# Grand canonical Hamiltonian

How to explain introducing "grand canonical" Hamiltonian $$\hat{H'}= \hat{H}-\mu \hat{N}$$ when we study a quantum system with fixed chemical potential? I understand such a substitution in a partition function but it's completely strange to see this in a pure quantum mechanics, e.g. writing Heisenberg operators or Green's functions.

I encountered that when redaing about Green's functions for interacting Bose- and Fermi-gases.

• This extra term only add a energy in function of number of particles. If the first halmitonian commute with number operator then this new halmitonian is simple a change of zero energy, without physical relevance for dynamics. – Nogueira Feb 16 '15 at 18:14