Sudden Perturbation Approximation Question

[jsc314159]

Homework Statement

In a beta decay H3 -> He3+, use the sudden perturbation approximation to determine the probability of that an electron initially in the 1s state of H3 will end up in the |n=16,l=3,m=0> state of He3+

Homework Equations

|<n'l'm'|nlm>|^2

The Attempt at a Solution

I actually know the answer to this but I am not clear as to why and I am wondering if there is an easier way to determine the solution.

The answer comes out to be 0. When integrating the wavefunctions |n'l'm'> and |nlm> in spherical coordinates to calculate the inner product (i.e. the probability amplitude), the integral over d\theta returns 0. Is there an easier way to see this other than going through the calculations?

How is this result interpreted?

Thanks,

jsc

 

[nrqed]

Homework Statement

In a beta decay H3 -> He3+, use the sudden perturbation approximation to determine the probability of that an electron initially in the 1s state of H3 will end up in the |n=16,l=3,m=0> state of He3+

Homework Equations

|<n'l'm'|nlm>|^2

The Attempt at a Solution

I actually know the answer to this but I am not clear as to why and I am wondering if there is an easier way to determine the solution.

The answer comes out to be 0. When integrating the wavefunctions |n'l'm'> and |nlm> in spherical coordinates to calculate the inner product (i.e. the probability amplitude), the integral over d\theta returns 0. Is there an easier way to see this other than going through the calculations?

How is this result interpreted?

Thanks,

jsc

The spherical harmonics are orthonormal so <l' m' | l m> is zero unless l=l' and m= m'. Clearly here the result is zero since |3,0> is orthogonal to |0,0>

Physically, it simply says that a particle in an s state has no angular momentum so the probability of it being observed with l=3 is zero. The sudden approximation simply assumes that the transition was so quick that the orbital angular momentum of the electron remained unchanged.

 

[jsc314159]

Thanks nrged.

That makes it clear.