# Qualitative question about the Stark effect

• Clara Chung
In summary, the conversation discusses the number of sublevels for n=7 and n=8 in the Hydrogen and Rydberg atoms. The speaker mentions that for n=9 Rydberg states, only the highest angular momentum states are approximately degenerate, and there are many levels that can be mixed. However, it is not easy to determine the mixing matrix elements and some levels may still remain degenerate. The speaker also mentions that they have not learned about degenerate perturbation theory and it is not clear how it simplifies the problem. Ultimately, the conversation ends with the speaker thanking for the help and deciding to leave the problem for now.
Clara Chung
Homework Statement
Attached below
Relevant Equations
Attached below

For e ii) The answer is

Why are there only 4 sublevels?
We haven't learned about degenerate perturbation theory, the only thing mention in lecture is

which I don't understand so I only memorize the good eigenfunctions for n=2. Could you explain why there are still only 4 sublevels for n=7 and 8?

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Clara Chung said:
which I don't understand so I only memorize the good eigenfunctions for n=2. Could you explain why there are still only 4 sublevels for n=7 and 8?
You do understand the Hydrogen atom Stark effect and not the Rydberg atom? (I'm not sure what you need here)...

Clara Chung
hutchphd said:
You do understand the Hydrogen atom Stark effect and not the Rydberg atom? (I'm not sure what you need here)...
For n=7 and n=8 there will be n^2 eigenfunction... if the states that will be mix are the states that satisfy the selection rules... i.e. delta l=+-1 and delta m=0, there will be so many... not only the two eigenstates in the hydrogen atom...?

Clara Chung said:
For n=7 and n=8 there will be n^2 eigenfunction... if the states that will be mix are the states that satisfy the selection rules... i.e. delta l=+-1 and delta m=0, there will be so many... not only the two eigenstates in the hydrogen atom...?
He is talking about the n=9 Rydberg states only. Within this are the allowed states L=0,1,...,8 each with multiplicity 2L+1. Only the highest angular momentum states are "far enough away" from the "nucleus/inner electron complex" to be approximately degenerate. So we are down to L=7,8. So how will this subset be affected by perturbation?

Clara Chung
hutchphd said:
He is talking about the n=9 Rydberg states only. Within this are the allowed states L=0,1,...,8 each with multiplicity 2L+1. Only the highest angular momentum states are "far enough away" from the "nucleus/inner electron complex" to be approximately degenerate. So we are down to L=7,8. So how will this subset be affected by perturbation?
If only L=7,8 are approximately degenerate... There are still many levels that can be mixed... Like l=7, ml=0,1,2,3,4,5,6,7 with l=8, ml=1,2,3,4,5,6,7...??

Clara Chung said:
If only L=7,8 are approximately degenerate... There are still many levels that can be mixed... Like l=7, ml=0,1,2,3,4,5,6,7 with l=8, ml=1,2,3,4,5,6,7...??
Yes all those states (let's be precise ml=0, ±1,..., ±L for L=7,8). But what about the (mixing) matrix elements. Are some zero? Are some the same size? This is not an easy exercise...but it is good. Clearly some levels will still be degenerate.

hutchphd said:
Yes all those states (let's be precise ml=0, ±1,..., ±L for L=7,8). But what about the (mixing) matrix elements. Are some zero? Are some the same size? This is not an easy exercise...but it is good. Clearly some levels will still be degenerate.
How to know whether they have the same size? :(

There are two states you know don't connect (create any off diagonal elements). What are they?

You say you have not had degenerate perturbation theory...why are you (we) looking at this problem? It is not obvious to me exactly how it simplifies completely as advertised.

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Clara Chung
hutchphd said:
There are two states you know don't connect (create any off diagonal elements). What are they?

You say you have not had degenerate perturbation theory...why are you (we) looking at this problem? It is not obvious to me exactly how it simplifies completely as advertised.
Thank you for your help.. This appears in an exam of an introductory course on atomic physics... I guess I will leave it...

Perhaps someone else on the Forum sees the route to the requested solution. Also you can keep this problem in your head and, as you learn more, re-examine the possible solutions. To have such mental touchstones is not a bad thing.

## 1. What is the Stark effect?

The Stark effect is a phenomenon in physics where the energy levels of atoms or molecules are shifted when placed in an external electric field. This effect was first discovered by Johannes Stark in 1913.

## 2. How does the Stark effect work?

The Stark effect occurs due to the interaction between the electric field and the electric dipole moment of the atom or molecule. The electric field causes a separation of charge within the atom or molecule, resulting in a shift in energy levels.

## 3. What are the applications of the Stark effect?

The Stark effect has various applications in fields such as spectroscopy, laser technology, and quantum computing. It is used to study the energy levels and properties of atoms and molecules, as well as to control and manipulate their behavior.

## 4. What is the difference between the normal and anomalous Stark effect?

The normal Stark effect refers to the energy level shifts that occur in atoms or molecules with a permanent electric dipole moment, while the anomalous Stark effect occurs in atoms or molecules with a temporary or induced electric dipole moment.

## 5. How is the Stark effect related to the Zeeman effect?

The Stark effect and the Zeeman effect are both examples of the broader phenomenon of electromagnetic interactions with atoms and molecules. While the Stark effect involves electric fields, the Zeeman effect involves magnetic fields. Both effects can cause shifts in energy levels and have various applications in physics and technology.

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