Statistical Physics: Quantum ideal gas

[aburriu]

Homework Statement

I'm reading the book about Statistical Physics from W. Nolting, specifically the chapter about quantum gas.
In the case of a classical ideal gas, we can get the state functions with the partition functions of the three ensembles (microcanonical, canonical and grand canonical). However, in the case of a quantum ideal gas, we can only apply the grand canonical ensemble. Why?

Homework Equations

The Hamilton operator for the whole system is additive:
[ tex ] H = \sum_{i=1}^N H^{(i)} [ /tex ]
where (i) denotes the particle number.
Each particle can be described by the Schrödinger equation:
[ tex ] H^{(i)} |\varphi_k^{(i)}> = \varepsilon_{k} |\varphi_k^{(i)}> [ /tex ]
where the subscript k characterizes the set of quantum numbers (n, l, m_l, m_s )

The Attempt at a Solution

I'm guessing it has to do with the indistinguishability of the particles. I've read something about the Fock states, but I didn't grasp the concept.

 

[Daniel Gallimore]

The partition function is built with the assumption that particles are distinguishable: you must be able to identify a single particle within a large collection of particles and keep track of its state. This does not agree with quantum mechanics at all.

Compare this to the grand canonical partition function: we are instead observing a small collection of particles within a much larger collection of particles, and both collections are able to exchange particles. E.g., if there are four particles currently in a given state, we can't be sure if all four were originally part of our small collection, or if some came from the much larger collection. Even though this is a very classical way to approach a collection of particles, it has the same consequences as if we had just made all of our particles indistinguishable. So this generalizes nicely in quantum mechanics.

Unfortunately, I'm not familiar enough with the microcanonical ensemble to give you a straight answer on this one, but the Wikipedia article seems to suggest that you can treat quantum systems with the microcanonical ensemble, provided the stationary states of the system have a very small range of energies.