445lec22 - PHYS 445 Lecture 22 Bosons Lecture 22 Bosons...

PHYS 445 Lecture 22 - Bosons 22 - 1
© 2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.
Lecture 22 - Bosons
What's Important:
• partition function
• number distribution
• condensation
Text : Reif
Boson partition function
We now tackle the general problem of Bose-Einstein 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 rapidly-increasing function Z ( N ) by a rapidly-decreasing
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