- #1

- 6

- 1

I have a problem in understanding the quantum operators in grand canonical ensemble.

The grand partition function is the trace of the operator: [tex]e^{\beta(\mu N-H)}[/tex] (N is the operator Number of particle)

and the trace is taken on the extended phase space:

[tex]\Gamma_{es}= \Gamma_1 \times \Gamma_2 \times .... \Gamma_N ...[/tex]

(product of all the phase spaces with arbitrary N).

But N is a constant of motion: [tex] [H,N]=0 [/tex] (if it were not so, it would be impossible to define the grand partition function in this way).

So my question is: how fluctuations of the avarage value of N can be justified with the fact that N is a constant of motion?

I understand that if we fix the chemical potential (that is exactly the hypothesis of the grand canonical ensemble), there are many corresponding values of N, but their avarage is fixed . But if N (operator) is constant, the time evolution should be only in a phase space [tex]\Gamma_N[/tex] with fixed N and so no fluctuations should be allowed (the system cannot change its own phase space during time evolution).

The grand partition function is the trace of the operator: [tex]e^{\beta(\mu N-H)}[/tex] (N is the operator Number of particle)

and the trace is taken on the extended phase space:

[tex]\Gamma_{es}= \Gamma_1 \times \Gamma_2 \times .... \Gamma_N ...[/tex]

(product of all the phase spaces with arbitrary N).

But N is a constant of motion: [tex] [H,N]=0 [/tex] (if it were not so, it would be impossible to define the grand partition function in this way).

So my question is: how fluctuations of the avarage value of N can be justified with the fact that N is a constant of motion?

I understand that if we fix the chemical potential (that is exactly the hypothesis of the grand canonical ensemble), there are many corresponding values of N, but their avarage is fixed . But if N (operator) is constant, the time evolution should be only in a phase space [tex]\Gamma_N[/tex] with fixed N and so no fluctuations should be allowed (the system cannot change its own phase space during time evolution).

Last edited: