- #1

Muthumanimaran

- 81

- 2

## 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

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