Question

Draw picture or pictures to explain:

The meanings of the two terminologies: ‘nondegenerate’ and ‘degenerate’ Hilbert spaces. Can a Hilbert space have both nondegenerate and degenerate states? If it can, how? If it cannot, why? Can an orthonormal set be constructed for this Hilbert space? How? (No mathematical derivations)

Answer 1

If a Hilbert space has multiple states with a common eigen values for a operator they are known degenerate.

Yes a Hilbert space can have both type of states. For example, take 2D harmonic oscillator. Ground energy eigen state is non degenerate while higher energy states are degenerate.

Yes a orthonormal set can be constructed for this Hilbert space.States corresponding to different eigen values are already orthonormal. To achieve this set, a simple procedure has to be followed. First identify degenrate subspaces. In every degenrate subspaces construct a orthonormal basis. Now form a set consisting of these basis elements of every subspace. This set will be the required set.