# why is the chemical potential included in the hamiltonian for a systeme coupeled to a particle reservoir

I am beginning with second quantification language so i saw that if we are in grand canonical ensemble then:
$$H=H_0 - \mu N$$

naturally i thought that this $\mu$ would be included in the weight factor of the statistical average but i am surprised too find it in H, is it equivalent or is does this have a physical sens that i am not seeing ?

• hint: what is the weight factor? hint2: how would you compute the internal energy? – Bort Jan 15 '16 at 11:35

## 1 Answer

Usually we will include the term −μN into the definition of the Hamiltonian Ĥ, so that expressions for the thermal average in the canonical and grand canonical ensembles are formally identical.