# Symmetry of eigenstates

1. Jun 28, 2009

It is common knowledge in Physics that eigenstates share the symmetries of the Hamiltonian.

And it is trivial to show that this is true for the eigenspaces. Let g be an element of a symmetry group of Hamiltonian H, $$M_g$$ its representation, $$\left| \phi \right>$$ an eigenvector and $$\lambda$$ the corresponding eigenvalue. If g is an element from a symmetry group of H it is given that:

$$M_gHM_{g^{-1}}\left|\phi\right>=\lambda\left|\phi\right>$$
Thus
$$HM_{g^{-1}}\left|\phi\right> = \lambda M_{g^{-1}}\left|\phi\right>$$

So we see that the eigenspace for $$\lambda$$ is closed under the operations of $$M_g$$
Is there some theorem, that I can decompose this eigenspace into eigenstates of M or how do I proceed from here? This is important for Bloch functions and crystallography. I read something about Schur's lemma being involved.

So to phrase a proper question:
Is every Eigenfunction of a Hamiltonian,invariant up to a scale factor of unity magnitude under the operation of the Hamiltonian's symmetry groups? And if this is so how do I show this?

Last edited: Jun 28, 2009
2. Jun 28, 2009