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Chapter 10: Quantum physics
53
For a macroscopic system in equilibrium holds
[
H,
ρ
]=0
. If all quantumstates with the same energy are
equally probable:
P
i
=
P
(
E
i
)
, one can obtain the distribution:
P
n
(
E
)=
ρ
nn
=
e

E
n
/kT
Z
with the state sum
Z
=
±
n
e

E
n
/kT
The thermodynamic quantities are related to these de±nitions as follows:
F
=

kT
ln(
Z
)
,
U
=
±
H
²
=
∑
n
p
n
E
n
=

∂
∂
kT
ln(
Z
)
,
S
=

k
∑
n
P
n
ln(
P
n
)
. For a mixed state of
M
orthonormal quantum states with
probability
1
/M
follows:
S
=
k
ln(
M
)
.
The distribution function for the internal states for a system in thermal equilibrium is the most probable func
tion. This function can be found by taking the maximum of the function which gives the number of states with
Stirling’s equation:
ln(
n
!)
≈
n
ln(
n
)

n
, and the conditions
∑
k
n
k
=
N
and
∑
k
n
k
W
k
=
W
. For identical,
indistinguishable particles which obey the Pauli exclusion principle the possible number of states is given by:
P
=
²
k
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.
 Spring '10
 Ye
 Energy, Quantum Physics

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