Quantum grand canonical ensemble

The quantum grand canonical ensemble is the quantum extension of the grand canonical ensemble. Its density matrix is

\hat{\rho}=z^{N}Q_{N}(V,T)

where $Q_{N}$ is the quantum canonical partition function.

\mathcal{Z}(z,V,T)=\sum_{N=0}^{\infty}z^{N}Q_{N}(V,T)=\text{Tr}(e^{-\beta(\hat{H}-\mu \hat{N})})

where $z$ is the fugacity, $V$ is the volume, $\beta=1/k_{B}T$ with $k_{B}$ is the Boltzmann constant and $T$ is the temperature, $\hat{H}$ is the Hamiltonian of the system, $\mu$ is the chemical potential and $\hat{N}$ is the particle number operator.

All properties derived from the partition function have identical expressions to the classical grand canonical.