14.820 --> 00:02:23.190
is to conjugate it and construct
the matrix whose elements are
00:02:23.190 --> 00:02:27.470
obtained-- a particular rho
vector by taking element
00:02:27.470 --> 00:02:31.260
and complex conjugate
elements of the vector.
00:02:31.260 --> 00:02:38.360
And if we were to then sum over
all members of this ensemble,
00:02:38.360 --> 00:02:42.410
this would give us an object
that we call the density
00:02:42.410 --> 00:02:45.220
matrix, rho.
00:02:45.220 --> 00:02:48.340
And the property of
this density matrix
00:02:48.340 --> 00:02:53.970
was that if I had some
observable for a pure state
00:02:53.970 --> 00:02:56.640
I would be able to
calculate an expectation
00:02:56.640 --> 00:03:00.010
value for this observable
by sandwiching it
00:03:00.010 --> 00:03:03.910
between the state in the way
that quantum mechanic has
00:03:03.910 --> 00:03:05.460
taught us.
00:03:05.460 --> 00:03:10.450
In our mech state, we would
get an ensemble average
00:03:10.450 --> 00:03:16.420
of this quantity by
multiplying this density matrix
00:03:16.420 --> 00:03:21.100
with the matrix that represents
the operator whose average we
00:03:21.100 --> 00:03:24.630
are trying to calculate,
and then taking
00:03:24.630 --> 00:03:27.155
the trace of the product
of these two matrices.
00:03:32.290 --> 00:03:35.980
Now, the next statement
was that in the same way
00:03:35.980 --> 00:03:40.100
that the micro-state classically
changes as a function of time,
00:03:40.100 --> 00:03:42.630
the vector that we have
in quantum mechanics
00:03:42.630 --> 00:03:46.300
is also changing as
a function of time.
00:03:46.300 --> 00:03:49.570
And so at a different
instant of time,
00:03:49.570 --> 00:03:52.230
our state vectors have changed.
00:03:52.230 --> 00:03:56.970
And in principle, our
density has changed.
00:03:56.970 --> 00:04:03.085
And that would imply a change
potentially in the average
00:04:03.085 --> 00:04:04.320
that we have over here.
00:04:06.870 --> 00:04:10.410
We said, OK, let's look
at this time dependence.
00:04:10.410 --> 00:04:15.540
And we know that these psis
obey Schrodinger equation.
00:04:15.540 --> 00:04:19.480
I h bar d psi by dt is h psi.
00:04:19.480 --> 00:04:22.630
So we did the same
operation, I h bar d
00:04:22.630 --> 00:04:26.850
by dt of this density matrix.
00:04:26.850 --> 00:04:29.290
And we found that
essentially, what we
00:04:29.290 --> 00:04:33.410
would get on the other
side is the commutator
00:04:33.410 --> 00:04:35.840
of the density
matrix and the matrix
00:04:35.840 --> 00:04:39.690
corresponding to
the Hamiltonian,
00:04:39.690 --> 00:04:47.100
which was reminiscent of
the Liouville statement
00:04:47.100 --> 00:04:49.020
that we had classically.
00:04:49.020 --> 00:04:53.090
Classically, we had
that d rho by dt
00:04:53.090 --> 00:04:59.391
was the Poisson bracket of a
rho with H versus this quantum
00:04:59.391 --> 00:04:59.890
formulation.
00:05:03.300 --> 00:05:07.410
Now, in both cases
we are looking
00:05:07.410 --> 00:05:10.940
for some kind of a density
that represents an equilibrium
00:05:10.940 --> 00:05:13.180
ensemble.
00:05:13.180 --> 00:05:14.920
And presumably,
the characteristic
00:05:14.920 --> 00:05:17.650
of the equilibrium ensemble
is that various measurements
00:05:17.650 --> 00:05:21.500
that you make at one time and
another time are the same.
00:05:21.500 --> 00:05:24.390
And hence, we want
this rho equilibrium
00:05:24.390 --> 00:05:27.920
to be something that does not
change as a function of time,
00:05:27.920 --> 00:05:31.600
which means that if we put it in
either one of these equations,
00:05:31.600 --> 00:05:35.560
the right-hand side for
d rho by dt should be 0.
00:05:35.560 --> 00:05:39.000
And clearly, a
simple way to do that
00:05:39.000 --> 00:05:45.040
is to make rho equilibrium
be a function of H.
00:05:45.040 --> 00:05:50.930
In the classical context, H
was a function in phase space.
00:05:50.930 --> 00:05:54.000
Rho equilibrium was,
therefore, made a function
00:05:54.000 --> 00:05:57.000
of various points in
phase space implicitly
00:05:57.000 --> 00:06:01.450
through its dependence on
H. In the current case,
00:06:01.450 --> 00:06:04.450
H is a matrix in Hilbert space.
00:06:04.450 --> 00:06:07.070
A function of a matrix
is another function,
00:06:07.070 --> 00:06:09.610
and that's how we
construct rho equilibrium.
00:06:09.610 --> 00:06:12.510
And that function will
also commute with H
00:06:12.510 --> 00:06:15.740
because clearly H with H is 0.
00:06:15.740 --> 00:06:19.350
Any function of H
with H will be 0.
00:06:19.350 --> 00:06:23.400
So the prescription that
we have in order now
00:06:23.400 --> 00:06:27.460
to construct the quantum
statistical mechanics
00:06:27.460 --> 00:06:33.960
is to follow what we had done
already for the classical case.
00:06:33.960 --> 00:06:36.485
And classically
following this equation,
00:06:36.485 --> 00:06:40.800
we made postulates
relating rho--
00:06:40.800 --> 00:06:42.710
what its functional
dependence was
00:06:42.710 --> 00:06:47.930
on H. We can now do the same
thing in the quantum context.
00:06:47.930 --> 00:06:51.360
So let's do that
following the procedure
00:06:51.360 --> 00:06:55.480
that we followed for
the classical case.
00:06:55.480 --> 00:06:57.820
So classically, we
said, well, let's look
00:06:57.820 --> 00:07:00.090
at an ensemble that
was micro-canonical.
00:07:05.940 --> 00:07:09.500
So in the
micro-canonical ensemble,
00:07:09.500 --> 00:07:13.630
we specified what the
energy of the system was,
00:07:13.630 --> 00:07:16.880
but we didn't allow either
work to be performed
00:07:16.880 --> 00:07:21.440
on it mechanically or chemically
so that the number of elements
00:07:21.440 --> 00:07:22.200
was fixed.
00:07:22.200 --> 00:07:25.060
The coordinates, such as
volume, length of the system,
00:07:25.060 --> 00:07:26.810
et cetera, were fixed.
00:07:26.810 --> 00:07:30.620
And in this ensemble,
classically the statement
00:07:30.620 --> 00:07:39.170
was that rho indexed by
E is a function of H.
00:07:39.170 --> 00:07:42.600
And clearly, what we
want is to only allow
00:07:42.600 --> 00:07:46.370
H's that correspond
to the right energy.
00:07:46.370 --> 00:07:50.070
So I will use a kind
of shorthand notation
00:07:50.070 --> 00:07:52.150
as kind of a delta H E.
00:07:52.150 --> 00:07:57.130
Although, when we were doing
it last time around, we allowed
00:07:57.130 --> 00:08:00.660
this function to allow a
range of possible E-values.
00:08:00.660 --> 00:08:03.850
I can do the same thing, it's
just that writing that out
00:08:03.850 --> 00:08:05.690
is slightly less convenient.
00:08:05.690 --> 00:08:08.630
So let's stick with
the convenient form.
00:08:08.630 --> 00:08:12.550
Essentially, it says
within some range delta A
00:08:12.550 --> 00:08:15.600
allow the Hamiltonians.
00:08:15.600 --> 00:08:18.240
Allow states, micro-states,
whose Hamiltonian
00:08:18.240 --> 00:08:22.530
would give the right energy
that is comparable to the energy
00:08:22.530 --> 00:08:27.280
that we have said exists
for the macro-state.
00:08:27.280 --> 00:08:30.040
Now clearly also,
this definition
00:08:30.040 --> 00:08:34.190
tells me that the trace
of rho has to be 1.
00:08:34.190 --> 00:08:38.679
Or classically, the integral
of rho E over the entire phase
00:08:38.679 --> 00:08:40.179
space has to be 1.
00:08:40.179 --> 00:08:43.210
So there is a
normalization condition.
00:08:43.210 --> 00:08:47.890
And this normalization condition
gave us the quantity omega
00:08:47.890 --> 00:08:50.290
of E, the number of
states of energy E,
00:08:50.290 --> 00:08:53.110
out of which we then
constructed the entropy
00:08:53.110 --> 00:08:56.870
and we were running
away calculating
00:08:56.870 --> 00:09:01.050
various thermodynamic
quantities.
00:09:01.050 --> 00:09:07.400
So now let's see how we evaluate
this in the quantum case.
00:09:07.400 --> 00:09:10.330
We will use the same
expression, but now me
00:09:10.330 --> 00:09:14.570
realize that rho is a matrix.
00:09:14.570 --> 00:09:18.090
So I have to evaluate, maybe
elements of that matrix
00:09:18.090 --> 00:09:22.390
to clarify what this matrix
looks like in some basis.
00:09:22.390 --> 00:09:25.190
What's the most
convenient basis?
00:09:25.190 --> 00:09:28.570
Since rho is expressed
as a function of H,
00:09:28.570 --> 00:09:40.880
the most convenient basis
is the energy basis,
00:09:40.880 --> 00:09:45.750
which is the basis that
diagonalizes your Hamiltonian
00:09:45.750 --> 00:09:46.790
matrix.
00:09:46.790 --> 00:09:54.250
Basically, there's some
vectors in this Hilbert space
00:09:54.250 --> 00:09:56.630
such that the
action of H on this
00:09:56.630 --> 00:09:59.400
will give us some
energy that I will
00:09:59.400 --> 00:10:01.885
call epsilon n, some eigenvalue.
00:10:01.885 --> 00:10:02.590
An n.
00:10:02.590 --> 00:10:06.730
So that's the definition
of the energy basis.
00:10:06.730 --> 00:10:11.930
Again, as all basis, we can
make these basis vectors
00:10:11.930 --> 00:10:17.660
n to be unit length and
orthogonal to each other.
00:10:17.660 --> 00:10:20.660
There is an orthonormal basis.
00:10:20.660 --> 00:10:25.590
If I evaluate rho E in
this basis, what do I find?
00:10:25.590 --> 00:10:32.075
I find that n rho m.
00:10:32.075 --> 00:10:34.310
Well, 1 over omega E
is just the constant.
00:10:34.310 --> 00:10:35.285
It comes out front.
00:10:39.280 --> 00:10:43.930
And the meaning of this delta
function becomes obvious.
00:10:43.930 --> 00:10:48.400
It is 1 or 0.
00:10:48.400 --> 00:11:06.840
It is 1 if, let's say, Em
equals to the right energy.
00:11:06.840 --> 00:11:11.970
Em is the right energy
for the ensemble E.
00:11:11.970 --> 00:11:15.150
And of course, there
is a delta function.
00:11:15.150 --> 00:11:21.460
So there is also
an m equals to n.
00:11:21.460 --> 00:11:26.560
And it is 0, clearly, for states
that have the wrong energy.
00:11:32.040 --> 00:11:35.710
But there is an
additional thing here
00:11:35.710 --> 00:11:39.795
that I will explain shortly
for m not equal to n.
00:11:43.520 --> 00:11:45.860
So let's parse
the two statements
00:11:45.860 --> 00:11:48.670
that I have made over here.
