Crystal Field Theory I am literally lost with this question:

Suppose that within the set of (2L+1)(2S+1) lowest-lying ionic states the crystal field can be represented in the form a(L_x)^2 + b(L_y)^2 + c(L_z)^2, with a, b, and c all different. Show in the special case L = 1 that if the crystal field is the dominant perturbation (compared with the spin-orbit coupling), then it will yield a (2S+1)-fold degenerate set of ground states in which every matrix element of every component of L vanishes.

 A: The physical context would have been helpful. I am guessing that this is some sort of once ionized atom, whose electron is in an L=1 state, sitting inside an atomic crystal, which breaks the rotational symmetry by the crystal axes.
First, the S is a decoupled electron spin, which is completely irrelevant. You can ignore the 2S+1 factors and all the misdirecting jawboning about spin in the question. This is a reformulation:
You have a quantum mechanical system with angular momentum 1. This means that there are 3 different rotated versions of the state, with $L_z=-1,0,+1$, which have exactly the same energy, they are degenerate, because the world is rotationally invariant.
Now you put this system inside a crystal that breaks the rotational symmetry, and the leading perturbation from the crystal between these three states and no other, is
$$ a L_x^2 + b L_y^2 + cL_z^2 $$
The operators $L_x$,$L_y$ and $L_z$ are specific 3 by 3 matrices acting on these three states. You probably were given their form from the raising and lowering operators for z-angular momentum: $L_x - iL_y$, $L_x + iL_y$.


*

*Find the new ground state,

*show that all the "matrix elements" of each component of L vanish in this ground state.


The first part is easy diagonalization of the 3 by 3 matrix given to you above. This is clearly a homework exercize, so you should do it yourself. It is straightforward.
The second depends on the intended interpretation of the word "matrix elements": after comments by jojo, I learn that the "matrix elements" are just supposed to be the expected value of the $L_x$, $L_y$ an $L_z$ between the ground state copies with different spin. This is a terrible abuse of the term "matrix elements", because these matrix elements are just the expectation value of $L_x$,$L_y$ and $L_z$ in the ground state, the spin is completely decoupled. So all you need to show is that the expected value of $L_x$,$L_y$ and $L_z$ are zero.
I misinterpreted the wording initially to mean something impossible, namely:


*

*Show that the matrix elements between the ground states and the other $L_z$ states are all zero, for each of the three operators $L_x,L_y,L_z$.


There are no states of L=1 system for which the matrix elements of each component of L vanishes. Such a state would be a rotationally invariant vector, which does not exist--- the only rotationally invariant objects are scalars.
Because the actual wording is potentially misleading, this problem is not very good. It is better to say explicitly: "Show that all the matrix elements between the components of L between the different ground states is zero."
