Lowest Energy State of Hydrogen Atom: Find A, Don't Miss the Obvious!

[Matthollyw00d]

A hydrogen atom is in the state \psi=Ar^2e^{-r/a}cos(\theta).
I need to find lowest energy state and etc. Obviously normalize to find A, but I'm not seeing the obvious linear combination of wave functions; and I really don't think my instructor wants me to do several inner products (plus actually calculate the several different wave functions) just to find the coefficients. Am I over looking something obvious?

 

[vela]

You should be able to at least identify the quantum numbers l and m_l. Start there.

 

[fzero]

You should be able to figure out which spherical harmonic is involved pretty quickly. This gives you two quantum numbers. The degree of the radial polynomial tells you the largest value of n involved. The rest involves looking at a table of solutions and finding the right linear combination.

 

[Matthollyw00d]

That was my first instinct, which gave me l=1, m=0. But the e^{(-r/a)} made me think that n=1, which then threw out my l (since l<n). Also the r^2, made me think n=2, but in either case I didn't see any good linear combination jumping out at me.(and it definitely wasn't just 210) I'll look at it again after I've had some coffee... it's been a long few days.

 

[vela]

I think you're going to have to calculate the inner products you were trying to avoid. If you use specific eigenfunctions, it's actually not too bad since all the integrands are just a polynomial multiplied by an exponential. You may be able to work out the general case, relying on some properties of the generalized Laguerre polynomials, like their orthogonality.

 

[Matthollyw00d]

Yah, I ended up using inner products and it all worked out. It's not that the problem was hard, the method just didn't seem of appropriate length. My professor is inconsistent with homework trends.