Degeneracy in Wave Function Energy States

[delee2]

Homework Statement

What is meant by the term degenerate when referring to wave function energy states? Do the wavefunctions for degenerate states necessarily look the same? Explain.

Homework Equations

The Attempt at a Solution

Degeneracy, in terms of wave function energy states, describes quantum states that occupy the same energy level . All orbitals of one shell have the same energy, meaning that they require the same energy to achieve this energy level. An example of this is in the hydrogen atom where the orbitals of its shell all have the same energy. This is likewise true for all one-electron ions such as He+ and C5+. Quantum numbers add up to the same energy

 

[gash789]

You appear to have answered the question very well.

The wave functions do not necessarily look the same, that would essentially make them the same wavefunction. However take as example the energy associated with wavefunctions of the hydrogen atom. It is a function of the quantum number n, but these wavefunctions also have two other quantum numbers, l and m. So for the set of wave functions that have the same n, but different l, they are degenerate. They have the same energy, but a different l.

The different shapes given by there probability density:

Horizontally the quantum number l changes, and vertically the quantum number n changes. So for the bottom row, all 3 have the same energy (when ignoring fine structure and some other thing I have forgotten) and hence are degenerate.