What Is the Correct Approach to Integrate the Wigner Function for Fock States?

[Muthumanimaran]

Homework Statement

I am currently doing a problem in Quantum optics, specifically the problem of finding Wigner Function for Number states or Fock states. I am actually did the problem in a different way and found that Wigner function for Number states is proportional to product of error function and Laguerre polynomials, now I finding the Wigner function from P-Glauber Sudarshan Function, where I encountered this Integral,

$$ \frac{2 exp(|α|^2)} {π^3 n!}\ ∫ \frac{exp(-|β|^2-4|α||β|)}{π^2*n!}\ \frac{∂^(2n)}{∂β^n∂(β*)^n}\ δ^2(β) d^2β $$

δ(β) is dirac delta function and α,β are complex

The Attempt at a Solution

I tried the solve the integral but shifting the index and got
$$ \frac{2(-1)^n (4)^(2n) exp(-|α|^2) |α|^(2n)}{π^3 n!}\ $$
But the correct answer is in terms of product of Laguerre polynomial and error function. Please help

 

[jedishrfu]

While I don't know the answer here, I was thinking what if you reversed your thinking and tried differentiating the Laguerre polynomial and see if you can reformulate it into the integrand you have. It might give you insight on how to do the integral.