Ionization of hydrogen atom by sinusoidal electric field

[pixatlazaki]

Homework Statement

"Suppose that a hydrogen atom, initially in its ground state, is placed in an oscillating electric field $\mathcal{E}_0 \cos(\omega t) \mathbf{\hat{z}}$, with $\hbar \omega \gg -13.6\text{eV}$. Calculate the rate of transitions to the continuum."

Homework Equations

$R = \frac{\pi}{2\hbar} |H_{fi}'|^2 \rho (E_f)$, where $R$ is the rate of transitions to the state with final energy $E_f$, $H_{fi}' = \langle \psi_f | H' | \psi_i \rangle$, and $\rho(E)$ is the density of states at an energy $E$. (Derived in course, though similar to equations in Griffiths.)

Density of states for an infinite cuboidal well of volume $V$:
$\rho(E) = \frac{V}{2\pi^2} (\frac{2m}{\hbar^2})^{\frac{3}{2}}\sqrt{E}$

The Attempt at a Solution

I am unclear on two points in the solution of this problem.

Firstly, for the density of states for the continuum, I would think that we may just take the limit $V \to \infty$, but that clearly will blow up (at least if done at the beginning).

Secondly, what is the most sensible way to approach determining the matrix element $H_{fi}'$? Our final state will be that of a free particle, i.e. $\psi_f(\mathbf{r},t) = A e^{\mathbf{k}\cdot\mathbf{r} - \omega_0 t} (\omega_0 \equiv \frac{E_f - E_i}{\hbar})$ (which is really only reasonable to manipulate in Cartesian coordinates), while our initial state will be the bound state $\psi_i(\mathbf{r},t) = \frac{1}{\sqrt{\pi} a_0^{3/2}} e^{\frac{r}{2a_0}}$, which only involves $r$, not $r^2$, making it suited for spherical coordinates.

 

[pixatlazaki]

Later in the week, we were allowed to assume that $\mathbf{k} = k\mathbf{\hat{z}}$, which simplified the problem a great deal. I ended up performing the integral for $H'_{fi}$ in spherical coordinates. I will post the solution in a week or two.