Statistical mechanics - Partition function of a system of N particles

Jalo

1. The problem statement, all variables and given/known data

Imagine a system with N distinguishable particles. Each particle may be in two states of energy: -ε and +ε.

Find the the partition function of the system

2. Relevant equations



3. The attempt at a solution

I know that I have to find the partition function for a single function, Z, and my final result will be Z^N. Now, I'll say that:

(Where it says ε it's meant to be ε(r) )

Z = Ʃ_r exp(-β(ε - ε) ) = Ʃ_r exp(-βε) * exp(βε) =
= Ʃ_r exp(-βε) * Ʃ_r exp(βε)



I'm sure this is incorrect. It doesn't make sense in my head.. E(r) is the energy associated with each microstate, therefore saying that E(r) = ε(r) - ε(r) can't make any sense! I know that the result is:

Z = ( exp(βε) + exp(-βε) )^N

I have no idea how to get there tho. How did it became a sum? How do I get rid of the summatories?

Any help will be appreciated!
Thanks.

klawlor419

The partition function is a summation over states. You simply are using the summation wrong. It is not a summation over the energy levels of within the exponent. It is a summation over e(-Es/T).

klawlor419

Look at any example problem in a thermo book for a 2-state system

Jalo

8ikmAm I not summing over the expoent of the energy of each microstate?

Do you know any good statistical mechanics book you'd advise me reading?

EDIT:
Is it a summation over all the states of energy instead of the energies of each microstate? Because then the solution would make sense!

klawlor419

No. You are summing over the exponential function raised to the -Es/T.

A good book is Thermal Physics by Kittel + Kroemer

klawlor419

Where Es is the energy of the s-th state