Difficult problem dealing with stationary states of hyrdrogen

1. Dec 13, 2006

holden

Really confused by this one.

1. The problem statement, all variables and given/known data

I'm given that a tritium atom, with one proton and two neutrons in the nucleus, decays by beta emission to a helium isotope with two protons and one neutron in the nucleus. During the decay, the atom changes from hydrogen to singly-ionized helium so Z doubles.

I need to find the probabilities that the helium atom is in the ground state or the first excited state (2s) immediately after the decay; I'm given that the tritium was in the 1s ground state before the decay.

I'm also told to ignore spin.

2. Relevant equations

I'm given that since I need to calculate the constants multiplying the eigenfunctions of helium (see below), that I need to use $$<\psi_{n'l'm'}|\psi_{nlm}> = \delta_{n'n}\delta_{l'l}\delta_{m'm}$$ to do that.

3. The attempt at a solution

Well, to solve it, I need to write the ground state of tritium as a superposition of stationary states of He+, because the square of the absolute value of the constant multiplying a particular helium eigenfunction gives the probability that the helium atom is in that state after the decay.

The trouble is I don't know what the stationary states of He+ would look like.

2. Dec 14, 2006

dextercioby

He+ is hydrogenoid ion, so the stationary states are computed in QM books when dealing with the Kepler problem for the Coulomb potential.

Daniel.

3. Dec 15, 2006

holden

Hm.. There's no section about that in my QM book, and we haven't mentioned any of the terms you used in class. We've discussed a little bit about hydrogen atoms.. I think I'm supposed to use perturbation theory to find them.. but it's hard when I have no examples or any instruction about it =/

4. Dec 15, 2006

dextercioby

C'mon, it's simply replacing e, the electron's charge (in absolute value), with Ze, the hydrogenoid ion's charge.

Daniel.

5. Dec 15, 2006

holden

Heh. I never would have known that. Thanks. I went back through (from the beginning) and calculated the wavefunctions for n=2,l=0 and n=1,l=0 (2s and 1s for He+, respectively).. but what do I do with those? I know I'm supposed to write the ground state of tritium as a superposition of stationary states of He+, so I added those together.. But how to find probabilities? I know the square of a constant multiplying a particular eigenstate of Helium is the probability that it will be in that state, but, uh.. does that mean I just need to square the constants out in front of each state?

My teacher said to calculate constants using inner products. TBH, I have no idea what he's talking about in this case since everything is in terms of r.