00:11:48.670 --> 00:11:54.070
The first one, it
says that if I know
00:11:54.070 --> 00:11:57.410
the energy of my
macro-state, I clearly
00:11:57.410 --> 00:12:02.290
have to find wave
functions to construct
00:12:02.290 --> 00:12:05.230
possible states of
what is in the box that
00:12:05.230 --> 00:12:08.060
have the right energy.
00:12:08.060 --> 00:12:12.700
States that don't have the
right energy are not admitted.
00:12:12.700 --> 00:12:16.380
States that are
the right energy,
00:12:16.380 --> 00:12:19.570
I have nothing against
each one of them.
00:12:19.570 --> 00:12:23.410
So I give them all
the same probability.
00:12:23.410 --> 00:12:36.740
So this is our whole
assumption of equal equilibrium
00:12:36.740 --> 00:12:39.940
a priori probabilities.
00:12:46.490 --> 00:12:52.480
But now I have a quantum system
and I'm looking at the matrix.
00:12:52.480 --> 00:12:58.750
And this matrix potentially
has off-diagonal elements.
00:12:58.750 --> 00:13:02.370
You see, if I am looking
at m equals to n,
00:13:02.370 --> 00:13:04.890
it means that I am looking
at the diagonal elements
00:13:04.890 --> 00:13:07.280
of the matrix.
00:13:07.280 --> 00:13:11.720
What this says is that if the
energies are degenerate-- let's
00:13:11.720 --> 00:13:16.610
say I have 100 states, all of
them have the right energy,
00:13:16.610 --> 00:13:19.080
but they are orthonormal.
00:13:19.080 --> 00:13:23.670
These 100 states, when I
look at the density matrix
00:13:23.670 --> 00:13:26.930
in the corresponding basis,
the off-diagonal elements
00:13:26.930 --> 00:13:29.760
would be 0.
00:13:29.760 --> 00:13:31.310
So this is the "or."
00:13:31.310 --> 00:13:34.210
So even if this
condition is satisfied,
00:13:34.210 --> 00:13:37.400
even if Em equals to
E, but I am looking
00:13:37.400 --> 00:13:41.370
at off-diagonal elements,
I have to put 0.
00:13:41.370 --> 00:13:44.255
And this is sometimes called
the assumption of random phases.
00:14:00.260 --> 00:14:02.030
Before telling you
why that's called
00:14:02.030 --> 00:14:05.380
an assumption of random
phases, let's also
00:14:05.380 --> 00:14:09.340
characterize what
this omega of E is.
00:14:09.340 --> 00:14:17.640
Because trace of rho has
to be 1, clearly omega of E
00:14:17.640 --> 00:14:22.010
is the trace of this
delta h E. Essentially
00:14:22.010 --> 00:14:26.510
as I scan all of my
possible energy levels
00:14:26.510 --> 00:14:30.910
in this energy
basis, I will get 1
00:14:30.910 --> 00:14:34.310
for those that have the
right energy and 0 otherwise.
00:14:34.310 --> 00:14:43.540
So basically, this is simply
number of states of energy E,
00:14:43.540 --> 00:14:45.960
potentially with minus
plus some delta E
00:14:45.960 --> 00:14:47.203
if I want to include that.
00:14:50.990 --> 00:14:56.220
Now, what these assumptions
mean are kind of like this.
00:14:56.220 --> 00:15:00.160
So I have my box
and I have been told
00:15:00.160 --> 00:15:04.350
that I have energy E. What's
a potential wave function
00:15:04.350 --> 00:15:07.060
that I can have in this box?
00:15:07.060 --> 00:15:12.270
What's the psi given that I
know what the energy E is?
00:15:12.270 --> 00:15:17.310
Well, any superposition
of these omega E states
00:15:17.310 --> 00:15:21.460
that have the right
energy will work fine.
00:15:21.460 --> 00:15:26.370
So I have a sum
over, let's say, mu
00:15:26.370 --> 00:15:32.090
that belongs to the
state such that H-- such
00:15:32.090 --> 00:15:36.770
that this energy En
is equal to the energy
00:15:36.770 --> 00:15:38.010
that I have specified.
00:15:38.010 --> 00:15:41.920
And there are omega
sub E such states.
00:15:41.920 --> 00:15:46.880
I have to find some kind of
an amplitude for these states,
00:15:46.880 --> 00:15:50.397
and then the
corresponding state mu.
00:15:53.260 --> 00:15:55.561
I guess I should
write here E mu.
00:15:59.980 --> 00:16:05.460
Now, the first
statement over here
00:16:05.460 --> 00:16:08.540
is that as far as
I'm concerned, I
00:16:08.540 --> 00:16:12.850
can put all of these a
mu's in whatever proportion
00:16:12.850 --> 00:16:13.690
that I want.
00:16:13.690 --> 00:16:16.830
Any linear combination
will work out fine.
00:16:16.830 --> 00:16:19.780
Because ultimately, psi
has to be normalized.
00:16:19.780 --> 00:16:23.680
Presumably, the
typical magnitude a m
00:16:23.680 --> 00:16:27.380
squared, if I average over
all members of the ensemble,
00:16:27.380 --> 00:16:30.340
should be 1 over omega.
00:16:30.340 --> 00:16:32.400
So this is a superposition.
00:16:32.400 --> 00:16:37.930
Typically, all of
them would contribute.
00:16:37.930 --> 00:16:41.800
But since we are thinking
about quantum mechanics,
00:16:41.800 --> 00:16:44.040
these amplitudes can,
in fact, be complex.
00:16:47.700 --> 00:16:51.440
And this statement
of random phases
00:16:51.440 --> 00:16:54.030
is more or less
equivalent to saying
00:16:54.030 --> 00:17:00.440
that the phases of
the different elements
00:17:00.440 --> 00:17:03.680
would be typically
uncorrelated when you average
00:17:03.680 --> 00:17:06.308
over all possible
members of this ensemble.
00:17:11.790 --> 00:17:15.140
Just to emphasize this
a little bit more,
00:17:15.140 --> 00:17:18.849
let's think about the
very simplest case
00:17:18.849 --> 00:17:23.520
that we can have for
thinking about probability.
00:17:23.520 --> 00:17:26.810
And you would think
of, say, a coin that
00:17:26.810 --> 00:17:30.090
can have two possibilities,
head or tail.
00:17:30.090 --> 00:17:33.290
So classically, you
would say, head or tail,
00:17:33.290 --> 00:17:37.120
not knowing anything
else, are equally likely.
00:17:37.120 --> 00:17:40.480
The quantum analog of
that is a quantum bit.
00:17:43.270 --> 00:17:47.770
And the qubit can have,
let's say, up or down states.
00:17:47.770 --> 00:17:52.540
It's a Hilbert space that
is composed of two elements.
00:17:52.540 --> 00:17:58.870
So the corresponding matrix that
I would have would be a 2 by 2.
00:17:58.870 --> 00:18:01.260
And according to the
construction that I have,
00:18:01.260 --> 00:18:02.880
it will be something like this.
00:18:07.220 --> 00:18:11.530
What are the possible wave
functions for this system?
00:18:15.110 --> 00:18:17.440
I can have any
linear combination
00:18:17.440 --> 00:18:26.760
of, say, up and down,
with any amplitude here.
00:18:26.760 --> 00:18:29.740
And so essentially, the
amplitudes, presumably,
00:18:29.740 --> 00:18:35.980
are quantities that I will
call alpha up and alpha down.
00:18:35.980 --> 00:18:40.820
That, on average, alpha squared
up or down would be 1/2.
00:18:40.820 --> 00:18:43.470
That's what would appear here.
00:18:43.470 --> 00:18:45.560
The elements that
appear here according
00:18:45.560 --> 00:18:48.440
to the construction
that I have up there,
00:18:48.440 --> 00:18:51.020
I have to really
take this element,
00:18:51.020 --> 00:18:54.940
average it against the complex
conjugate of that element.
00:18:54.940 --> 00:18:58.310
So what will appear here
would be something like e
00:18:58.310 --> 00:19:04.910
to the i phi of up minus phi
of down, where, let's say,
00:19:04.910 --> 00:19:07.201
I put here phi of up.
00:19:10.640 --> 00:19:14.840
And what I'm saying
is that there
00:19:14.840 --> 00:19:17.655
are a huge number
of possibilities.
00:19:17.655 --> 00:19:23.810
Whereas, the classical coin has
really two possibilities, head
00:19:23.810 --> 00:19:27.000
or tail, the quantum
analog of this
00:19:27.000 --> 00:19:30.010
is a huge set of possibilities.
00:19:30.010 --> 00:19:33.270
These phis can be
anything, 0 to 2 pi.
00:19:33.270 --> 00:19:35.580
Independent of each
other, the amplitudes
00:19:35.580 --> 00:19:38.000
can be anything as long as
the eventual normalization
00:19:38.000 --> 00:19:39.720
is satisfied.
00:19:39.720 --> 00:19:42.760
And as you sort of sum
over all possibilities,
00:19:42.760 --> 00:19:44.737
you would get something
like this also.
00:19:56.920 --> 00:20:05.580
Now, the more convenient
ensemble for calculating things
00:20:05.580 --> 00:20:13.830
is the canonical one, where
rather than specifying
00:20:13.830 --> 00:20:16.410
what the energy
of the system is,
00:20:16.410 --> 00:20:18.510
I tell you what
the temperature is.
00:20:18.510 --> 00:20:21.150
I still don't allow
work to take place,
00:20:21.150 --> 00:20:24.750
so these other
elements we kept fixed.
00:20:24.750 --> 00:20:30.850
And our classical
description for rho sub T
00:20:30.850 --> 00:20:35.880
was e to the minus
beta H divided
00:20:35.880 --> 00:20:40.611
by some partition function,
where beta was 1 over kT,
00:20:40.611 --> 00:20:41.110
of course.
00:20:51.000 --> 00:20:56.640
So again, in the energy
basis, this would be diagonal.
00:20:56.640 --> 00:21:00.720
And the diagonal elements would
be e to the minus beta epsilon
00:21:00.720 --> 00:21:02.960
n.
00:21:02.960 --> 00:21:06.760
I could calculate the
normalization Z, which
00:21:06.760 --> 00:21:10.270
would be trace of e
to the minus beta H.
00:21:10.270 --> 00:21:13.605
This trace is calculated
most easily the basis
00:21:13.605 --> 00:21:15.980
in which H is diagonal.
00:21:15.980 --> 00:21:19.590
And then I just pick out all
of the diagonal elements, sum
00:21:19.590 --> 00:21:21.670
over n e to the
minus beta epsilon.
00:21:28.790 --> 00:21:32.140
Now if you recall, we already
did something like this
00:21:32.140 --> 00:21:34.900
without justification
where we said
00:21:34.900 --> 00:21:38.510
that the states of a harmonic
oscillator, we postulated
00:21:38.510 --> 00:21:42.870
to be quantized h
bar omega n plus 1/2.
00:21:42.870 --> 00:21:45.460
And then we calculated
the partition function
00:21:45.460 --> 00:21:48.310
by summing over all states.
00:21:48.310 --> 00:21:54.460
You can see that
this would provide
00:21:54.460 --> 00:21:56.120
the justification for that.
00:22:04.950 --> 00:22:11.200
Now, various things that we have
in classical formulations also
00:22:11.200 --> 00:22:17.180
work, such that
classically if we had Z,
00:22:17.180 --> 00:22:19.500
we could take the
log of Z. We could
00:22:19.500 --> 00:22:23.210
take a derivative of log
Z with respect to beta.
00:22:23.210 --> 00:22:26.180
It would bring
down a factor of H.
00:22:26.180 --> 00:22:33.880
And then we show that this
was equal to the average
00:22:33.880 --> 00:22:36.210
of the energy.
