Probability of helium isotope in ground or first excited state after beta decay

[holden]

Really confused by this one.

Homework Statement

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.

Homework 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.

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.

 

[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.

 

[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 =/

 

[dextercioby]

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

Daniel.

 

[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.