Exploring Quantum Physics - Week 6 Question 10

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.