00:22:36.210 --> 00:22:38.770
It was the average of
Hamiltonian, which we then
00:22:38.770 --> 00:22:42.850
would identify with the
thermodynamic energy.
00:22:42.850 --> 00:22:49.030
It is easy to show that if you
take the same set of operations
00:22:49.030 --> 00:22:56.960
over here, what you would
get is trace of H rho, which
00:22:56.960 --> 00:23:00.888
is the definition of the
average that you find.
00:23:15.070 --> 00:23:21.550
Now, let's do one example
in the canonical ensemble,
00:23:21.550 --> 00:23:27.526
which is particle in box.
00:23:31.590 --> 00:23:36.530
Basically, this is the kind of
thing that we did all the time.
00:23:36.530 --> 00:23:40.780
We assume that there is
some box of volume v.
00:23:40.780 --> 00:23:46.120
And for the time being, I
just put one particle in it.
00:23:46.120 --> 00:23:48.900
Assume that there
is no potential.
00:23:48.900 --> 00:23:52.290
So the Hamiltonian
for this one particle
00:23:52.290 --> 00:23:56.250
is just its kinetic
energy p1 squared over 2m,
00:23:56.250 --> 00:23:58.160
maybe plus some kind
of a box potential.
00:24:03.410 --> 00:24:06.430
Now, if I want to think about
this quantum mechanically,
00:24:06.430 --> 00:24:09.000
I have to think about
some kind of a basis
00:24:09.000 --> 00:24:11.950
and calculating things
in some kind of basis.
00:24:11.950 --> 00:24:14.450
So for the time
being, let's imagine
00:24:14.450 --> 00:24:17.215
that I do things in
coordinate basis.
00:24:22.730 --> 00:24:27.930
So I want to have
where the particle is.
00:24:27.930 --> 00:24:30.050
And let's call
the coordinates x.
00:24:30.050 --> 00:24:33.140
Let's say, a vector x that
has x, y, and z-components.
00:24:36.480 --> 00:24:40.020
Then, in the
coordinate basis, p is
00:24:40.020 --> 00:24:44.940
h bar over i, the
gradient vector.
00:24:44.940 --> 00:24:51.240
So this becomes minus h bar
squared Laplacian over 2m.
00:24:54.900 --> 00:25:01.690
And if I want to
look at what are
00:25:01.690 --> 00:25:07.860
the eigenstates of his
Hamiltonian represented
00:25:07.860 --> 00:25:11.270
in coordinates basis,
as usual I want
00:25:11.270 --> 00:25:17.725
to have something like H1 acting
on some function of position
00:25:17.725 --> 00:25:20.370
giving me the energy.
00:25:20.370 --> 00:25:29.250
And so I will indicate that
function of position as x k.
00:25:33.696 --> 00:25:37.740
And I want to know
what that energy is.
00:25:41.780 --> 00:25:44.710
And the reason I do
that is because you all
00:25:44.710 --> 00:25:51.510
know that the correct form
of these eigenfunctions
00:25:51.510 --> 00:25:55.350
are things like e
to the i k dot x.
00:25:59.560 --> 00:26:04.500
And since I want these to be
normalized when I integrate
00:26:04.500 --> 00:26:14.950
over all coordinates, I have
to divide by square root of V.
00:26:14.950 --> 00:26:18.340
So that's all nice and fine.
00:26:18.340 --> 00:26:23.420
What I want to do is to
calculate the density matrix
00:26:23.420 --> 00:26:27.040
in this coordinate basis.
00:26:27.040 --> 00:26:33.175
So I pick some
point x prime rho x.
00:26:35.740 --> 00:26:37.440
And it's again, a one particle.
00:26:37.440 --> 00:26:37.940
Yes?
00:26:37.940 --> 00:26:40.290
AUDIENCE: I was thinking
the box to be infinite
00:26:40.290 --> 00:26:45.610
or-- I mean, how to treat a
boundary because [INAUDIBLE].
00:26:45.610 --> 00:26:48.810
PROFESSOR: So I wasn't
careful about that.
00:26:48.810 --> 00:26:52.530
The allowed values of k
that I have to pick here
00:26:52.530 --> 00:26:55.550
will reflect the
boundary conditions.
00:26:55.550 --> 00:26:59.520
And indeed, I will be able
to use these kinds of things.
00:26:59.520 --> 00:27:03.530
Let's say, by using
periodic boundary condition.
00:27:07.390 --> 00:27:10.190
And if I choose periodic
boundary conditions,
00:27:10.190 --> 00:27:15.470
the allowed values of
k will be quantized.
00:27:15.470 --> 00:27:19.475
So that, say, kx
would be multiples
00:27:19.475 --> 00:27:24.195
of 2 pi over Lx
times some integer.
00:27:27.660 --> 00:27:32.000
If I use really box boundary
conditions such as the wave
00:27:32.000 --> 00:27:34.970
function having to
vanish at the two ends,
00:27:34.970 --> 00:27:38.270
in reality I should use
the sine and cosine type
00:27:38.270 --> 00:27:42.990
of boundary wave functions,
which are superpositions
00:27:42.990 --> 00:27:46.600
of these e to the i kx
and e to the minus i kx.
00:27:46.600 --> 00:27:50.950
And they slightly change
the normalization--
00:27:50.950 --> 00:27:53.706
the discretization.
00:27:53.706 --> 00:27:57.210
But up to, again, these kinds
of superpositions and things
00:27:57.210 --> 00:28:01.590
like that, this is a
good enough statement.
00:28:01.590 --> 00:28:03.790
If you are really doing
quantum mechanics,
00:28:03.790 --> 00:28:07.650
you have to clearly
be more careful.
00:28:07.650 --> 00:28:12.550
Just make sure that I'm
sufficiently careful, but not
00:28:12.550 --> 00:28:16.595
overly careful in
what I'm doing next.
00:28:20.750 --> 00:28:23.590
So what do I need to do?
00:28:23.590 --> 00:28:31.350
Well, rho 1 is
nicely diagonalized
00:28:31.350 --> 00:28:36.880
in the basis of
these plane waves.
00:28:36.880 --> 00:28:41.110
In these eigenstates
of the Hamiltonian.
00:28:41.110 --> 00:28:48.250
So it makes sense for me to
switch from this representation
00:28:48.250 --> 00:28:52.670
that is characterized
by x and x prime
00:28:52.670 --> 00:28:58.390
to a representation
where I have k.
00:28:58.390 --> 00:29:03.770
And so I can do that by
inserting a sum over k here.
00:29:03.770 --> 00:29:05.680
And then I have rho 1.
00:29:05.680 --> 00:29:08.755
Rho 1 is simply e
to the minus beta.
00:29:08.755 --> 00:29:13.870
It is diagonal in
this basis of k's.
00:29:13.870 --> 00:29:19.420
There is a normalization that
we call Z1, partition function.
00:29:19.420 --> 00:29:22.190
And because it is
diagonal, in fact
00:29:22.190 --> 00:29:26.044
I will use the same k
on both sides of this.
00:29:28.950 --> 00:29:33.930
So the density matrix is, in
fact, diagonal in momentum.
00:29:33.930 --> 00:29:36.400
This is none other than
momentum, of course.
00:29:39.090 --> 00:29:43.175
But I don't want to calculate
the density matrix in momentum
00:29:43.175 --> 00:29:47.000
where it is diagonal because
that's the energy basis we
00:29:47.000 --> 00:29:51.360
already saw, but in
coordinate basis.
00:29:51.360 --> 00:29:54.560
So now let's write
all of these things.
00:29:54.560 --> 00:29:59.020
First of all, the sum over
k when l becomes large,
00:29:59.020 --> 00:30:01.770
I'm going to replace
with an integral over k.
00:30:05.230 --> 00:30:07.860
And then I have to
worry about the spacing
00:30:07.860 --> 00:30:10.710
that I have between k-values.
00:30:10.710 --> 00:30:14.650
So that will multiply with
a density of states, which
00:30:14.650 --> 00:30:18.740
is 2 pi cubed product
of lx, ly, lz,
00:30:18.740 --> 00:30:20.680
which will give me
the volume of the box.
00:30:23.390 --> 00:30:29.790
And then I have these factors
of-- how did I define it?
00:30:29.790 --> 00:30:36.310
kx is e to the i k dot x.
00:30:36.310 --> 00:30:44.880
And this thing where k and
x prime are interchanged
00:30:44.880 --> 00:30:46.480
is its complex conjugate.
00:30:46.480 --> 00:30:49.410
So it becomes x minus x prime.
00:30:49.410 --> 00:30:52.940
I have two factors of 1
over square root of V,
00:30:52.940 --> 00:30:55.650
so that will give me
a factor of 1 over V
00:30:55.650 --> 00:30:57.760
from the product
of these things.
00:30:57.760 --> 00:31:01.940
And then I still
have minus beta h bar
00:31:01.940 --> 00:31:03.990
squared k squared over 2m.
00:31:16.282 --> 00:31:17.670
And then I have the Z1.
00:31:29.390 --> 00:31:32.130
Before proceeding, maybe
it's worthwhile for me
00:31:32.130 --> 00:31:36.530
to calculate the
Z1 in this case.
00:31:36.530 --> 00:31:45.160
So Z1 is the trace of rho 1.
00:31:45.160 --> 00:31:46.550
It's essentially the same thing.
00:31:46.550 --> 00:31:53.870
It's a sum over k e
to the minus beta h
00:31:53.870 --> 00:31:55.450
bar squared k squared over 2m.
00:31:59.400 --> 00:32:01.920
I do the same thing
that I did over there,
00:32:01.920 --> 00:32:08.010
replace this with V integral
over k divided by 2 pi cubed
00:32:08.010 --> 00:32:12.150
e to the minus beta h bar
squared k squared over 2m.
00:32:17.040 --> 00:32:20.465
I certainly recognize h bar k
as the same thing as momentum.
00:32:20.465 --> 00:32:24.630
This is really p squared.
00:32:24.630 --> 00:32:28.900
So just let me write
this after rescaling
00:32:28.900 --> 00:32:31.730
all of the k's by h bar.
00:32:31.730 --> 00:32:37.210
So then what I would have is
an integral over momentum.
00:32:37.210 --> 00:32:41.120
And then for each
k, I essentially
00:32:41.120 --> 00:32:44.110
bring down a factor of h bar.
00:32:44.110 --> 00:32:45.700
What is 2 pi h bar?
00:32:45.700 --> 00:32:48.050
2 pi h bar is h.
00:32:48.050 --> 00:32:51.960
So I would have a
factor of h cubed.
00:32:51.960 --> 00:32:57.910
I have an integration over
space that gave me volume.
00:32:57.910 --> 00:33:01.650
e to the minus beta h
bar-- p squared over 2m.
00:33:07.570 --> 00:33:10.660
Why did I write it this way?
00:33:10.660 --> 00:33:15.580
Because it should remind
you of how we made
00:33:15.580 --> 00:33:18.890
dimensionless the
integrations that we
00:33:18.890 --> 00:33:22.150
had to do for
classical calculations.
00:33:22.150 --> 00:33:24.000
Classical partition
function, if I
00:33:24.000 --> 00:33:27.380
wanted to calculate
within a single particle,
00:33:27.380 --> 00:33:29.500
for a single
particle Hamiltonian,
00:33:29.500 --> 00:33:31.150
I would have this
Boltzmann weight.
00:33:31.150 --> 00:33:35.450
I have to integrate over all
coordinates, over all momenta.
00:33:35.450 --> 00:33:40.110
And then we divided by h
to make it dimensionless.
00:33:40.110 --> 00:33:43.320
So you can see that that
naturally appears here.
