Advanced Quantum Mechanics Schwabl · Problem 1.9

Spin-1/2 Fermions

Physics — Quantum Mechanics · thermal-expectation-value-occupation-number

Problem

Determine, for bosons and fermions, the thermal expectation value of the occupation-number operator

\hat{n}_i

for the state

|i\rangle

in the grand canonical ensemble, whose density matrix is given by

\rho_G = \frac{1}{Z_G} e^{-\beta(H - \mu\hat{N})}

with

Z_G = \mathrm{Tr}\left(e^{-\beta(H-\mu\hat{N})}\right)