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