00:33:43.320 --> 00:33:47.290
And so the answer to
this is V over lambda
00:33:47.290 --> 00:33:51.180
cubed with the lambda
that we had defined
00:33:51.180 --> 00:33:54.590
before as h over root 2 pi m kT.
00:34:00.630 --> 00:34:12.969
So you can see that this
traces that we write down
00:34:12.969 --> 00:34:17.090
in this quantum
formulations are clearly
00:34:17.090 --> 00:34:19.420
dimensionless quantities.
00:34:19.420 --> 00:34:22.489
And they go over to
the classical limit
00:34:22.489 --> 00:34:25.780
where we integrate over
phase space appropriately
00:34:25.780 --> 00:34:29.040
dimensional-- made
dimensionless by dividing
00:34:29.040 --> 00:34:33.590
by h per combination of p and q.
00:34:33.590 --> 00:34:39.260
So this V1 we already
know is V-- Z1
00:34:39.260 --> 00:34:42.070
we already know is
V over lambda cubed.
00:34:42.070 --> 00:34:44.080
The reason I got a
little bit confused
00:34:44.080 --> 00:34:47.489
was because I saw that
the V's were disappearing.
00:34:47.489 --> 00:34:49.739
And then I remembered,
oh, Z1 is actually
00:34:49.739 --> 00:34:52.050
proportional to mu
over lambda cubed.
00:34:52.050 --> 00:34:53.239
So that's good.
00:34:53.239 --> 00:34:58.690
So what we have here from
the inverse of Z1 is a 1
00:34:58.690 --> 00:35:00.620
over V lambda cubed here.
00:35:04.500 --> 00:35:08.430
Now, then I have to complete
this kind of integration,
00:35:08.430 --> 00:35:11.780
which are Gaussian integrations.
00:35:11.780 --> 00:35:15.700
If x was equal to
x prime, this would
00:35:15.700 --> 00:35:20.380
be precisely the integration
that I'm doing over here.
00:35:20.380 --> 00:35:23.910
So if x was equal to x
prime, that integration,
00:35:23.910 --> 00:35:26.020
the Gaussian integration,
would have given me
00:35:26.020 --> 00:35:29.570
this 1 over lambda cubed.
00:35:29.570 --> 00:35:34.350
But because I have this
shift-- in some sense
00:35:34.350 --> 00:35:40.950
I have to shift k to remove
that shift in x minus x prime.
00:35:40.950 --> 00:35:43.170
And when I square
it, I will get here
00:35:43.170 --> 00:35:48.360
a factor, which is exponential,
of minus x minus x prime
00:35:48.360 --> 00:35:51.700
squared-- the Square
of that factor-- times
00:35:51.700 --> 00:35:58.070
the variance of k, which
is m kT over h bar squared.
00:35:58.070 --> 00:35:59.440
So what do I get here?
00:35:59.440 --> 00:36:01.540
I will get 2.
00:36:01.540 --> 00:36:05.430
And then I would get
essentially m kT.
00:36:08.780 --> 00:36:10.874
And then, h bar squared here.
00:36:19.550 --> 00:36:26.910
Now, this combination m kT
divided by h, same thing
00:36:26.910 --> 00:36:29.010
as here, m kT and h.
00:36:29.010 --> 00:36:33.840
Clearly, what that is, is
giving me some kind of a lambda.
00:36:33.840 --> 00:36:36.130
So let me write the answer.
00:36:36.130 --> 00:36:40.250
Eventually, what I
find is that evaluating
00:36:40.250 --> 00:36:45.210
the one-particle density
matrix between two points, x
00:36:45.210 --> 00:36:50.825
and x prime, will
give me 1 over V.
00:36:50.825 --> 00:36:54.320
And then this
exponential factor,
00:36:54.320 --> 00:36:58.050
which is x minus
x prime squared.
00:36:58.050 --> 00:37:02.185
The coefficient here has to be
proportional to lambda squared.
00:37:02.185 --> 00:37:04.205
And if you do the
algebra right, you
00:37:04.205 --> 00:37:10.590
will find that the
coefficient is pi over 2.
00:37:10.590 --> 00:37:11.370
Or is it pi?
00:37:11.370 --> 00:37:14.650
Let me double check.
00:37:14.650 --> 00:37:17.640
It's a numerical factor
that is not that important.
00:37:17.640 --> 00:37:19.860
It is pi, pi over
lambda squared.
00:37:28.900 --> 00:37:33.750
OK, so what does this mean?
00:37:33.750 --> 00:37:36.150
So I have a particle in the box.
00:37:36.150 --> 00:37:40.990
It's a quantum mechanical
particle at some temperature T.
00:37:40.990 --> 00:37:44.150
And I'm trying to
ask, where is it?
00:37:44.150 --> 00:37:45.970
So if I want to ask
where is it, what's
00:37:45.970 --> 00:37:49.230
the probability it
is at some point x?
00:37:49.230 --> 00:37:52.400
I refer to this
density matrix that I
00:37:52.400 --> 00:37:54.920
have calculated in
coordinate space.
00:37:54.920 --> 00:37:57.090
Put x prime and
x to be the same.
00:37:57.090 --> 00:38:00.800
If x prime and x are the
same, I get 1 over V.
00:38:00.800 --> 00:38:03.550
Essentially, it says
that the probability
00:38:03.550 --> 00:38:08.530
that I find a particle is
the same anywhere in the box.
00:38:08.530 --> 00:38:10.250
So that makes sense.
00:38:10.250 --> 00:38:12.581
It is the analog of
throwing the coin.
00:38:12.581 --> 00:38:14.080
We don't know where
the particle is.
00:38:14.080 --> 00:38:16.940
It's equally likely
to be anywhere.
00:38:16.940 --> 00:38:20.560
Actually, remember that I used
periodic boundary conditions.
00:38:20.560 --> 00:38:22.970
If I had used sine
and cosines, there
00:38:22.970 --> 00:38:27.980
would have been some dependence
on approaching the boundaries.
00:38:27.980 --> 00:38:31.150
But the matrix tells
me more than that.
00:38:31.150 --> 00:38:33.630
Basically, the matrix
says it is true
00:38:33.630 --> 00:38:37.250
that the diagonal
elements are 1 over V.
00:38:37.250 --> 00:38:41.280
But if I go off-diagonal,
there is this [INAUDIBLE] scale
00:38:41.280 --> 00:38:44.540
lambda that is
telling me something.
00:38:44.540 --> 00:38:46.660
What is it telling you?
00:38:46.660 --> 00:38:49.460
Essentially, it is telling you
that through the procedures
00:38:49.460 --> 00:38:55.460
that we have outlined here, if
you are at the temperature T,
00:38:55.460 --> 00:39:01.620
then this factor tells you
about the classical probability
00:39:01.620 --> 00:39:03.660
of finding a momentum p.
00:39:03.660 --> 00:39:07.700
There is a range of momenta
because p squared over 2m,
00:39:07.700 --> 00:39:09.680
the excitation is
controlled by kT.
00:39:09.680 --> 00:39:12.410
There is a range of
momenta that is possible.
00:39:12.410 --> 00:39:14.680
And through a
procedure such as this,
00:39:14.680 --> 00:39:19.820
you are making a
superposition of plane waves
00:39:19.820 --> 00:39:22.450
that have this
range of momentum.
00:39:22.450 --> 00:39:25.840
The superposition of plane
waves, what does it give you?
00:39:25.840 --> 00:39:29.230
It essentially gives
you a Guassian blob,
00:39:29.230 --> 00:39:32.650
which we can position
anywhere in the space.
00:39:32.650 --> 00:39:35.290
But the width of
that Gaussian blob
00:39:35.290 --> 00:39:38.180
is given by this
factor of lambda,
00:39:38.180 --> 00:39:40.790
which is related to
the uncertainty that
00:39:40.790 --> 00:39:43.010
is now quantum in character.
00:39:43.010 --> 00:39:46.321
That is, quantum mechanically
if you know the momentum,
00:39:46.321 --> 00:39:47.695
then you don't
know the position.
00:39:47.695 --> 00:39:50.870
There is the Heisenberg
uncertainty principle.
00:39:50.870 --> 00:39:53.250
Now we have a
classical uncertainty
00:39:53.250 --> 00:39:56.500
about the momentum because
of these Boltzmann weights.
00:39:56.500 --> 00:40:00.590
That classical uncertainty
that we have about the momentum
00:40:00.590 --> 00:40:07.280
translates to some kind of a
wave packet size for these--
00:40:07.280 --> 00:40:10.960
to how much you can
localize a particle.
00:40:10.960 --> 00:40:13.360
So really, this
particle you should
00:40:13.360 --> 00:40:17.800
think of quantum mechanically
as being some kind of a wave
00:40:17.800 --> 00:40:20.873
packet that has some
characteristic size lambda.
00:40:24.260 --> 00:40:25.680
Yes?
00:40:25.680 --> 00:40:29.880
AUDIENCE: So could you
interpret the difference
00:40:29.880 --> 00:40:35.430
between x and x prime
appearing in the [INAUDIBLE]
00:40:35.430 --> 00:40:41.040
as sort of being the probability
of finding a particle at x
00:40:41.040 --> 00:40:44.270
having turned to x
prime within the box?
00:40:49.250 --> 00:40:50.950
PROFESSOR: At this
stage, you are
00:40:50.950 --> 00:40:54.510
sort of introducing
something that is not there.
00:40:54.510 --> 00:40:57.900
But if I had two
particles-- so very soon we
00:40:57.900 --> 00:41:00.330
are going to do
multiple particles--
00:41:00.330 --> 00:41:03.070
and this kind of structure
will be maintained
00:41:03.070 --> 00:41:07.490
if I have multiple particles,
then your statement is correct.
00:41:07.490 --> 00:41:10.310
That this factor will
tell me something
00:41:10.310 --> 00:41:13.160
about the probability of
these things being exchanged,
00:41:13.160 --> 00:41:17.650
one tunneling to the other
place and one tunneling back.
00:41:17.650 --> 00:41:22.180
And indeed, if you went to
the colloquium yesterday,
00:41:22.180 --> 00:41:25.180
Professor [? Block ?]
showed pictures
00:41:25.180 --> 00:41:27.950
of these kinds of
exchanges taking place.
00:41:32.630 --> 00:41:35.250
OK?
00:41:35.250 --> 00:41:38.490
But essentially,
really at this stage
00:41:38.490 --> 00:41:42.290
for the single particle
with no potential,
00:41:42.290 --> 00:41:45.630
the statement is that the
particle is, in reality,
00:41:45.630 --> 00:41:47.602
some kind of a wave packet.
00:41:51.320 --> 00:41:53.460
OK?
00:41:53.460 --> 00:41:54.400
Fine.
00:41:54.400 --> 00:41:56.750
So any other questions?
00:41:59.950 --> 00:42:02.190
The next thing is, of
course, what I said.
00:42:02.190 --> 00:42:05.760
Let's put two particles
in the potential.
00:42:05.760 --> 00:42:19.180
So let's go to two particles.
00:42:23.460 --> 00:42:26.390
Actually, not necessarily
even in the box.
00:42:26.390 --> 00:42:28.970
But let's say what
kind of a Hamiltonian
00:42:28.970 --> 00:42:31.265
would I write for two particles?
00:42:34.010 --> 00:42:36.770
I would write something
that has a kinetic energy
00:42:36.770 --> 00:42:42.580
for the first particle, kinetic
energy for the second particle,
00:42:42.580 --> 00:42:48.020
and then some kind of a
potential of interaction.
00:42:48.020 --> 00:42:50.500
Let's say, as a
function of x1 minus x2.
