Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.3.

Probably the most common continuous probability density is the gaussian distribution, specified by

First, we need to normalize the distribution by finding . That is, we must have

The gaussian integral is very common, and the result is that

Although there is no closed form indefinite integral, the definite integral can be found by a cute trick.

We can now transform to polar coordinates using

Therefore

Using Maple (or integration by parts) we can work out the average and variance.

The distribution has the standard bell shape. Here’s a plot for and :