# Prove the energy eigenstates are degenerate

1. Oct 12, 2016

### Philethan

1. The problem statement, all variables and given/known data

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 $A_{1}\rightarrow L_{z}$, $A_{2}\rightarrow L_{x}$.

2. Relevant equations
$[A_{1},A_{2}]\neq0,$
$[A_{1},H]=0\text{, }[A_{2},H]=0.$

3. 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!!

#### Attached Files:

• ###### Problem_set_2.pdf
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Last edited: Oct 12, 2016
2. Oct 12, 2016

### 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?

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