- #1
Xemnas92
- 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).
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