# Hydrogen wave function

1. Jan 4, 2009

### toqp

1. The problem statement, all variables and given/known data
A problem from an examination:
A hydrogen atom is in the state
$$\Psi=A(\sqrt{6}\psi_{100}+\sqrt{2}\psi_{200}+\psi_{211}+2\psi_{31-1}+\sqrt{3}\psi_{321}+3\psi_{32-2})$$
where $$\psi_{nlm}$$ are the eigenfunctions of hydrogen. Find A so that the equation is normalized.

2. Relevant equations
$$\psi_{nlm}=R_{nl}Y^m_l,\ \ \ Y^m_l=AP^m_l(cos\theta)$$

3. The attempt at a solution
Well I can get the angular parts for Y from some handbook. But in the test no additional data is provided, so should I just remember the equations for R to get the problem solved.

I mean... in another problem it was told:
"Remember that $$a_{+}\psi=\sqrt(n+1)\psi_{n+1}$$"

and then suddenly I have to remember how to get Rnl?
Well, ok.

I can remember that (according to Griffiths)
$$R_{nl}=\frac{1}{\rho}(\rho)^{l+1}\nu(\rho),\ \ \ \rho=\frac{r}{an}$$

But then I also have to remember the recursion formula for the coefficients of $$\nu(\rho)$$?

I can understand if that stuff is really something one needs to memorize, but what I find confusing is that in several cases stuff far easier to remember is given along with the problem. Which leads me to think I've gotten something wrong... that it shouldn't be this complex.

2. Jan 4, 2009

### LMZ

offtop: totally agree with you about memorizing all this coeffs and formulas.

normalization condition is:
$$\int_{-\infty}^{\infty}\Psi \Psi^{*} = 1$$
where $$\Psi^{*}$$ is http://en.wikipedia.org/wiki/Complex_conjugation" [Broken] of $$\Psi$$

so, introduce your $$\Psi$$ in integral, and get $$A$$ from there.

Last edited by a moderator: May 3, 2017