Derivation of Sudarshan-Glauber P function

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Homework Statement


It is not a problem, But I am doing a coursework on Quantum Optics on my own. The following derivation is for P-function in Quantum optics from the book Quantum Optics by Marlan O.Scully and M. Suhail Zubairy. I attached the Image of the derivation with this Question.

Homework Equations


Relevant Equations are given in the Image

The Attempt at a Solution


I expanded the trace in the (3.1.11) in |α> basis. But I am unable to get (3.1.13).
 

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Muthumanimaran said:

Homework Statement


It is not a problem, But I am doing a coursework on Quantum Optics on my own. The following derivation is for P-function in Quantum optics from the book Quantum Optics by Marlan O.Scully and M. Suhail Zubairy. I attached the Image of the derivation with this Question.

Homework Equations


Relevant Equations are given in the Image

The Attempt at a Solution


I expanded the trace in the (3.1.11) in |α> basis. But I am unable to get (3.1.13).
They are just saying that (by definition),

$$ \int d^2 \alpha ~~\delta(a- \alpha) \delta(a^\dagger - \alpha^*) f(\alpha) f(\alpha^*) = f(a) f(a^\dagger) $$
 
Yeah I absolutely understand that, but I don't know how to expand f(a)f(a†). Can you help me with this?
 
Muthumanimaran said:
Yeah I absolutely understand that, but I don't know how to expand f(a)f(a†). Can you help me with this?
Ah, but then 3.1.13 follows directly from 3.1.11 with the choice ## f(a) = a^m## and ## f(a^\dagger) = (a^\dagger)^n##. That's all there is to it. I am not sure what the question is, then.
 
Oh yes, I confused myself a bit. Now I got. Its pretty straightforward. Thank you.
 

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