About mixed integral for perturbation in hydrogen atom

[ntusg]

Homework Statement

During my calculation of hydrogen atom perturbation, I need to integral below in cartesian coordinate. It is given that below integral can be transformed.

Homework Equations

Anyone could help to see what will the transformed integral in polar coordinate if the left-hand-side integral were changed to r*x^2*exp(-r) instead of x^2*exp(-r).

Thanks.

The Attempt at a Solution

 

[Dick]

They are just using that the integral of x^2*exp(-r), y^2*exp(-r) and z^2*exp(-r) are all equal by symmetry. So if you add them all together you get r^2*exp(-r). Hence the '1/3'. Putting another r in front of x^2*exp(-r) doesn't stop you from doing the same thing.

 

[ntusg]

Hi Dick, I just got it, and it only about rewriting dV in Cartesian to the polar coordinate.
Thanks for the help!