Solving Troublesome Integral: Quantum Mechanics Problem

[genxhis]

A quantum mechanics problem calls for the reader to find the momentum space wave function of $$\Psi(x,0) = A/(x^2 + a^2)$$. But I do not know how to resolve the Fourier transform:

$$\Phi(p, 0) = \frac{1}{\sqrt{2 \pi \hbar}}\int_{-\infty}^\infty e^{-i p x/\hbar} \frac{A}{x^2+a^2}dx.$$​

The problem implies an exact solution can be found since it subsequantly asks you to check normalization and compute the expected values of p and p² using the transformed fn. Mathematica evaluates the transform in terms of a special fn MeijerG.

 

[Tide]

The integral has singularities at x = ia and x = -ia. To evaluate the integral you need evaluate the residues resulting from closing the path either in the upper half plane (Im z > 0) or the lower half plane (Im x < 0) depending on the sign of x (i.e. with a semicircle whose radius approaches infinity).

 

[genxhis]

Singularities? I don't understand. The integrand is well-defined for all real values x and the integral almost certainly exists over any range. Sorry, I have not had a course in complex analysis.

 

[Tide]

genxhis,

Sorry about that - I edited my original post to reflect the correct locations of the singularities.

I assumed you were familiar with complex analysis but, since you're not, what I said won't make sense to you. I'll have to think a bit about how to do it without invoking complex analysis.