Q & A: what is degeneracy?

What is Degeneracy? and how is it related to Zeeman effect? and why several difference states(?) of electron(?) have the same energy level when there is no magnetic field? and why these states split into (take?) several different energy levels when magnetic field is present?
- Anonymous

"Degeneracy" here just means that there are more than one quantum states with the same sharply-defined energy. For example, there can be a state where an electron is rotating one way around the nucleus and another state of the same energy where it rotates the opposite way.

Such rotating states are kind of like current loops. They have magnetic moments. In a magnetic field, the energy of the state depends on whether the moment is aligned with or against the magnetic moment, or in-between. The split between the energies of course depends on how big the magnetic field is. That's called the Zeeman effect.

There was a puzzle in the early days of quantum mechanics when it was realized that a Zeeman effect (splitting of a degeneracy via a magnetic field) occurs in situations where no explanation based on electron orbits could apply.  This was an early piece pf evidence for electron "spin" which gives the electron a magnetic moment even when it's not in any sort of rotating orbit.

Mike W.

(published on 08/19/2011)

I see. Thank you, now it is less puzzling except one part:"the moment is aligned with or against the magnetic moment, or in-between" "aligned" with or "against" the magnetic field: these can take distinct energy levels because of their one or the other clear, distinct, separate, configuration. but how can "in-between" take distinct energy levels? maybe "in-between" configuration can transiently happen while moving toward either "along" or "against" configuration but eventually the magnetic moment fall into only one of two configurations (i.e. "aligned" with or "against" the magnetic field)? Why this is not the case? and "stable(?)" "in-between" states(or electron spin direction/configuration) can exist? And as externally applied magnetic field increases (including extreme intensity range), do numbers of splitting energy states change(increase, decrease, or stay constant after initial splitting)? If so, why and what is the mechanism?
- Anonymous

You were very sharp in catching my attempt to get past the complications for quantum magnetic moments. For an electron orbital moment with the smallest non-zero value, there are actually three states with definite values of the component of the magnetic moment along the field axis. The values are -1, 0, and +1 times a natural unit of magnetic moment, the Bohr magneton. So the 0 is the in-between value. The next higher possible total angular momentum allows values of -2, -1, 0, +1, +2 for that component. Etc. These orbital moments always have an odd number of possible distinct values for one component. The electron spin gives two possible values (-1/2 and +1/2).  I can't off-hand think of a simple very short explanation, but these issues are discussed in almost any first-year quantum book.

The numbers of different energy values generally do not change further after the introduction of a small field. Increasing the field just increases the energy splits between them. You can see why that would be so rather easily in your simple case of a spin with just two possible values of the magnetic moment on the field axis.

In more complicated cases, with dissimilar orbital states (e.g. ones that start at different energies even in zero field) the pattern of how the energies change as a function of applied field becomes more complicated. The general rule that the number of different possible energy values doesn't change once the field isn't zero remains, however.

I realize that this answer doesn't really give the explanations, but perhaps you can follow up again. I suspect that when you think about it the explanation of the last part will be sort of obvious. The first part, about the possible discrete values of angular momentum,  probably requires more reading.

Mike W.

(published on 08/25/11)