Is every eigenfunction of angular momentum magnitude necessarily also an eigenfunction of total energy? Is every eigenfunction of angular momentum magnitude necessarily also an eigenfunction of total energy? Is the reciprocal statement true?
This question is from Eisberg and it sounds very confuse to me. 
$\psi$, as we know, it is related to the eigenfunction of the hamiltonian, so it is related to a defined energy state and a angular momenta. The reciprocal statement sound untrue to me, but I don't know how to explain it correctly.
 A: There are cases when the first part of the quote is true, this part is not necessarily true. (see nice counterexample for hydrogen in the comment by @spaceisdarkgreen below).  If the potential is not central then this is certainly not true.  Examples of the latter case are molecules: the wavefunctions are LCAO or linear combination of atomic orbitals.  In molecules, eigenstates of the angular momentum squared do not necessarily commute with the Hamiltonian (in fact rarely do) so are not eigenfunctions of the total energy.  Putting atoms in electric or magnetic fields also produces energy wavefunctions that are not eigenfunctions of the angular momentum squared operator (the amount of $\ell$ mixing typically depends on the strengths of the external fields.)
The reciprocal is also false, i.e. eigenfunctions of the total energy need not be eigenfunctions of the square of angular momentum, even when the potential is central. There are quantum systems, such as the hydrogen atom and the 3d harmonic oscillator - for which there is more than one possible value of $\ell$ for a given energy.  In such cases, it is possible to construct linear combinations of different $\ell$ states that will have definite energy but will not be eigenstates of the angular momentum squared operator.

This answer edited after a comment from @speceisdarkgreen
A: An eigenfunction of angular momentum is an eigenfunction of the Hamiltonian if angular momentum is conserved i.e if $[H,L]=0$ . If I turn on a magnetic field then this will no longer be true. More generally anything that breaks conservation of angular momentum will cause the statement not to be true.