00:42:59.290 --> 00:43:02.930
Now, I could have done many
things for two particles.
00:43:02.930 --> 00:43:05.600
I could, for example,
have had particles
00:43:05.600 --> 00:43:07.780
that are different
masses, but I clearly
00:43:07.780 --> 00:43:10.910
wanted to write things
that were similar.
00:43:10.910 --> 00:43:13.090
So this is Hamiltonian
that describes
00:43:13.090 --> 00:43:16.820
particles labeled by 1 and 2.
00:43:16.820 --> 00:43:19.340
And if they have the same
mass and the potential
00:43:19.340 --> 00:43:21.870
is a function of
separation, it's
00:43:21.870 --> 00:43:27.540
certainly h that is
for 2 and 1 exchange.
00:43:27.540 --> 00:43:30.590
So this Hamiltonian
has a certain symmetry
00:43:30.590 --> 00:43:33.660
under the exchange
of the labels.
00:43:33.660 --> 00:43:37.300
Now typically, what
you should remember
00:43:37.300 --> 00:43:41.240
is that any type of symmetry
that your Hamiltonian has
00:43:41.240 --> 00:43:46.120
or any matrix has will
be reflected ultimately
00:43:46.120 --> 00:43:49.130
in the form of the
eigenvectors that you
00:43:49.130 --> 00:43:51.250
would construct for that.
00:43:51.250 --> 00:43:54.370
So indeed, I already
know that for this, I
00:43:54.370 --> 00:43:57.535
should be able to construct
wave functions that
00:43:57.535 --> 00:44:01.590
are either symmetrical or
anti-symmetric under exchange.
00:44:01.590 --> 00:44:04.170
But that statement
aside, let's think
00:44:04.170 --> 00:44:08.200
about the meaning
of the wave function
00:44:08.200 --> 00:44:10.630
that we have for
the two particles.
00:44:10.630 --> 00:44:16.230
So presumably, there is a psi
as a function of x1 and x2
00:44:16.230 --> 00:44:19.050
in the coordinate space.
00:44:19.050 --> 00:44:22.870
And for a particular
quantum state,
00:44:22.870 --> 00:44:26.610
you would say that the
square of this quantity
00:44:26.610 --> 00:44:36.840
is related to quantum
probabilities of finding
00:44:36.840 --> 00:44:41.540
particles at x1 and x2.
00:44:47.810 --> 00:44:53.370
So this could, for example,
be the case of two particles.
00:44:53.370 --> 00:44:57.780
Let's say oxygen and nitrogen.
00:44:57.780 --> 00:45:00.180
They have almost the same mass.
00:45:00.180 --> 00:45:02.760
This statement would be true.
00:45:02.760 --> 00:45:05.530
But the statement
becomes more interesting
00:45:05.530 --> 00:45:10.120
if we say that the
particles are identical.
00:45:10.120 --> 00:45:24.070
So for identical
particles, the statement
00:45:24.070 --> 00:45:30.050
of identity in quantum mechanics
is damped at this stage.
00:45:30.050 --> 00:45:33.660
You would say that I can't
tell apart the probability
00:45:33.660 --> 00:45:36.610
that one particle,
number 1, is at x1,
00:45:36.610 --> 00:45:40.220
particle 2 is at
x2 or vice versa.
00:45:44.540 --> 00:45:48.080
The labels 1 and
2 are just thinks
00:45:48.080 --> 00:45:50.750
that I assign for
convenience, but they really
00:45:50.750 --> 00:45:52.400
don't have any meaning.
00:45:52.400 --> 00:45:53.650
There are two particles.
00:45:53.650 --> 00:45:56.650
To all intents and purposes,
they are identical.
00:45:56.650 --> 00:46:02.420
And there is some probability
for seeing the events
00:46:02.420 --> 00:46:04.250
where one particle--
I don't know
00:46:04.250 --> 00:46:06.950
what its label is--
at this location.
00:46:06.950 --> 00:46:09.180
The other particle-- I don't
know what its label is--
00:46:09.180 --> 00:46:10.790
is at that location.
00:46:10.790 --> 00:46:12.110
Labels don't have any meaning.
00:46:15.260 --> 00:46:16.800
So this is different.
00:46:16.800 --> 00:46:19.940
This does not have
a classic analog.
00:46:19.940 --> 00:46:23.080
Classically, if I put
something on the computer,
00:46:23.080 --> 00:46:26.820
I would say that
particle 1 is here at x1,
00:46:26.820 --> 00:46:29.280
particle 2 is here at x2.
00:46:29.280 --> 00:46:32.630
I have to write some
index on the computer.
00:46:32.630 --> 00:46:36.790
But if I want to
construct a wave function,
00:46:36.790 --> 00:46:38.280
I wouldn't know what to do.
00:46:38.280 --> 00:46:41.050
I would just essentially
have a function of x1 and x2
00:46:41.050 --> 00:46:44.680
that is peaked here and there.
00:46:44.680 --> 00:46:52.740
And we also know that somehow
quantum mechanics tells us
00:46:52.740 --> 00:46:55.130
that there is
actually a stronger
00:46:55.130 --> 00:47:01.470
version of this statement,
which is that psi of x1, x2 x1
00:47:01.470 --> 00:47:07.400
is either plus psi of
x1, x2 or minus psi
00:47:07.400 --> 00:47:13.320
of x1, x2 for
identical particles
00:47:13.320 --> 00:47:16.235
depending on whether they
are boson or fermions.
00:47:31.900 --> 00:47:43.910
So let's kind of build
upon that a little bit more
00:47:43.910 --> 00:47:45.730
and go to many particles.
00:48:00.680 --> 00:48:14.150
So forth N particles
that are identical,
00:48:14.150 --> 00:48:22.770
I would have some
kind of a state psi
00:48:22.770 --> 00:48:24.580
that depends on the coordinates.
00:48:24.580 --> 00:48:26.840
For example, in the
coordinate representation,
00:48:26.840 --> 00:48:30.140
but I could choose any
other representation.
00:48:30.140 --> 00:48:32.080
Coordinate kind of
makes more sense.
00:48:32.080 --> 00:48:34.420
We can visualize it.
00:48:34.420 --> 00:48:38.430
And if I can't tell
them apart, previously I
00:48:38.430 --> 00:48:41.800
was just exchanging
two of the labels
00:48:41.800 --> 00:48:45.480
but now I can permute
them in any possible way.
00:48:45.480 --> 00:48:56.690
So I can add the permutation
P. And of course, there
00:48:56.690 --> 00:48:58.570
are N factorial in number.
00:49:06.130 --> 00:49:13.050
And the generalization of the
statement that I had before
00:49:13.050 --> 00:49:15.820
was that for the
case of bosons, I
00:49:15.820 --> 00:49:18.529
will always get the
same thing back.
00:49:27.160 --> 00:49:31.380
And for the case of fermions,
I will get a number back
00:49:31.380 --> 00:49:35.910
that is either minus
or plus depending
00:49:35.910 --> 00:49:40.549
on the type of permutation that
I apply of what I started with.
00:49:48.380 --> 00:49:55.010
So I have introduced here
something that I call minus 1
00:49:55.010 --> 00:49:58.580
to the power of
P, which is called
00:49:58.580 --> 00:50:00.300
a parity of the permutation.
00:50:06.220 --> 00:50:10.420
A permutation is either even--
in which case, this minus 1
00:50:10.420 --> 00:50:14.620
to the power of P is plus--
or is odd-- in which case,
00:50:14.620 --> 00:50:18.520
this minus 1 to P is minus 1.
00:50:18.520 --> 00:50:22.710
How do I determine the
parity of permutation?
00:50:22.710 --> 00:50:27.250
Parity of the
permutation is the number
00:50:27.250 --> 00:50:34.020
of exchanges that lead
to this permutation.
00:50:42.110 --> 00:50:44.765
Basically, take any permutation.
00:50:44.765 --> 00:50:48.770
Let's say we stick
with four particles.
00:50:48.770 --> 00:50:53.010
And I go from 1, 2, 3,
4, which was, let's say,
00:50:53.010 --> 00:50:56.920
some regular ordering,
to some other ordering.
00:50:56.920 --> 00:51:00.082
Let's say 4, 2, 1, 2.
00:51:00.082 --> 00:51:06.000
And my claim is that I can
achieve this transformation
00:51:06.000 --> 00:51:08.140
through a series of exchanges.
00:51:08.140 --> 00:51:11.192
So I can get here as follows.
00:51:11.192 --> 00:51:15.390
I want 4 to come all
the way back to here,
00:51:15.390 --> 00:51:19.560
so I do an exchange of 1 and 4.
00:51:19.560 --> 00:51:22.050
I call the exchange
in this fashion.
00:51:22.050 --> 00:51:23.970
I do exchange of 1 and 4.
00:51:23.970 --> 00:51:30.640
The exchange of 1 and 4
will make for me 4, 2, 3, 1.
00:51:30.640 --> 00:51:31.140
OK.
00:51:31.140 --> 00:51:33.620
I compare this with this.
00:51:33.620 --> 00:51:37.560
I see that all I need to
do is to switch 3 and 1.
00:51:37.560 --> 00:51:41.290
So I do an exchange of 1 and 3.
00:51:41.290 --> 00:51:45.480
And what I will get
here is 4, 2, 1, 3.
00:51:45.480 --> 00:51:49.340
So I could get to
that permutation
00:51:49.340 --> 00:51:50.870
through two exchanges.
00:51:50.870 --> 00:51:52.123
Therefore, this is even.
00:51:55.830 --> 00:51:57.670
Now, this is not
the only way that I
00:51:57.670 --> 00:51:59.810
can get from one to the other.
00:51:59.810 --> 00:52:04.470
I can, for example, sit and do
multiple exchanges of this 4,
00:52:04.470 --> 00:52:08.980
2 a hundred times, but
not ninety nine times.
00:52:08.980 --> 00:52:11.040
As long as I do
an even number, I
00:52:11.040 --> 00:52:12.540
will get back to the same thing.
00:52:12.540 --> 00:52:15.510
The parity will be conserved.
00:52:15.510 --> 00:52:18.540
And there's another way
of calculating parity.
00:52:18.540 --> 00:52:22.570
You just start with this
original configuration
00:52:22.570 --> 00:52:25.550
and you want to get to
that final configuration.
00:52:25.550 --> 00:52:27.670
You just draw lines.
00:52:27.670 --> 00:52:37.370
So 1 goes to 1, 2 goes to
2, 3 goes to 3, 4 goes to 4.
00:52:37.370 --> 00:52:41.160
And you count how many
crossings you have,
00:52:41.160 --> 00:52:42.757
and the parity of
those crossings
00:52:42.757 --> 00:52:44.590
will give you the parity
of the permutation.
00:52:50.350 --> 00:52:56.480
So somehow within
quantum mechanics,
00:52:56.480 --> 00:53:01.720
the idea of what is
identical particle is stamped
00:53:01.720 --> 00:53:04.710
into the nature of
the wave vectors,
00:53:04.710 --> 00:53:07.460
in the structure of the Hilbert
space that you can construct.
00:53:10.940 --> 00:53:16.550
So let's see how that leads to
this simple example of particle
00:53:16.550 --> 00:53:20.610
in the box, if we continue to
add particles into the box.
00:53:36.780 --> 00:53:39.485
So we want to now put
N particles in box.
00:53:43.910 --> 00:53:48.340
Otherwise, no interaction
completely free.
00:53:48.340 --> 00:53:56.170
So the N particle Hamiltonian is
some alpha running from 1 to N
00:53:56.170 --> 00:53:59.560
p alpha squared over
2m kinetic energies
00:53:59.560 --> 00:54:00.630
of all of these things.
