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

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Homework Help Overview

The discussion revolves around the partition function of a system of N particles, specifically questioning whether the partition function Z for N independent particles can be expressed as Z=(Z_1)^N, where Z_1 is the partition function for a single particle. The context includes considerations of classical versus quantum systems and the implications of particle indistinguishability.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the validity of the partition function expression in classical systems and question its applicability in quantum systems, particularly in relation to the Pauli exclusion principle and over counting issues related to identical particles.

Discussion Status

The discussion is active, with participants providing insights into the differences between classical and quantum treatments of the partition function. Some guidance has been offered regarding the use of the grand-partition function in quantum cases, and there is an acknowledgment of the need to address over counting in classical systems.

Contextual Notes

Participants reference Gibbs' Paradox and the implications of particle indistinguishability, indicating that assumptions about particle identity and statistical mechanics are under examination.

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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?
 
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Even without quantum considerations, you end up with over counting if the particles are identical. See Gibb's Paradox.
 
Right. Sorry, I meant to write
[tex]Z={1\over {N!}}(Z_1)^N[/tex]
Does that take care of over counting?
What about the quantum case?
 
It is not true for the quantum case. The quantum case is easier handled via grand-partition function.
 
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?
 
Yep. But like I said, usually, you calculate the grand-partition function (which factorises neatly into a function of single particle states).
 

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