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## Homework Statement

"The numerator of this fraction:

[tex]\overline{E}=\frac{\int \! E N(E)D(E)dE}{\int \! N(E)D(E)dE}[/tex]

(N(E) is the number of particles in an energy state, D(E) is the density of states)

is the total (as opposed to the average particle) energy, which we'll call [tex]U_{total}[/tex] here (In other words, the total system energy U is the average particle energy [tex]\overline{E}[/tex] times the total number of particles N.) Calculate [tex]U_{total}[/tex] as a function of [tex]E_{Fermi}[/tex] and use this to show that the minimum (T=0) energy of a gas of spin-1/2 fermions may be written as:

[tex]U_{total}=\frac{3}{10} (\frac{3\pi^2 \hbar^3}{m^{3/2}V})^{2/3} N^{5/3}[/tex]

## Homework Equations

- The above.

[tex]D(E) = \frac{(2s+1)m^{3/2}V}{\pi^2 \hbar^3 \sqrt{2}}E^{1/2}[/tex]

[tex]N(E)_{FD} = \frac{1}{e^{(E-E_{f})/k_{B}T}+1}[/tex]

[tex]N(E_{F})_{FD} = \frac{1}{2}[/tex]

[tex]\overline{E} = k_{B}T[/tex]

## The Attempt at a Solution

I'm having a hard time deciding what should be multiplied together, and in which ways.

I thought it would be [tex]N(E_{F})_{FD} * \overline{E}[/tex] (the ones above), yet that obviously isn't a function of [tex]E_{Fermi}[/tex], it's just a function of temperature.

I guess I'm having a hard time deciding which equations I should be using.