As the title suggests, I am interested in symmetries of QM systems. Assume we have a stationary nonrelativistic quantum mechanical system [itex]H\psi = E\psi[/itex] where we have a unique ground state. I am interested in the conditions under which the stationary states of the system inherit the symmetries of the hamiltonian. I am aware of some examples. If the hamiltionian is symmetric in one axis, the wave function will be symmetric or antisymmetric. In particular, the ground state will be symmetric. Similarly, if H is invariant under interchange of two coordinates, so will the ground state be (for example in the hydrogen atom). In general, if the Hamiltonian is invariant under the action of a group, will the ground state also be? Can anyone supply a proof? Also, even if there is not a unique ground state, will the action of such a group on one ground state always produce another ground state (such that the set of all ground states is invariant under the group action)? Thank you for your time.