Transition Probability of Hydrogen atom in an electric field

[jmm5872]

A hydrogen atom is in its ground state and is subject to an external electric field of

E = ε(\hat{x}+\hat{y}+2\hat{z})e^-t/\tau

I'm confused as to how to compute the matrix elements of the perturbed hamiltonian since this is not in the z direction.

Would I have to do something like this?

H'_ba = -pE = -qεe^-t/\tau<\psi_b|(\hat{x}+\hat{y}+2\hat{z})|\psi_a>

Thanks

 

[diazona]

If I remember this stuff correctly, then yes. You're just using a unit vector in the direction of the electric field rather than in the z direction. Alternatively, you could rotate your coordinate system so that the electric field points in the z direction, solve the problem, and then rotate your solution back to the original coordinates.

 

[TSny]

Shouldn't you take the dot product of the electric field with the dipole moment vector operator: $\vec{p} = q\vec{r}= q(x \hat{x} + y \hat{y}+z \hat{z})$?

 

[diazona]

Oh yes, somehow I completely missed the fact that there was no dot product in the original post.