Atom in electric field

[yevgeny]

What is the effect of a uniform constant electric field on the energy levels of an atom? ?

[ecm]

The so called "Stark effect":

The interaction with the electric field E causes the splitting of the atomic spectral lines. In the case of hydrogen this is a first order correction, since the atomic spectra of hydrogen is degenerated in l. The interaction with the electric field breaks this degeneration. For all the other atoms there is no such degeneration, and the Stark effect results in a second order correction (the correction of each energy level E is of the order of E^2).

The splitting of the energy levels by an electric field first requires that the field polarizes the atom and then it interacts with the resulting electric dipole moment.


I hope that helps.

By the way, yevgeny, how did your GRE go?

[yevgeny]

You are great, ecm!!!

Do you know how the magnitude of the effect can be estimated (theoretically, I mean)? Or at least how one can see whether the effect is of 1st or 2nd order?

My GRE was OK. The Quantitative part was more difficult than in the examples, but I still got 800. The Verbal part was hard as usual, and I got 470. I don't know the Analytical Writing grade yet, but it does not seem to be a failure.

[ecm]

Way to go, yevgeny, nice GRE!! Now you only need a fine GRE subject and you are all ready!

Let's see, the Stark effect. Man, it's been a long time since I studied this... don't you think the Zeeman effect is more important??

LOL, ok, enough excuses. Let's see.
The effect of the electric field can be considered as a perturbation to the original hamiltonian of the atom. (At least if the electric field is not too strong, which we will assume). The additional term that you have to add to the hamiltonian, let's call it H_s, is something like this:
H_s = - mu · E
(I hate writting equations like this, but my keyboard isn't too equation-friendly... Perhaps we could write equations as if we were going to compile them using latex... do you know how to use latex?)

Anyway, in that equation "mu" is supposed to be the dipolar electric momentum of the atom (a vector), and E is the electric field (a vector too). So H_s is the scalar product of these two vectors. Well, then you could write it as:
H_s = - |mu||E| cos(theta),
where theta is the angle between mu and E. If you remember the spherical harmonics (I don't know if that's the way they are called in English, but Im sure that it's close enough), they contain a part that is a polynomial of degree L of cos(theta). So, the effect of H_s acting on the eigenstate of your atom is to increase its L in 1.

Then, we look at perturbation theory. The change of energy (Delta(E)) of a non-degenerated state of a system would be:

Delta(E)= <L S J M | H_s | L S J M>

By the way, here E refers to energy, not the electric field. Sorry about that.

Well, the thing is, H_s acting on the eigenstate increases L, and two states with different values of L are orthogonal (at least if the other quantum numbers are all the same):
<L S J M | L+1 S J M>=0

So that's the reason why for every non-degenerated state, the first order correction of the energy cancels out. So, what's so hot about hydrogen? Hydrogen eigenstates are degenerated in L. You can't use the formula I wrote for Delta(E), you have to use the perturbation theory for degenerated states, which includes a series of terms that involve two different eigenstates, so when you calculate <1|H_s|2> it does not cancel out. That's why the Stark effect breaks the L-degeneration of hydrogen.

Now, if you want to calculate the second order correction to the energy (which is called cuadratic Stark effect) you have to use the expresion for the second order correction in perturbation theory. By the way, if they make you do all this in your GRE you may as well skip the question and use the time to work thru another 10 questions.... Well, anyway, the correction involves the product of two terms like
<1 |H_s| 2>. Each one is proportional to the electric field E (since H_s is proportional to E). So, the correction of energy, delta(E) is also proportional to E^2 (*Please, notice that Im using the letter E to refer to two different things!*). I think I made a mistake there in my last post.

Well, do you want more or can we leave it at that? I don't see how they could ever possibly ask this.... No, but seriously, if you want more information let me know. It may take me a while to answer, though, cause I'd have to work out all the expressions........ and we'd certainly need an improved way to write formulaes.

Well, anyway thanks for making me re-study all the atomic physics! The more I study now, the less I'll have to study later.......

Cheers!

[ecm]

My last post turned out a bit too long...... if you get bored midway I'll understand... ;-)

[yevgeny]

Hello ecm!

I am happy to see your answer.
Amazing explanation! It's such a pity that you are so far from me...

But how can we calculate the dipole moment mu? Do atoms have intrinsic dipole moments (at zero field)? If the dipole moment is only induced by the field, it is probably proportional to E, so that more powers of E are added to the final result.

(You may use Latex if you wish, but I love more the way you did it now.)

By the way, what is your field of specialization?

Thank you very much!

[ecm]

Yes, you're right. Atoms in general should not have dipole moments when there is no electric field. The induced polarization is proportional to E as well. So the change in energy would be proportional to E^4.

My specialization... hmm. Well, the physics faculty of my university no longer allows students to choose specialization. But, considering the optative subjects Im taking, my specialization would be fundamental physics. (The other option was electronics....which I dont really like much).
So whats yours?

[ecm]

Hey, I just found this page that has a couple solved problems regarding the Stark effect on Hydrogen, in case you're interested:

http://electron6.phys.utk.edu/phys59...oximation2.htm

[yevgeny]

Thank you again, ecm!

If I could choose my specialization this would also be fundametal physics . But this is difficult because there are not so many people working on these topics, and not in every university. It becomes even more difficult if you want to avoid durty stuff such as string theory.
But it makes me think: I don't need many people, I need one. So why don't I just go to a university where such group exists? The answer in the M.Sc. case was the circumstances. In the Ph.D. case the answer may happen to be similar...
Meanwhile, the research experience which I have is mostly in the field of experimental solid state physics. I can give you more details if you like.
Probably I will choose a different field in the graduate studies, probably theoretical.

Have you been involved in research?