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I have a confusion that in an Homogeneous electron gas.

When $T=0$, we know that $\mu=E_F$.

So Kinetic Energy per electron=$\frac{3}{5}E_F=\frac{3}{5}\mu$.

I thought that Temperature is related to kinetic energy. How can $T=0$ condition can have kinetic energy?

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Actually what you say is a quantum mechanical problem. Electron gas model quantised the energy level, and Pauli exclusive principle told us fermions can't condesate to k=0 state all. T=0 gives step-function-like distribution of fermions so obviously they have kinetic energy to quantum view.

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