I need some help with this problem:
Consider a system of N distinguishable particles. Each of these particles can be in a state with energy ϵ or −ϵ. Consider that the states with energy ϵ are M-fold degenerate, and those with energy −ϵ are P-fold degenerate. Using the micro canonical ensemble, determine the entropy S(E,N)
I haven't been able to correctly calculate the number of microstates Ω(E,N). I know that in the case where the states are not degenerate, it's just:
Ω(E,N)=N!N+!(N−N+)!
where N+ is the number of particles with energy ϵ, but in this case I do not know how to take into account the degeneracies.
Any help will be truly appreciated.
Answer
Every one of N+ (N−=N−N+) particles with the energy ϵ (−ϵ) is in one of P (M) possible states. Hence there is the additional factor PN+MN− and the answer is Ω(E,N)=N!N+!N−!PN+MN−
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