PHYS 445 Lecture 22  Bosons
22  1
© 2001 by David Boal, Simon Fraser University.
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Lecture 22  Bosons
What's Important:
•
partition function
•
number distribution
•
condensation
Text
: Reif
Boson partition function
We now tackle the general problem of BoseEinstein statistics.
The situation is more
difficult than pure photons because the sum over
n
i
is now restricted by the fixed value
of the particle number
N
.
It should not surprise us to see a chemical potential appear
to enforce the constraint on
N
.
As usual, we start with the generic partition function
Z
=
exp(

ßn
1
1

ßn
2
2
...)
R
∑
(22.1)
which is now subject to the constraint
Σ
i
n
i
=
N
.
(22.2)
A mathematical trick is introduced to enforce the constraint.
Viewing
Z
as a function of
N
, its behaviour is
Now, if we multiply the rapidlyincreasing function
Z
(
N
) by a rapidlydecreasing
function exp(
N
) we can produce a distribution with a sharp spike at the desired
value of
N
:
N
Z
(
N
)
rapidly
increasing
N'
exp(
N'
)
N'
Z
(
N'
)
times
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 Spring '08
 DavidBoal
 Physics, Photon, Quantum Field Theory, Fundamental physics concepts, David Boal

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