Statistical Physics: Partition function and fermions

[Niles]

Homework Statement

Hi all.

The partition function for fermions is (according to Wikipedia: http://en.wikipedia.org/wiki/Partit...hanics)#Relation_to_thermodynamic_variables_2) given by:

$$Z = \prod\limits_i {\left( {1 + \exp \left[ { - \beta \left( {\varepsilon _i - \mu } \right)} \right]} \right)},$$

where the product is over the different states. I cannot see how this works out correct:

Let's look at a system with 3 single-particle (energi 0, 1 and 2) states with two fermions. Each fermion can be in one state, so there is a total of 3 states. Using the above expression this should give us

$$Z = \left( {1 + \exp \left[ { - \beta \left( {(0 + 1) - \mu } \right)} \right]} \right)\left( {1 + \exp \left[ { - \beta \left( {(0 + 2) - \mu } \right)} \right]} \right)\left( {1 + \exp \left[ { - \beta \left( {(1 + 2) - \mu } \right)} \right]} \right).$$

Have I understood this correctly?

Thanks in advance.Niles.

 

[Jhero]

Hello Niles,

Yes, your understanding is correct. The partition function for fermions takes into account the different energy levels and number of particles in a system. In your example, with 3 states and 2 fermions, the first term in the product corresponds to the case where both fermions are in the lowest energy state (energy 0), the second term corresponds to one fermion in the lowest energy state and the other in the second energy state (energy 1), and the third term corresponds to both fermions in the highest energy state (energy 2). This is because fermions follow the Pauli exclusion principle and cannot occupy the same energy state.

I hope this helps clarify your understanding of the partition function for fermions. Let me know if you have any other questions.