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

• Muthumanimaran
In summary, the conversation is about finding the Wigner function for Number states or Fock states in Quantum Optics. The speaker shares their approach of using a different method and finding that the Wigner function is proportional to the product of an error function and Laguerre polynomials. However, they encountered an integral involving a Dirac delta function and complex variables, which they attempted to solve but did not get the correct answer. Another person suggests trying to differentiate the Laguerre polynomial to see if it can be reformulated into the integrand.
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!}\$$

Last edited:
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.

## 1. How do I determine the limits of integration for an integral?

The limits of integration for an integral are typically given in the problem. If they are not given, you can use the properties of the function or the shape of the graph to determine the appropriate limits.

## 2. What is the process for solving an integral?

The process for solving an integral involves first determining the appropriate integration technique, such as substitution, integration by parts, or partial fractions. Then, you must apply the chosen technique and follow through with the necessary algebraic steps until the integral is simplified and can be evaluated.

## 3. How do I know if I have solved an integral correctly?

You can check your work by differentiating your answer and seeing if it matches the original function that you were trying to find the integral of. You can also use online integration calculators or ask a fellow mathematician to check your work.

## 4. Can any integral be solved analytically?

No, not all integrals can be solved analytically. Some integrals have no closed-form solution and can only be approximated using numerical methods.

## 5. How do I know which integration technique to use for a given integral?

There is no one-size-fits-all approach for determining the appropriate integration technique. It often requires practice and experience to recognize which technique will be most effective for a particular integral. However, there are certain guidelines and patterns that can help in making this decision, such as looking for certain terms or functions that may indicate a specific technique to use.

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