Hydrogen Ionization Rates Using Time Dependent Perturbation

[ianmgull]

Homework Statement

[/B]
Calculate the rate of ionization of a hydrogen atom in the 2p state in a monochromatic external electric field, averaged over the component of angular momentum in the direction of the field. Ignore the spin of the particles. In this case we can write:
$$H'(t)=-eEx_3exp(-i\omega t) -eE^*x_3exp(i\omega t)$$
assuming that the electric field E is in the x₃ direction.

Homework Equations

The 2p hydrogen wave function: $\psi_{2p}=\frac{1}{2 \sqrt{6}a^{5/2}}re^{\frac{-r}{2a}} Y_1^m$
The wave function of a free particle: $\psi_f=\frac{1}{(2 \pi a)^{3/2}} e^{ik \cdot x}$

The Attempt at a Solution

I know the magnitude squared of the following relation will give me the probability of a transition:
$$\frac{dc_{2p}}{dt} = \frac{-i}{\hbar} H'_{(2p,e)}e^{-i \omega_0 t}c_f$$
where $\omega_0=\frac{E_f-E_{2p}}{\hbar}$ and $H'_{(2p,e)}= <\psi_{2p}|H'|\psi_f>$.

My problem at this point is to set up the integral with the perturbed Hamiltonian and the two wave functions. I feel like I'm doing something wrong because I'm integrating with respect to x, but the 2p wave function is in terms of r. I don't know if I should be using a 3d version of the free particle or not.

Also, I'm not sure what is meant by "averaged over the component of angular momentum in the direction of the field" but it sounds like it's relevant to my first concern.

Finally, does "Ignore the spin of the particles" mean I can ignore the spherical harmonic portion of the 2p wave function? If not, how do I choose which value of m (-1, 0, 1) to use.

Thanks,

Any help would be appreciated.

 

[TSny]

The wave function of a free particle: $\psi_f=\frac{1}{(2 \pi a)^{3/2}} e^{ik \cdot x}$

I don't understand the normalization constant here.

I don't know if I should be using a 3d version of the free particle or not.

Yes, your free particle wavefunction will be proportional to $e^{ i \bf k \cdot \bf r}$, where $\bf k$ and $\bf r$ are 3D vectors. There is a well-known expansion of this function in terms of spherical harmonics that you can use in this problem.

Also, I'm not sure what is meant by "averaged over the component of angular momentum in the direction of the field" but it sounds like it's relevant to my first concern.

The different components of angular momentum refer to the different values of $m$ for $l = 1$. States with different $m$ can have different ionization rates. They want you to take the average of the rates for the three $m$ values corresponding to $l = 1$.

Finally, does "Ignore the spin of the particles" mean I can ignore the spherical harmonic portion of the 2p wave function?

No. The spherical harmonics correspond to orbital angular momentum, not spin angular momentum. So, you can't ignore them. Ignoring spin means that you can work with the wavefunctions $\psi_{2p}$ as you wrote them and you don't need to include any additional spin part of the wavefunction.

If not, how do I choose which value of m (-1, 0, 1) to use.

You will calculate a rate for each value of $m$ and then average the three rates.

The calculations in this problem are lengthy. The chance of anyone getting all the numerical factors correct is pretty small. At a risk of giving away too much, here is a link to a similar calculation for the ground state of hydrogen.

http://scipp.ucsc.edu/~haber/ph216/NRQM5sol_12.pdf (See problem 2 starting on page 7.)