Question

In the context of calculating the partition function of a quantum ideal gas of N

indistinguishable particles, we introduced the notion of ocupation numbers np,sz
as the number of particles in the state |p,sz⟩
where p
is the momentum and sz
is the z
-projection of the spin. Then we imposed that
∑p,sznp,sz=N
where N
is the number of particles of the gas. But I found this odd, or perhaps I'm misunderstanding something. Couldn't some particle be in the state
|ψ⟩=12–√(|p1,1/2⟩+|p2,−1/2⟩)
or something like that? How would this count to the occupation number np1,1/2
for example?

Answer 1

Quantum particles can exist in superposition states, but for statistical mechanics calculations, occupation numbers represent particles in distinct quantum states. The Pauli Exclusion Principle ensures no two fermions occupy the same state, allowing for practical counting in large systems.

Step-by-step explanation:

The confusion regarding states such as |\u03c8\u003e = \u215c(|p1,\u2153\u003e + |p2,\u2212\u2153\u003e) arises because while quantum particles like electrons can indeed exist in superposition states, the occupation number np,sz in the context of calculating the partition function for a quantum ideal gas refers to the number of particles in a distinct quantum state.

In a quantum ideal gas, where particles are indistinguishable and the system is treated statistically, we assume that particles occupy distinct energy levels described by quantum numbers (including the z-component of the spin, ms).

The Pauli Exclusion Principle dictates that no two fermions (particles with half-integer spin such as electrons) can occupy the same quantum state simultaneously. Given this principle, for computational purposes in systems with a large number of particles, each state is either occupied by one particle or none, which is tractable when counting the total occupation using quantum numbers.

Superposition states do not directly contribute to the occupation numbers in this statistical representation because the particles are not in a definite state with specific quantum numbers. Therefore, in statistical mechanics, the notion of individual particles in superposition is replaced by an ensemble average where particles distribute among available quantum states.