Normalizing a radial wave function

[atarr3]

1. The problem statement, all variables and given/known data
Show that the radial function R_{31} is normalized.


2. Relevant equations
\frac{1}{a_{0}^{3/2}}\frac{4}{81\sqrt{6}}\left(6-\frac{r}{a_{0}}\right)\frac{r}{a_{0}}e^{-r/3a_{0}}

\int^{\infty}_{0}r^{2}R_{31}*R_{31}dr=1

3. The attempt at a solution
So I plugged that radial function in and got \int^{\infty}_{0}a_{0}^{2}u^{2}\left(6u-u^{2}\right)^{2}e^{-2u/3}du=1 all multiplied by some constant and u=\frac{r}{a_{0}}

I'm getting \frac{243}{4} times the constant, and that does not equal one. So I feel like I'm not using the right equation for this one.

[badphysicist]

Remember what coordinate system you are working in.

[atarr3]

Spherical coordinates. So there's an r^2 thrown in there. Did I not account for everything in my integral?

[badphysicist]

Well, it looks like you didn't in your equation with the u's.

[dotman]

I'm seeing an extra a_o^2, since there's an (\frac{1}{a_0^{3/2}})^2 = \frac{1}{a_0^3} term, an r^2 = a_0^2u^2 term, and a dr = a_0 du term:

\frac{1}{a_0^3}a_0^2u^2a_0 du = u^2 du

I'm curious as to how you're going about integrating

C\int_0^\infty (u^6 - 12u^5 + 36u^4)e^{-2u/3} du ?

Does this have to be done by repeated parts (after you split it up of course), or is there a trick?

[kuruman]

I'm curious as to how you're going about integrating

C\int_0^\infty (u^6 - 12u^5 + 36u^4)e^{-2u/3} du ?

Does this have to be done by repeated parts (after you split it up of course), or is there a trick?

A very useful definite integral that was given to me when I learned about the hydrogen atom is

\int^{\infty}_{0}x^n \; e^{-ax}dx=\frac{n!}{a^{n+1}}; \; (a>0;\; n\; integer \:>0)