Quantum Gibbs state
= Quantum Gibbs state
{title2=$\rho_\beta=e^{-\beta H}/\operatorname{Tr}(e^{-\beta H})$}
A <quantum Gibbs state> is the <density operator> of a <canonical ensemble> with inverse <temperature> $\beta=1/(k_BT)$. It exists when the <canonical partition function> $Z=\operatorname{Tr}(e^{-\beta H})$ is finite. The <spectral theorem> assigns <probabilities> $e^{-\beta E_j}/Z$ to an <orthonormal eigenbasis> of the <Hamiltonian>. An additive constant in the <Hamiltonian> cancels between numerator and denominator. For $H=-\mu B\sigma_z$, the <exponential of a Pauli vector>, continued to an imaginary argument, gives $\rho_\beta=[I+\tanh(\beta\mu B)\sigma_z]/2$.