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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.

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  • $\begingroup$ 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. $\endgroup$ – Nogueira Feb 16 '15 at 18:14
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In basic quantum mechanics this is indeed strange, because when ones studies basic QM the assumption is that the particle number is conserved, so such terms do not appear.

This term is more natural in the framework of 2nd quantization. This framework is natural for many-body quantum problems where the system is not described as having a fixed particle numbers (or excitations). In order to do statistics and thermodynamics on such systems, states with different particle numbers have to be taken into account. Now, the chemical potential has its usual meaning - the energy associated with adding a particle into the system.

For an introduction to 2nd quantization and quantum partition functions, see for example Condensed Matter Field Theory by Altland and Simons, chapters 2 and 4.

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