PHYS 445 Lecture 20  Quantum statistics
20  1
© 2001 by David Boal, Simon Fraser University.
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Lecture 20  Quantum statistics
What's Important:
•
quantum statistics
Text
: Reif
Quantum statistics
The quantum approach to mechanics can be formulated in a statistical form using
expectation values or means. For example, the mean energy of a classical gas
(without momentum or positiondependent forces) can be cast as
E
=
d
3
q
1
∫
...
d
3
q
N
∫
E
exp(

ßE
)
d
3
p
1
...
d
3
p
N
d
3
q
1
∫
...
d
3
q
N
d
3
p
1
...
d
3
p
N
=
E
∫
exp(

ßE
)
d
3
p
1
...
d
3
p
N
d
3
p
1
...
d
3
p
N
∫
(20.1)
For a single particle in contact with a heat bath, this just becomes
E
=
E
∫
exp(

ßE
)
d
3
p
d
3
p
∫
.
(20.2)
In other words, take the observable of interest (
E
here), evaluate it using some weight
function [
∫
exp(
ßE
) ], then make sure the weight function is properly normalized (divide
by
∫
d
3
p
).
Quantum mechanics can be formulated in the same way.
As you know from PHYS
285, Heisenberg's uncertainty principle prevents us from knowing the precise position
and momentum of particles.
Hence, we take averages or means:
E
=
E d
3
p
∫
d
3
p
∫
(20.3)
or
x
=
x d
3
x
∫
d
3
x
∫
(20.4)
The quantity
is called the density matrix and is analogous to the Boltzmann weight.
In terms of the quantum wavefunctions
,
is defined as
≡
*
.
(20.5)
Now, all that really counts as far as observables (
e.g.
<
E
> or <
x
>) are concerned is the
density matrix, not the wavefunction. Nevertheless, quantum mechanics was
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 Spring '08
 DavidBoal
 Physics, mechanics, Photon, Quantum Field Theory, Fundamental physics concepts, David Boal

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