Partition Function of N Particles: Is Z=(Z_1)^N?

[Pacopag]

Homework Statement

If we have a system of N independent particles and the partition function for one particle is Z_1, then is the partition function for the N particle system Z=(Z_1)^N?

Homework Equations

The Attempt at a Solution

I'm pretty sure that this is true for a classical system, but I'm not sure if it's true for a quantum system. Does the Pauli exclusion principle spoil this somehow?

 

[genneth]

Even without quantum considerations, you end up with over counting if the particles are identical. See Gibb's Paradox.

 

[Pacopag]

Right. Sorry, I meant to write
Z={1\over {N!}}(Z_1)^N
Does that take care of over counting?
What about the quantum case?

 

[genneth]

It is not true for the quantum case. The quantum case is easier handled via grand-partition function.

 

[Pacopag]

So in the quantum case, if we want to use the canonical ensemble, we have to calculate the whole partition function all in one shot?

 

[genneth]

Yep. But like I said, usually, you calculate the grand-partition function (which factorises neatly into a function of single particle states).