A integral about expotential function

[athrun200]

Homework Statement

Homework Equations

Unknown

The Attempt at a Solution

How to calculate this integral?
I have tried substitution and by parts. But they fail to get the answer.

 

[hunt_mat]

There is a trick for this, square the integral:
$$\left(\int_{-\infty}^{\infty}e^{-x^{2}}dx\right)^{2}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^{2}+y^{2})}dxdy$$
Now the idea is to make a change of co-ordinates from cartesian to polar...

 

[dextercioby]

Are you taking quantum mechanics before knowing what a gaussian integral is ? Because you shouldn't...

 

[athrun200]

Are you taking quantum mechanics before knowing what a gaussian integral is ? Because you shouldn't...

Oh you are right.
In fact, I am self-learning quantum mechanics, and I don't know what maths skills I need to have.

 

[hunt_mat]

So move on from my explanation:
$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^{2}+y^{2})}dxdy=\int_{0}^{2\pi}\int_{0}^{\infty}re^{-r^{2}}drd\theta =2\pi\int_{0}^{\infty}re^{-r^{2}}dr$$
I think I will leave the last bit to you as it's standard integration.