- #1

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If you have a discrete collection of single-particle energy levels [itex]\epsilon_i[/itex], then the grand canonical partition function (for noninteracting particles) is defined by:

[itex]\mathcal{Z}(\mu, \beta) = \sum_i \sum_{N_i} exp(N_i \beta(\mu - \epsilon_i))[/itex]

[itex]= \sum_i \sum_{N_i} exp(\beta(\mu - \epsilon_i))^{N_i} [/itex]

where [itex]N_i[/itex] is the occupancy number: the number of particles in state [itex]i[/itex]. To get Bose statistics, the allowable values for [itex]N_i[/itex] are [itex]N_i = 0, 1, 2, ...[/itex], leading to

[itex]\mathcal{Z}(\mu, \beta) = \sum_i \dfrac{1}{1 - exp(\beta(\mu - \epsilon_i))}[/itex]

(because [itex]1+x+x^2 + ... = \frac{1}{1-x}[/itex])

For Fermi statistics, the only possible values for [itex]N_i[/itex] are [itex]N_i = 0, 1[/itex], leading to:

[itex]\mathcal{Z}(\mu, \beta) = \sum_i (1 + exp(\beta(\mu - \epsilon_i)))[/itex]

Here's the question: Suppose that the energy levels for an electron are independent of spin direction. That means that for every single-particle state, there is a second state with the same energy and opposite spin state. Then it seems to me that there are two different ways to take this degeneracy into account:

(1) Replace [itex]\sum_i ...[/itex] by [itex]\sum_i g_i ...[/itex], where [itex]g_i[/itex] is the degeneracy of energy level [itex]i[/itex], and where the index [itex]i[/itex] ranges only over states with distinct energies. In this case, [itex]g_i = 2[/itex], so the result is just to multiple [itex]\mathcal{Z}[/itex] by 2.

(2) Modify the allowable occupancy number [itex]N_i[/itex] to range from [itex]0[/itex] to [itex]g_i[/itex], rather than just 0 or 1.

These two approaches give different answers, but I don't understand, physically, why. It seems that the only thing that should be important is how many electrons can have energy [itex]\epsilon_i[/itex].