00:54:04.760 --> 00:54:05.850
fine.
00:54:05.850 --> 00:54:11.830
Now, note that this Hamiltonian,
since it is in some sense
00:54:11.830 --> 00:54:17.230
built up of lots of
non-interacting pieces.
00:54:17.230 --> 00:54:19.940
And we saw already classically,
that things are not
00:54:19.940 --> 00:54:23.400
interacting-- calculating
probabilities,
00:54:23.400 --> 00:54:26.650
partition functions, et
cetera, is very easy.
00:54:26.650 --> 00:54:28.520
This has that same structure.
00:54:28.520 --> 00:54:29.910
It's the ideal gas.
00:54:29.910 --> 00:54:31.690
Now, quantum mechanically.
00:54:31.690 --> 00:54:34.170
So it should be
sufficiently easy.
00:54:34.170 --> 00:54:39.110
And indeed, we can immediately
construct the eigenstates
00:54:39.110 --> 00:54:39.940
for this.
00:54:39.940 --> 00:54:42.250
So we can construct
the basis, and then
00:54:42.250 --> 00:54:44.790
do the calculations
in that basis.
00:54:44.790 --> 00:54:49.670
So let's look at something
that I will call product state.
00:54:56.530 --> 00:55:01.780
And let's say that
I had this structure
00:55:01.780 --> 00:55:05.890
that I described on the board
above where I have a plane
00:55:05.890 --> 00:55:09.320
wave that is characterized
by some wave number
00:55:09.320 --> 00:55:11.950
k for one particle.
00:55:11.950 --> 00:55:15.010
For N particles, I pick
the first one to be k1,
00:55:15.010 --> 00:55:18.476
the second one to be k2,
the last one to be kN.
00:55:21.740 --> 00:55:25.890
And I will call this a
product state in the sense
00:55:25.890 --> 00:55:31.940
that if I look at the positional
representation of this product
00:55:31.940 --> 00:55:38.380
state, it is simply the
product of one-particle states.
00:55:38.380 --> 00:55:43.892
So this is a product
over alpha of x alpha
00:55:43.892 --> 00:55:49.110
k alpha, which is
this e to the i kx
00:55:49.110 --> 00:55:51.480
k dot x over square root of it.
00:55:51.480 --> 00:55:55.460
So this is perfectly
well normalized
00:55:55.460 --> 00:55:58.860
if I were to integrate
over x1, x2, x3.
00:55:58.860 --> 00:56:01.335
Each one of the integrals
is individually normalized.
00:56:01.335 --> 00:56:03.560
The overall thing is normalized.
00:56:03.560 --> 00:56:08.030
It's an eigenstate because
if I act Hn on this product
00:56:08.030 --> 00:56:16.290
state, what I will get is a
sum over alpha h bar squared
00:56:16.290 --> 00:56:20.050
k alpha squared over 2m, and
then the product state back.
00:56:26.480 --> 00:56:30.510
So it's an eigenstate.
00:56:30.510 --> 00:56:33.100
And in fact, it's a
perfectly good eigenstate
00:56:33.100 --> 00:56:36.550
as long as I want to
describe a set of particles
00:56:36.550 --> 00:56:37.620
that are not identical.
00:56:40.660 --> 00:56:43.200
I can't use this state
to describe particles
00:56:43.200 --> 00:56:46.280
that are identical
because it does not
00:56:46.280 --> 00:56:50.160
satisfy the symmetries that
I set quantum mechanics
00:56:50.160 --> 00:56:53.420
stamps into identical particles.
00:56:53.420 --> 00:56:59.025
And it's clearly the case that,
for example-- so this is not
00:56:59.025 --> 00:57:11.210
symmetrized since clearly,
if I look at k1, k2,
00:57:11.210 --> 00:57:15.390
k1 goes with x1,
k2 goes with x2.
00:57:15.390 --> 00:57:19.270
And it is not the
same thing as k2, k1,
00:57:19.270 --> 00:57:24.970
where k2 goes with x1
and k1 goes with x2.
00:57:24.970 --> 00:57:31.090
Essentially, the two particles
can be distinguished.
00:57:31.090 --> 00:57:33.530
One of them has
momentum h bar k1.
00:57:33.530 --> 00:57:35.880
The other has
momentum h bar k 2.
00:57:35.880 --> 00:57:40.370
I can tell them apart because
of this unsymmetrized nature
00:57:40.370 --> 00:57:43.360
of this wave function.
00:57:43.360 --> 00:57:46.480
But you know how to make
things that are symmetric.
00:57:46.480 --> 00:57:52.770
You basically add k1 k2 plus
k2 k1 or k1 k2 minus k2 k1
00:57:52.770 --> 00:57:54.390
to make it anti-symmetrized.
00:57:54.390 --> 00:57:57.820
Divide by square root
of 2 and you are done.
00:57:57.820 --> 00:58:00.490
So now, let's generalize
that to the case
00:58:00.490 --> 00:58:03.220
of the N-particle system.
00:58:03.220 --> 00:58:09.360
So I will call a-- let's
start with the case
00:58:09.360 --> 00:58:16.322
of the fermionic state.
00:58:16.322 --> 00:58:19.310
In fermionic state,
I will indicate
00:58:19.310 --> 00:58:29.030
by k1, k2, kN with a
minus index because
00:58:29.030 --> 00:58:31.360
of the asymmetry
or the minus signs
00:58:31.360 --> 00:58:33.540
that we have for fermions.
00:58:33.540 --> 00:58:38.530
The way I do that is I
sum over all N factorial
00:58:38.530 --> 00:58:41.860
permutations that I have.
00:58:41.860 --> 00:58:46.480
I let p act on
the product state.
00:58:50.800 --> 00:58:54.950
And again, for two particles,
you have the k1 k2,
00:58:54.950 --> 00:58:57.400
then you do minus k2 k1.
00:58:57.400 --> 00:59:02.800
For general particles, I do
this minus 1 to the power of p.
00:59:02.800 --> 00:59:06.800
So all the even permutations
are added with plus.
00:59:06.800 --> 00:59:09.630
All the odd permutations
are added with minus.
00:59:13.110 --> 00:59:17.210
Except that this is a whole
bunch of different terms
00:59:17.210 --> 00:59:18.770
that are being added.
00:59:18.770 --> 00:59:20.580
Again, for two
particles, you know
00:59:20.580 --> 00:59:22.780
that you have to divide
by a square root of 2
00:59:22.780 --> 00:59:25.120
because you add 2 vectors.
00:59:25.120 --> 00:59:27.715
And the length of
the overall vector
00:59:27.715 --> 00:59:30.140
is increased by a
square root of 2.
00:59:30.140 --> 00:59:34.330
Here, you have to divide in
general by the number of terms
00:59:34.330 --> 00:59:36.930
that you have, square
root of N factorial.
00:59:42.090 --> 00:59:47.310
The only thing that you have
to be careful with is that you
00:59:47.310 --> 00:59:51.370
cannot have any two of
these k's to be the same.
00:59:51.370 --> 00:59:54.820
Because let's say these two are
the same, then along the list
00:59:54.820 --> 00:59:56.830
here I have the
exchange of these two.
00:59:56.830 --> 00:59:59.800
And when I exchange them,
I go from even to odd.
00:59:59.800 --> 01:00:03.170
I put a minus sign and
I have a subtraction.
01:00:03.170 --> 01:00:07.590
So here, I have to
make sure that all k
01:00:07.590 --> 01:00:09.744
alpha must be distinct.
01:00:20.876 --> 01:00:27.770
Now, say the bosonic
one is simpler.
01:00:27.770 --> 01:00:36.890
I basically construct it,
k1, k2, kN with a plus.
01:00:36.890 --> 01:00:38.740
By simply adding
things together,
01:00:38.740 --> 01:00:42.450
so I will have a sum over p.
01:00:42.450 --> 01:00:44.400
No sign here.
01:00:44.400 --> 01:00:49.955
Permutation k1 through kN.
01:00:49.955 --> 01:00:52.940
And then I have to divide
by some normalization.
01:00:59.930 --> 01:01:03.420
Now, the only tricky
thing about this
01:01:03.420 --> 01:01:11.460
is that the normalization
is not N factorial.
01:01:11.460 --> 01:01:15.860
So to give you an
example, let's imagine
01:01:15.860 --> 01:01:20.460
that I choose to start
with a product state
01:01:20.460 --> 01:01:23.720
where two of the k's are
alpha and one of them is beta.
01:01:26.320 --> 01:01:33.265
So let's sort of put here 1,
1, 1, 2, 3 for my k1, k2, k3.
01:01:33.265 --> 01:01:37.730
I have chosen that k1
and k2 are the same.
01:01:37.730 --> 01:01:42.940
And what I have to do is to sum
over all possible permutations.
01:01:42.940 --> 01:01:46.920
So there is a permutation
here that is 1, 3, 2.
01:01:46.920 --> 01:01:50.410
So I get here
alpha, beta, alpha.
01:01:50.410 --> 01:01:59.230
Then I will have 2, 1, 3,
2, 3, 1, 3, 1, 2, 3, 2, 1,
01:01:59.230 --> 01:02:05.440
which basically would be alpha,
alpha, beta, alpha, beta,
01:02:05.440 --> 01:02:10.580
alpha, beta, alpha,
alpha, beta, alpha, alpha.
01:02:13.930 --> 01:02:20.270
So there are for three things,
3 factorial or 6 permutations.
01:02:20.270 --> 01:02:26.700
But this entity is
clearly twice alpha,
01:02:26.700 --> 01:02:32.670
alpha, beta, plus alpha, beta,
alpha, plus beta, alpha, alpha.
01:02:35.370 --> 01:02:41.270
And the norm of this is going
to be 4 times 1 plus 1 plus 1.
01:02:41.270 --> 01:02:44.410
4 times 3, which is 12.
01:02:44.410 --> 01:02:46.540
So in order to
normalize this, I have
01:02:46.540 --> 01:02:50.300
to divide not by 1/6, but 1/12.
01:02:53.170 --> 01:02:55.560
So the appropriate
normalization here
01:02:55.560 --> 01:02:57.110
then becomes 1 over root 3.
01:03:02.190 --> 01:03:05.335
Now, in general what
would be this N plus?
01:03:07.990 --> 01:03:15.460
To calculate N plus,
I have to make sure
01:03:15.460 --> 01:03:18.230
that the norm of
this entity is 1.
01:03:18.230 --> 01:03:23.270
Or, N plus is the
square of this quantity.
01:03:23.270 --> 01:03:26.320
And if I were to
square this quantity,
01:03:26.320 --> 01:03:28.205
I will get two sets
of permutations.
01:03:28.205 --> 01:03:31.910
I will call them p and p prime.
01:03:31.910 --> 01:03:39.522
And on one side, I would
have the permutation p of k1
01:03:39.522 --> 01:03:41.420
through kN.
01:03:41.420 --> 01:03:45.720
On the other side, I would have
a permutation k prime of k1
01:03:45.720 --> 01:03:46.220
through kN.
01:03:54.320 --> 01:03:59.260
Now, this is clearly N
factorial square terms.
01:03:59.260 --> 01:04:04.090
But this is not N factorial
squared distinct terms.
01:04:04.090 --> 01:04:08.530
Because essentially,
over here I could
01:04:08.530 --> 01:04:12.080
get 1 of N factorial
possibilities.
01:04:12.080 --> 01:04:15.220
And then here, I would
permute over all the other N
01:04:15.220 --> 01:04:17.410
factorial possibilities.
01:04:17.410 --> 01:04:20.290
Then I would try the next
one and the next one.
01:04:20.290 --> 01:04:23.910
And essentially,
each one of those
01:04:23.910 --> 01:04:25.600
would give me the same one.
