- #1
pixatlazaki
- 9
- 1
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.