As the title suggests, I am interested in symmetries of QM systems.(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# When does a wavefunction inherit the symmetries of the hamiltonian?

**Physics Forums | Science Articles, Homework Help, Discussion**