01:04:25.600 --> 01:04:29.780
So that is, once I have
fixed what this one is,
01:04:29.780 --> 01:04:35.015
permuting over them will
give me one member of N
01:04:35.015 --> 01:04:37.780
factorial identical terms.
01:04:37.780 --> 01:04:42.370
So I can write this as N
factorial sum over Q. Let's say
01:04:42.370 --> 01:04:49.700
I start with the
original k1 through kN,
01:04:49.700 --> 01:04:54.300
and then I go and do a
permutation of k1 through kN.
01:04:59.250 --> 01:05:02.680
And the question
is, how many times
01:05:02.680 --> 01:05:05.600
do I get something
that is non-zero?
01:05:05.600 --> 01:05:09.390
If these two lists are
completely distinct,
01:05:09.390 --> 01:05:12.310
except for the identity
any transformation
01:05:12.310 --> 01:05:14.830
that I will make
here will make me
01:05:14.830 --> 01:05:19.490
a vector that is orthogonal
to this and I will get 0.
01:05:19.490 --> 01:05:23.200
But if I have two of
them that are identical,
01:05:23.200 --> 01:05:25.140
then I do a
permutation like this
01:05:25.140 --> 01:05:27.540
and I'll get the
same thing back.
01:05:27.540 --> 01:05:32.800
And then I have two things that
are giving the same result.
01:05:32.800 --> 01:05:39.080
And in general, this
answer is a product
01:05:39.080 --> 01:05:44.600
over all multiple
occurrences factorial.
01:05:44.600 --> 01:05:46.740
So let's say here
there was something
01:05:46.740 --> 01:05:49.050
that was repeated twice.
01:05:49.050 --> 01:05:52.270
If it had been
repeated three times,
01:05:52.270 --> 01:05:56.100
then all six possibilities
would've been the same.
01:05:56.100 --> 01:05:58.620
So I would have had 3 factorial.
01:05:58.620 --> 01:06:01.851
So if I had 3 and then 2
other ones that were the same,
01:06:01.851 --> 01:06:03.600
then the answer would
have been multiplied
01:06:03.600 --> 01:06:06.050
by 3 factorial 2 factorial.
01:06:06.050 --> 01:06:10.440
So the statement here
is that this N plus
01:06:10.440 --> 01:06:16.210
that I have to put here is N
factorial product over k nk
01:06:16.210 --> 01:06:20.190
factorial, which is essentially
the multiplicity that
01:06:20.190 --> 01:06:21.617
is associated with repeats.
01:06:25.030 --> 01:06:30.340
Oh
01:06:30.340 --> 01:06:38.810
So we've figured out what
the appropriate bosonic
01:06:38.810 --> 01:06:44.160
and fermionic eigenstates are
for this situation of particles
01:06:44.160 --> 01:06:47.310
in the box where I put
multiple particles.
01:06:47.310 --> 01:06:50.820
So now I have N
particles in the box.
01:06:50.820 --> 01:06:54.680
I know what the appropriate
basis functions are.
01:06:54.680 --> 01:07:00.640
And what I can now try
to do is to, for example,
01:07:00.640 --> 01:07:04.220
calculate the analog of
the partition function,
01:07:04.220 --> 01:07:07.780
the analog of what I did
here for N particles,
01:07:07.780 --> 01:07:10.970
or the analog of
the density matrix.
01:07:10.970 --> 01:07:13.581
So let's calculate the
analog of the density matrix
01:07:13.581 --> 01:07:14.455
and see what happens.
01:07:33.395 --> 01:07:39.500
So I want to calculate
an N particle density
01:07:39.500 --> 01:07:43.840
matrix, completely
free particles
01:07:43.840 --> 01:07:47.010
in a box, no interactions.
01:07:47.010 --> 01:07:51.270
For one particle, I
went from x to x prime.
01:07:51.270 --> 01:07:57.830
So here, I go from x1, x2, xN.
01:07:57.830 --> 01:08:03.665
To here, x1 prime,
x2 prime, xN prime.
01:08:07.110 --> 01:08:10.150
Our answer ultimately
will depend on
01:08:10.150 --> 01:08:13.980
whether I am dealing
with fermions or bosons.
01:08:13.980 --> 01:08:19.630
So I introduce an
index here eta.
01:08:19.630 --> 01:08:28.800
Let me put it here, eta is
plus 1 for bosons and minus 1
01:08:28.800 --> 01:08:29.784
for fermions.
01:08:33.509 --> 01:08:38.149
So kind of goes with
this thing over here
01:08:38.149 --> 01:08:43.689
whether or not I'm using bosonic
or fermionic symmetrization
01:08:43.689 --> 01:08:46.069
in constructing the
wave functions here.
01:08:49.090 --> 01:08:50.960
You say, well, what is rho?
01:08:50.960 --> 01:08:55.270
Rho is e to the minus this beta
hn divided by some partition
01:08:55.270 --> 01:08:58.430
function ZN that I don't know.
01:08:58.430 --> 01:09:03.510
But what I certainly know about
it is because I constructed,
01:09:03.510 --> 01:09:09.130
in fact, this basis so that I
have the right energy, which
01:09:09.130 --> 01:09:12.479
is sum over alpha
h bar squared alpha
01:09:12.479 --> 01:09:14.870
squared k alpha squared over 2m.
01:09:14.870 --> 01:09:21.960
That is, this rho is diagonal
in the basis of the k's.
01:09:21.960 --> 01:09:26.319
So maybe what I should do
is I should switch from this
01:09:26.319 --> 01:09:30.590
representation to the
representation of k's.
01:09:30.590 --> 01:09:33.770
So the same way that
for one particle
01:09:33.770 --> 01:09:37.380
I sandwiched this
factor of k, I am
01:09:37.380 --> 01:09:39.930
going to do the same
thing over here.
01:09:39.930 --> 01:09:42.595
Except that I will have
a whole bunch of k's.
01:09:45.850 --> 01:09:50.484
Because I'm doing
a multiple system.
01:09:55.730 --> 01:09:56.630
What do I have?
01:09:56.630 --> 01:10:06.510
I have x1 prime to xN
prime k1 through kN,
01:10:06.510 --> 01:10:10.830
with whatever appropriate
symmetry it is.
01:10:10.830 --> 01:10:17.410
I have e to the minus
sum over alpha beta h
01:10:17.410 --> 01:10:21.400
bar squared k alpha
squared over 2m.
01:10:21.400 --> 01:10:25.080
Essentially, generalizing
this factor that I have here,
01:10:25.080 --> 01:10:28.480
except that I have to
normalize it with some ZN
01:10:28.480 --> 01:10:32.180
that I have yet to calculate.
01:10:32.180 --> 01:10:41.760
And then I go back
from k1, kN to x1, xN.
01:10:41.760 --> 01:10:44.580
Again, respecting
whatever symmetry I
01:10:44.580 --> 01:10:46.038
want to have at
the end of the day.
01:10:50.720 --> 01:10:55.090
Now, let's think a little
bit about this summation
01:10:55.090 --> 01:10:57.820
that I have to do over here.
01:10:57.820 --> 01:11:00.170
And I'll put a prime
up there to indicate
01:11:00.170 --> 01:11:03.370
that it is a restricted sum.
01:11:03.370 --> 01:11:05.040
I didn't have any
restriction here.
01:11:05.040 --> 01:11:08.400
I said I integrate over--
or sum over all possible k's
01:11:08.400 --> 01:11:12.080
that were consistent with
the choice of 2 pi over l,
01:11:12.080 --> 01:11:14.130
or whatever.
01:11:14.130 --> 01:11:18.970
Now, here I have to make sure
that I don't do over-counting.
01:11:18.970 --> 01:11:20.840
What do I mean?
01:11:20.840 --> 01:11:24.110
I mean that once I do
this symmetrization,
01:11:24.110 --> 01:11:31.940
let's say I start with two
particles and I have k1, k2.
01:11:31.940 --> 01:11:33.560
Then, I do the symmetrization.
01:11:33.560 --> 01:11:37.110
I get something that is a
symmetric version of k1, k2.
01:11:37.110 --> 01:11:39.030
I would have gotten
exactly the same thing
01:11:39.030 --> 01:11:42.670
if I started with k2, k1.
01:11:42.670 --> 01:11:47.030
That is, a particular
one of these states
01:11:47.030 --> 01:11:51.410
corresponds to one
list that I have here.
01:11:51.410 --> 01:11:58.120
And so over here, I should
not sum over k1, k2, k3, et
01:11:58.120 --> 01:12:00.520
cetera, independently.
01:12:00.520 --> 01:12:05.110
Because then I will be
over-counting by N factorial.
01:12:05.110 --> 01:12:12.780
So I say, OK, let me sum
over things independently
01:12:12.780 --> 01:12:15.570
and then divide by
the over-counting.
01:12:18.600 --> 01:12:22.190
Because presumably,
these will give me
01:12:22.190 --> 01:12:25.180
N factorial similar states.
01:12:25.180 --> 01:12:29.320
So let's just sum over all
of them, forget about this.
01:12:29.320 --> 01:12:31.640
You say, well, almost.
01:12:31.640 --> 01:12:34.280
Not quite because you have
to worry about these factors.
01:12:37.390 --> 01:12:40.320
Because now when you
did the N factorial,
01:12:40.320 --> 01:12:42.750
you sort of did not
take into account
01:12:42.750 --> 01:12:47.010
the kinds of exchanges that I
mentioned that you should not
01:12:47.010 --> 01:12:53.510
include because essentially if
you have k1, k1, and then k3,
01:12:53.510 --> 01:12:56.330
then you don't have 6
different permutations.
01:12:56.330 --> 01:12:59.510
You only have 3
different permutations.
01:12:59.510 --> 01:13:03.380
So actually, the correction
that I have to make
01:13:03.380 --> 01:13:07.420
is to multiply here
by this quantity.
01:13:07.420 --> 01:13:14.900
So now that sum be the
restriction has gone into this.
01:13:14.900 --> 01:13:23.470
And this ensures that as I sum
over all k's independently,
01:13:23.470 --> 01:13:27.920
the over-counting of
states that I have made
01:13:27.920 --> 01:13:31.344
is taken into account by
the appropriate factor.
01:13:34.960 --> 01:13:39.740
Now, this is actually
very nice because when
01:13:39.740 --> 01:13:44.550
I look at these states, I have
the normalization factors.
01:13:44.550 --> 01:13:48.610
The normalizations depend
on the list of k's that I
01:13:48.610 --> 01:13:49.360
have over here.
01:13:53.540 --> 01:13:58.530
So the normalization of these
objects will give me a 1
01:13:58.530 --> 01:14:03.950
over N factorial product
over k nk factorial.
01:14:03.950 --> 01:14:08.390
And these nk
factorials will cancel.
01:14:08.390 --> 01:14:10.420
Now you say, hang on.
01:14:10.420 --> 01:14:11.760
Going too fast.
01:14:11.760 --> 01:14:16.350
This eta says you can do this
both for fermions and bosons.
01:14:16.350 --> 01:14:19.180
But this calculation
that you did over here
01:14:19.180 --> 01:14:22.142
applies only for
the case of bosons.
01:14:22.142 --> 01:14:26.390
You say, never mind because for
fermions, the allowed values
01:14:26.390 --> 01:14:30.140
of nk are either 0 or 1.
01:14:30.140 --> 01:14:32.710
I cannot put two of them.
01:14:32.710 --> 01:14:37.740
So these nk factorials are
either-- again, 1 or 1.
01:14:37.740 --> 01:14:40.775
So even for fermions,
this will work out fine.
