Stark effect: find ground state

[JulienB]

Homework Statement

Hi everybody! I have a problem related to first-order perturbation theory, and I'm not sure I'm tackling the problem correctly. Here is the problem:

Consider a hydrogen atom in an externally applied electric field ##\vec{F}##. Use first-order perturbation theory to find the perturbed ground state wavefunction. (Take ##\vec{F}=F\hat{z}## and, just to make things easier, include only the ##n = 2## states.)

Homework Equations

(I use Griffiths as a source)

Correction of wave function: ##\psi_n^1 = \sum_{m\neq n} \frac{\langle \psi_m^0 | H' | \psi_n^0 \rangle}{E_n^0 - E_m^0}##

The Attempt at a Solution

I think that the perturbation due to ##\vec{F}## is ##H'=eFz##. My main difficulty is to interpret the equation given in Griffiths for a wave function in spherical coordinates. Here is my attempt:

##\psi_{100}^1= \sum_{nlm\neq 100}^{\infty} \frac{|nlm \rangle \langle nlm | H' | 100 \rangle}{E_1^0 - E_n^0}##

Is that the correct way to deal with this equation? Then I calculated ##\langle nlm | H' | 100 \rangle## (generally), and using ##z=r \cos \theta##, ##Y_0^0 \cos \theta = \frac{1}{\sqrt{3}} Y_1^0## and the orthogonality of spherical harmonics I get:

##\langle nlm | H' | 100 \rangle = \frac{eF}{\sqrt{3}} \delta_{l1} \delta_{m0} \frac{2}{\sqrt{a^3}} \int dr\ r^3 R_{nl}^* e^{-r/a}##

which means that ##\langle nlm | H' | 100 \rangle=0## for ##l\neq 1##, ##m\neq 0##. Then the exercise asks to consider only ##n=2## states, so the only ##nlm## triplet for which the correction is not ##0## is ##|210\rangle##. So now I can integrate the radial function and get:

##\langle 210 | H' 100 \rangle = \frac{128 \sqrt{2}}{243} eFa##

which takes me to a correction:

##\psi_0^1=\frac{|210\rangle \langle 210 | H' | 100 \rangle}{E_1^0 - E_2^0}##
##= - \frac{256 eF}{729 \sqrt{3}} \frac{1}{Ry} \frac{r}{\sqrt{a^3}} e^{-r/2a}##

(The ##-## comes from the energy difference: ##E_1^0 - E_2^0 = -Ry + Ry/4 = -3Ry/4##)

Finally I can add the correction to the unperturbated ground state wave function and get:

##|100\rangle_\text{perturbated} = \frac{1}{\sqrt{\pi a^3}} e^{-r/a} + \psi_0^1##

Does that make sense? The result is kind of ugly, but from what I read on the internet and in Griffiths it would be surprising if that wasn't the case. I've also only very recently started to use the Dirac notation, hopefully I didn't get confused along the way.

Thank you very much in advance for your answers.Julien.

 

[kuruman]

I did not check your numbers but the method looks right. Note that the perturbed wavefunction you found is not normalized. It's not pretty but that's PT for you.