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phos19

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- Fermi energy for arbitrary multiplicity

I ran across the following problem :

Consider a gas of ## N ## fermions and suppose that each energy level ## \varepsilon_n## has a multiplicity of ## g_n = (n+1)^2 ##. What is the Fermi energy and the average energy of this gas when ## N \rightarrow \infty## ?

The average occupation number for a state of the ##n##th level is:

$$\langle N_n \rangle = \dfrac{1}{ e^{\beta(\varepsilon_n + \mu)} + 1 }$$

Usually if the system has a fixed degeneracy, say only the spin degeneracy ##g = 2s +1## , one can write the total number of particles ##N## as an integral over ##\vec{p}##:

$$

N = \sum_n \langle N_n \rangle = \dfrac{gV}{h^3} \int d^3 p \ \dfrac{1}{ e^{\beta(\varepsilon_p + \mu)} + 1 }

$$

One can than find the Fermi energy in the limit ##T \rightarrow 0##.

But this is not the case when ##g = g(n)##... Any hints on how to do this ?

**Statement:**Consider a gas of ## N ## fermions and suppose that each energy level ## \varepsilon_n## has a multiplicity of ## g_n = (n+1)^2 ##. What is the Fermi energy and the average energy of this gas when ## N \rightarrow \infty## ?

**My attempt:**The average occupation number for a state of the ##n##th level is:

$$\langle N_n \rangle = \dfrac{1}{ e^{\beta(\varepsilon_n + \mu)} + 1 }$$

Usually if the system has a fixed degeneracy, say only the spin degeneracy ##g = 2s +1## , one can write the total number of particles ##N## as an integral over ##\vec{p}##:

$$

N = \sum_n \langle N_n \rangle = \dfrac{gV}{h^3} \int d^3 p \ \dfrac{1}{ e^{\beta(\varepsilon_p + \mu)} + 1 }

$$

One can than find the Fermi energy in the limit ##T \rightarrow 0##.

But this is not the case when ##g = g(n)##... Any hints on how to do this ?

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