01:14:45.330 --> 01:14:48.720
There will be appropriate
cancellation of wave functions
01:14:48.720 --> 01:14:51.030
that should not
be included when I
01:14:51.030 --> 01:14:53.010
do the summation
over permutations
01:14:53.010 --> 01:14:58.240
with the corresponding
factors of plus and minus.
01:14:58.240 --> 01:15:01.530
So again, the minus signs--
everything will work.
01:15:01.530 --> 01:15:04.690
And this kind of
ugly factor will
01:15:04.690 --> 01:15:07.412
disappear at the end of the day.
01:15:07.412 --> 01:15:09.350
OK?
01:15:09.350 --> 01:15:11.246
Fine.
01:15:11.246 --> 01:15:15.440
Now, each one of these is
a sum over permutations.
01:15:15.440 --> 01:15:17.120
Actually, before
I write those, let
01:15:17.120 --> 01:15:22.240
me just write the factor of e to
the minus sum over alpha beta h
01:15:22.240 --> 01:15:26.094
bar squared k alpha
squared over 2m.
01:15:26.094 --> 01:15:27.810
And then I have ZN.
01:15:27.810 --> 01:15:29.250
I don't know what ZN is yet.
01:15:33.480 --> 01:15:40.130
Each one of these states
is a sum over permutations.
01:15:40.130 --> 01:15:44.970
I have taken care of the
overall normalization.
01:15:44.970 --> 01:15:47.270
I have to do the sum
over the permutations.
01:15:49.900 --> 01:15:56.150
And let's say I'm
looking at this one.
01:15:56.150 --> 01:15:59.400
Essentially, I have to sandwich
that with some combination
01:15:59.400 --> 01:16:00.060
of x's.
01:16:00.060 --> 01:16:01.520
What do I get?
01:16:01.520 --> 01:16:07.800
I will get a factor
of eta to the p.
01:16:07.800 --> 01:16:10.370
So actually, maybe
I should have really
01:16:10.370 --> 01:16:13.680
within the following
statement at some point.
01:16:30.270 --> 01:16:33.180
So we introduced these
different states.
01:16:33.180 --> 01:16:39.800
I can combine them together
and write k1, k2, kN
01:16:39.800 --> 01:16:41.890
with a symmetry that
is representative
01:16:41.890 --> 01:16:43.610
of bosons or fermions.
01:16:43.610 --> 01:16:47.810
And that's where I introduced a
factor of plus or minus, which
01:16:47.810 --> 01:16:53.730
is obtained by summing
over all permutations.
01:16:53.730 --> 01:16:58.235
This factor eta, which is plus
or minus, raised to the p.
01:16:58.235 --> 01:17:01.700
So for bosons, I would be
adding everything in phase.
01:17:01.700 --> 01:17:04.370
For fermions, I would
have minus 1 to the p.
01:17:04.370 --> 01:17:09.615
I have the permutation
of the set of indices k1
01:17:09.615 --> 01:17:14.283
through kN in the product state.
01:17:14.283 --> 01:17:16.010
Whoops.
01:17:16.010 --> 01:17:17.447
This should be product.
01:17:17.447 --> 01:17:20.820
This should be a product state.
01:17:20.820 --> 01:17:26.160
And then I have to divide
by square root of N
01:17:26.160 --> 01:17:29.700
factorial product
over k nk factorial.
01:17:34.250 --> 01:17:43.080
And clearly, and we will see a
useful way of looking at this.
01:17:43.080 --> 01:17:49.810
I can also look at the set of
states that are occupied among
01:17:49.810 --> 01:17:52.030
the original list of k's.
01:17:52.030 --> 01:17:54.060
Some k's do not
occur in this list.
01:17:54.060 --> 01:17:55.970
Some k's occurs once.
01:17:55.970 --> 01:17:58.110
Some k's occur more than once.
01:17:58.110 --> 01:18:00.225
And that's how I
construct these factors.
01:18:03.050 --> 01:18:06.130
And then the condition that
I have is that, of course,
01:18:06.130 --> 01:18:10.610
sum over k nK should
be the total number
01:18:10.610 --> 01:18:17.962
N. That nk is 0
or 1 for fermions.
01:18:17.962 --> 01:18:23.905
That nk is any value up to
that constraint for bosons.
01:18:28.900 --> 01:18:36.020
Now, I take these functions and
I substitute it here and here.
01:18:36.020 --> 01:18:37.990
What do I get?
01:18:37.990 --> 01:18:40.730
I get the product state.
01:18:40.730 --> 01:18:42.870
So I have eta to the p.
01:18:42.870 --> 01:18:48.580
The product state is
e to the i sum over,
01:18:48.580 --> 01:18:59.965
let's say, beta k of
p beta x beta prime.
01:19:03.260 --> 01:19:10.610
And then from the next one, I
will get e to the minus i sum
01:19:10.610 --> 01:19:13.240
over-- again, some other beta.
01:19:13.240 --> 01:19:15.470
Or the same beta,
it doesn't matter.
01:19:15.470 --> 01:19:18.128
p prime beta x beta.
01:19:22.290 --> 01:19:25.390
I think I included everything.
01:19:25.390 --> 01:19:25.890
Yes?
01:19:25.890 --> 01:19:26.858
AUDIENCE: Question.
01:19:26.858 --> 01:19:31.214
Why do you only have one
term of beta to the p?
01:19:35.086 --> 01:19:36.070
PROFESSOR: OK.
01:19:36.070 --> 01:19:38.980
So here I have a list.
01:19:38.980 --> 01:19:45.740
It could be k1, k2, k2--
well, let's say k3, k4.
01:19:45.740 --> 01:19:48.080
Let's say I have four things.
01:19:48.080 --> 01:19:50.080
And the product
means I essentially
01:19:50.080 --> 01:19:52.430
have a product of these things.
01:19:52.430 --> 01:19:54.940
When I multiply them
with wave functions,
01:19:54.940 --> 01:19:58.530
I will have e to the
i, k1, x1 et cetera.
01:19:58.530 --> 01:20:00.700
Now, I do some permutation.
01:20:00.700 --> 01:20:04.450
Let's say I go from
1, 2, 3 to 3, 2, 1.
01:20:04.450 --> 01:20:07.160
So there is a permutation
that I do like that.
01:20:07.160 --> 01:20:10.010
I leave the 4 to be the same.
01:20:10.010 --> 01:20:14.310
This permutation
has some parity.
01:20:14.310 --> 01:20:18.584
So I have to put a plus or minus
that depends on the parity.
01:20:18.584 --> 01:20:19.392
AUDIENCE: Yeah.
01:20:19.392 --> 01:20:24.230
But my question is why in
that equation over there--
01:20:24.230 --> 01:20:24.980
PROFESSOR: Oh, OK.
01:20:24.980 --> 01:20:28.445
AUDIENCE: --we have a
term A to the p for the k.
01:20:28.445 --> 01:20:29.070
PROFESSOR: Yes.
01:20:29.070 --> 01:20:29.750
AUDIENCE: k factor
and not for the bra.
01:20:29.750 --> 01:20:30.580
PROFESSOR: Good.
01:20:30.580 --> 01:20:34.822
You are telling me that I forgot
to put p prime and eta of p
01:20:34.822 --> 01:20:35.322
prime.
01:20:35.322 --> 01:20:35.730
AUDIENCE: OK.
01:20:35.730 --> 01:20:37.320
I wasn't sure if
there was a reason.
01:20:37.320 --> 01:20:37.920
PROFESSOR: No.
01:20:37.920 --> 01:20:40.270
I was going step by step.
01:20:40.270 --> 01:20:46.250
I had written the bra part and I
was about to get to the k part.
01:20:46.250 --> 01:20:48.000
I put that part
of the ket, and I
01:20:48.000 --> 01:20:49.850
was about to do the rest of it.
01:20:49.850 --> 01:20:51.120
AUDIENCE: Oh, sorry.
01:20:51.120 --> 01:20:54.110
PROFESSOR: It's OK.
01:20:54.110 --> 01:20:56.390
OK.
01:20:56.390 --> 01:21:03.760
So you can see that what happens
at the end of the day here
01:21:03.760 --> 01:21:07.670
is that a lot of these
nk factorials disappear.
01:21:07.670 --> 01:21:10.695
So those were not things
that we would have liked.
01:21:13.530 --> 01:21:19.230
There is a factor of
N factorial from here
01:21:19.230 --> 01:21:21.950
and a factor of N factorial
from there that remains.
01:21:21.950 --> 01:21:26.390
So the answer will be 1
over ZN N factorial squared.
01:21:29.120 --> 01:21:34.530
I have a sum over two
permutations, p and p prime,
01:21:34.530 --> 01:21:36.260
of something.
01:21:36.260 --> 01:21:40.260
I will do this more
closing next time around,
01:21:40.260 --> 01:21:42.720
but I wanted to
give you the flavor.
01:21:42.720 --> 01:21:45.880
This double sum at
the end of the day,
01:21:45.880 --> 01:21:50.640
just like what we did
before, becomes one sum up
01:21:50.640 --> 01:21:54.090
to repetition of N factorial.
01:21:54.090 --> 01:21:58.430
So the N factorial
will disappear.
01:21:58.430 --> 01:22:02.060
But what we will find
is that the answer here
01:22:02.060 --> 01:22:07.550
is going to be something that
is a bunch of Gaussians that
01:22:07.550 --> 01:22:09.340
are very similar
to the integration
01:22:09.340 --> 01:22:11.800
that I did for one particle.
01:22:11.800 --> 01:22:15.060
I have e to the minus
k squared over 2m.
01:22:15.060 --> 01:22:18.220
I have e to the i
k x minus x prime.
01:22:18.220 --> 01:22:21.120
Except that all of
these k's and x's and x
01:22:21.120 --> 01:22:25.310
primes have been permuted in
all kinds of strange ways.
01:22:25.310 --> 01:22:29.280
Once we untangle that, we
find that the answer is going
01:22:29.280 --> 01:22:34.400
to end up to be eta
of Q x alpha minus x
01:22:34.400 --> 01:22:40.910
prime Q alpha squared divided
by pi 2 lambda squared.
01:22:40.910 --> 01:22:45.200
And we have a sum
over alpha, which
01:22:45.200 --> 01:22:49.260
is really kind of
like a sum of things
01:22:49.260 --> 01:22:52.940
that we had for the
case of one particle.
01:22:52.940 --> 01:22:55.630
So for the case of
one particle, we
01:22:55.630 --> 01:22:59.390
found that the
off-diagonal density matrix
01:22:59.390 --> 01:23:03.760
had elements that's
reflected this wave packet
01:23:03.760 --> 01:23:05.600
nature of this.
01:23:05.600 --> 01:23:09.946
If we have multiple
particles that are identical,
01:23:09.946 --> 01:23:15.370
then the thing is if
I have 1, 2 and then
01:23:15.370 --> 01:23:20.280
I go 1, 1 prime, 2, 2 prime
for the two different locations
01:23:20.280 --> 01:23:23.840
that I have x1 prime,
x2 prime, et cetera.
01:23:23.840 --> 01:23:27.065
And 1 and 2 are identical,
then I could have really
01:23:27.065 --> 01:23:30.860
also put here 2 prime, 1 prime.
01:23:30.860 --> 01:23:33.360
And I wouldn't have
known the difference.
01:23:33.360 --> 01:23:36.960
And this kind of sum
will take care of that,
01:23:36.960 --> 01:23:39.000
includes those
kinds of exchanges
01:23:39.000 --> 01:23:42.140
that I mentioned
earlier and is something
01:23:42.140 --> 01:23:44.630
that we need to
derive more carefully
01:23:44.630 --> 01:23:47.580
and explain in more
detail next time.