Good question

Dear Das

degeneracy means the same energy with different eigenstate. The degenerate states of the hydrogen atom are equal to square of the principal quantum number. However, the degenerate states can be partially lost due to spin-orbit coupling and Zeeman effect.

Dear Hazrat Ali,

Thanks for your nice explanation of number of degenerate states.

But,

My question is , about the cause of arising degeneracy .

How symmetry of potential is related to degeneracy ?

It will be helpful, if you explain this root of arising degeneracy.

Thanks

N Das

Nityananda Das

Generally, degeneracy of quantum states emerges when,

1. There exist transformations which do not change the quantum system, (i.e., its Hamiltonian) while

2. There exist quantum states of the system which do change under the same transformations.

Commonly, the said transformations form a group, in which case property 1 is a definition of the symmetry group of the system. For instance, the Hydrogen atom is spherically symmetric; for it transformations mentioned in 1 are arbitrary 3D rotations. In what follows, I stick to this case.

Rotations may be regarded as a change of the reference frame. This means that a rotated state (cf. 2) is the same state but viewed from a different angle. It therefore must have the same energy. Here enters the degeneracy, i.e., sets of different states with the same energies. Normally states comprising such set are chosen orthogonal.

Now, the answer to your question (symmetry as the cause of degeneracy) depends on what you call "symmetry". I am not aware of any physical system where symmetry (i.e. property 1) would not result in degeneracy (property 2). The opposite is however not exactly true: there exist examples of degeneracy which cannot be attributed to obvious symmetries. Such degeneracy is called accidental. (E.g., accidental degeneracy in Hydrogen atom.) Mathematically, accidental degeneracy is always related to some group of transformations, but it may be highly non-intuitive; calling it symmetry is a matter of taste. The example is again accidental degeneracy in Hydrogen atom.

This is more or less all that can be said without serious mathematics.

Prof. Nityananda Das, I think the right answer is given in previous posts, but to complement atomic physics has different "atomic scales":

• The gross atomic structure, predicted by the orbital kinetic and potential energies, from 1 − 10 eV.
• The fine atomic structure predicted by relativistic corrections (spin-orbit) splits degenerate multiplets leading to small shifts in energy - 10−4 − 10−5 eV.
• The hyperfine atomic structure predicted by the interaction of electron magnetic moment with the field generated by nuclear spin, leads to further splitting of multiplets, from 10−7 − 10−8 eV.

The atomic scales lift/break degeneracy, as small correction terms are added to the potential, for example, the spin-orbit coupling term, the Dirac term. the lamb shift, the hyperfine structure, and so on.

Interesting to see how these atomic scales change the symmetries of the potentials.

https://blog.cupcakephysics.com/quantum%20mechanics/2014/11/09/breaking-degeneracy-with-the-spin-orbit-interaction.html