- #1

andre220

- 75

- 1

## Homework Statement

I need to evaluate the following integral: [tex]\sqrt{\frac{2}{\pi}}\frac{\sigma}{\hbar}\int\limits_{-\infty}^{\infty}p^2 e^{-32\sigma^2(p-p_0)^2/\hbar^2}\,dp[/tex]

## Homework Equations

Integrals of the form: [tex] \int\limits_{-\infty}^{\infty}x^2e^{-ax^2}\,dx=\frac{1}{2}\sqrt{\frac{\pi}{a^3}}[/tex], taken from a table

## The Attempt at a Solution

Ok so, obviously the integral provided is not of the form of the one given in the table, however, it is close. My thought was to change the integral to be such that we let some new variable [itex] t = (p-p_0)^2[/itex] so then we would have the following new integral [tex]\sqrt{\frac{2}{\pi}}\frac{\sigma}{\hbar}\int\limits_{-\infty}^{\infty}(t+p_0)^2 e^{-32\sigma^2 t^2/\hbar^2}\,dt[/tex]. Then I would expand it so that there are three integral that need to be evaluated i.e. (by expanding the [itex](t + p_0)^2[/itex] term.

My question is... is this mathematically sound (or allowable), I can't see a reason why this shouldn't work but I wanted to get some feedback on my proposed method. Thank you