How Do You Determine Rotational Invariance in a Two-Fermion System?

[jjohnson]

Homework Statement

Homework Equations

The Attempt at a Solution

I suppose to determine if a hamiltonian is rotational invariant, we check if [H(1),L^2], however, I am not sure how to do it if the hamiltonian is operate on a two particle wave function.
Is it just to evaluate [S1z Z2 +S2z Z1, L1^2+L2^2]?

What is the ground state wave function for this system?
I suppose it is

A (|0,1,1/2,-1/2> - |0,1,-1/2,1/2>), if you apply S1z, you get 0. It is also true for S2z. I think I am just lost here.

 

[blue_leaf77]

The rotation operator in this system will be associated with the angular momentum operator $J^z = L_1^z + S_1^z + L_2^z + S_2^z$. To check if the perturbation alters the rotation invariant of the system, calculate the commutation $[H^{(1)},J^z]$.

 

[jjohnson]

The rotation operator in this system will be associated with the angular momentum operator $J^z = L_1^z + S_1^z + L_2^z + S_2^z$. To check if the perturbation alters the rotation invariant of the system, calculate the commutation $[H^{(1)},J^z]$.

Thank you fore replying.
How should I reprent $L_z$? is it $XP_y - YP_x$? Get r in spherical coordinate using raising and lowering operator and projection z axis? seems like a lot of work.

 

[blue_leaf77]

is it $XP_y - YP_x$ ?

Yes. $L_z$ is the orbital angular momentum of one of the fermions.

Get r in spherical coordinate using raising and lowering operator and projection z axis?

That's too much than needed. Just plug in each expression for $H^{(1)}$ and $J_z$ into the commutator $[H^{(1)},J_z]$. It should be simple if you use the property of commutator between angular momentum and position.