Prove the energy eigenstates are degenerate

[Philethan]

Homework Statement

Two observables ##A_{1}## and ##A_{2}## which do not involve time explicitly, are known not to commute, ## [A_{1},A_{2}]\neq0, ##
yet we also know that ##A_{1}## and ##A_{2}## both commute with the Hamiltonian: ## [A_{1},H]=0\text{, }[A_{2},H]=0. ##
Prove that the energy eigenstates are, in general, degenerate. Are there exceptions? As an example, you may think of the central-force problem ##H=\textbf{p}^{2}/2m+V(r)##, with ##A_{1}\rightarrow L_{z}##, ##A_{2}\rightarrow L_{x}##.

Homework Equations

## [A_{1},A_{2}]\neq0, ##
## [A_{1},H]=0\text{, }[A_{2},H]=0. ##

The Attempt at a Solution

Please read my attached file. I type in latex. I really don't understand why I'm incorrect.

Thanks in advance!

 

[kuruman]

What do you know about the eigenstates of two operators that commute?
What do you know about the eigenstates of two operators that do not commute?