Consider a fermion gas of N electrons in volume V

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Consider a fermion gas of N electrons in volume V. Using the density of momentum
states , show that the Fermi energy can be written as

(h^2)/(2m (subscript e)) ((3N)/(8pi V))^(3/2)

Homework Equations

g(p)=(V/h^3) (4pi) p^2

The Attempt at a Solution

N=integral from 0 to k(subsript F) g(k) dk

N=(V/h^3)4pi (p^3)/3

N=(V/h^3)(4/3)pi ((k(subsript F))^3)

N=(V/h^3)(4/3)pi ((2mE(subscriptF))/((h-bar)^2)) ^(1/2)) ^3

N=(V/h^3)(4/3)pi ((2mE(subscriptF))/((h-bar)^2)) ^(3/2))

(3N*h^3)/(V*4pi)= ((2mE(subscriptF))/((h-bar)^2)) ^(3/2))

((3N*h^3)/(V*4pi))^(2/3)= ((2mE(subscriptF)*4pi^2)/(h^2))

 

[member 553083]

^(1/2)((3N*h^2)/(V*4pi))= (2m (subscript e)) E(subscriptF)E(subscriptF)=(h^2)/(2m (subscript e)) ((3N)/(8pi V))^(3/2)