Question:
Do all states of this new Hamiltonian that have the same energy also have the same parity?
Solution:
No, it is possible to have $ |1\rangle|1\rangle|2\rangle $ to have energy $ 6\hbar\omega $ and $ |1\rangle|3\rangle|1\rangle $ to have energy $ 6\hbar\omega $. But they have different parity. The former is even and the latter is odd.
Do all states of this new Hamiltonian that have the same energy also have the same parity?
Solution:
No, it is possible to have $ |1\rangle|1\rangle|2\rangle $ to have energy $ 6\hbar\omega $ and $ |1\rangle|3\rangle|1\rangle $ to have energy $ 6\hbar\omega $. But they have different parity. The former is even and the latter is odd.
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