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Ionization of hydrogen atom by sinusoidal electric field

  1. Oct 28, 2015 #1
    1. The problem statement, all variables and given/known data
    "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."

    2. Relevant 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}##

    3. 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.
     
  2. jcsd
  3. Nov 2, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Nov 2, 2015 #3
    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.
     
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