|1s> -> |2p> transition probabilities

[cscott]

Homework Statement

A hydrogen atom is placed in a time-dependent homogeneous electric field given by \epsilon = \epsilon_0 (t^2 + \tau^2)^{-1} where \epsilon_0,\tau are constants. If the atom is in the ground state at t=-\inf, obtain the probability that it ill be found in a 2p state at t=\inf. Assume that the electric field is along the \hat{z} direction.

Homework Equations

\dot{|2p>} = \int_{\inf}^{\inf} H'_{2p,1s} e^{-i \omega_{2p,1s}} dt

H'_{2p,1s} = <2p|H'|1s>

The attempt at a solution

Firstly I'm confused with the assumption of field direction and how I get it into the solution. Maybe this is the source of my troubles...

By taking H'(t)= q\epsilon_0 (t^2 + \tau^2)^{-1} I get zero for a transition between |1s> and |2p0>, |2p\pm1> because I get Hamiltonian elements of zero.

|2p0> has a \cos \theta factor so upon integration in spherical,

\int_0^\pi \sin \theta d\theta \cos \theta = 0

and |2p\pm 1> has the e^{\pm i\phi} factor,

\int_0^{2\pi} e^{\pm i\phi} d\phi = 0

Am I simply right or really missing something? The question gives the time integral result so it seems like Hamiltonian elements shouldn't be zero...

I'm using the wavefunctions from here: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html#c3

 

[fzero]

Firstly I'm confused with the assumption of field direction and how I get it into the solution. Maybe this is the source of my troubles...

By taking H'(t)= q\epsilon_0 (t^2 + \tau^2)^{-1} I get zero for a transition between |1s> and |2p0>, |2p\pm1> because I get Hamiltonian elements of zero.

The interaction term in the Hamiltonian should be such that the electron couples to the electric potential. The resulting matrix elements will no longer vanish.