Quantum mechanics, symmetry and degeneracy

[Chronos000]

Homework Statement

I'm struggling to understand the concept of symmetry in quantum mechanics. My notes state "In general if the probability density has lower symmetry than the hamiltonian, the wavefunction will be degenerate". I don't really get the connection with the hamiltonian.

It then says "if we reduce the symmetry of the hamiltonion, we lift the degeneracy".
If I imagine a sphere of full energy states -it is symmetric, so if it's less symmetric then the energy levels can't all be filled and then it's no longer degenerate right?

 

[member 553083]

Is that correct?Homework Equations N/AThe Attempt at a SolutionYes, that is correct. If the probability density has lower symmetry than the Hamiltonian, then the wavefunction will be degenerate, meaning that two or more states have the same energy. This is because the symmetry of the Hamiltonian allows for a certain level of degeneracy in the energy states. If we reduce the symmetry of the Hamiltonian, then the degeneracy is lifted, and the energy levels can no longer all be filled.