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Assume a low temperature regime in which levels up to the Fermi Level, $E_F$, are populated.

I have evaluated the density of states in energy space as $$D(E)=\frac{L^3}{\pi^2\hbar^3}(2m_e^2E)^{1/2},$$

and the Fermi Energy as $$E_F=\frac{(3\pi^2n_e)^{2/3}}{2m_e}\hbar^2,$$

where symbols have usual meanings and $n_e=\frac{N}{L^3}$. So far, so good. Now, I wish to evaluate the average kinetic energy of the electrons. Intuitively, I would integrate $$\int_0^{E_F}E\times D(E) dE$$ which would give me the expectation value of kinetic energy $\equiv \left< E_K\right>$ - or so I thought. Apparently, I need to divide by the number of molecules $N$ once more, such that $$\left< E_K\right> = \frac{\int_0^{E_F}E\times D(E) dE}{\int_0^{E_F} D(E) dE}.$$ I can understand that to find the average energy of an electron, we need to divide through by $N$. But then physically, what is the integral in the numerator above? As far as I remember, for other distribution functions, the expectation value or average of a quantity was only this integral in the numerator. Have I misunderstood something basic? Would be grateful if someone could quickly and logically clear this up for me.

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Did some more digging on this and answered myself. The integral in the numerator in fact determines the average energy for the entire system, in other words, the internal energy $U$.

By analogy relating to thermodynamics and kinetic theory for a monatomic ideal gas, each molecule is said to have a kinetic energy $E_K=\frac{3}{2}k_BT$ and internal energy $U$ is defined as $U=\frac{3}{2}Nk_BT$.

Pretty simple really... mods feel free to delete, but happy to leave this standing as it is.

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