Someone is working with a system built from an orbital angular momentum and a…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Adding angular momenta, tensor operators and Wigner-Eckart
Problem
Someone is working with a system built from an orbital angular momentum l=1 and a separate particle of spin s=1 (the two act on independent factors of a tensor-product space). They reason as follows."The orbital factor has 2l+1=3 states and the spin factor has 2s+1=3 states, so together they describe 3+3=6 states."They then turn to the operator triplet (L^x2,L^y2,L^z2), built entirely from the components of L^, and argue: "this has three Cartesian components, one for each axis, so it must transform under rotations the way (L^x,L^y,L^z) does, i.e. it is a rank-1 (vector) irreducible tensor operator."
((a)) State the correct dimension of the combined l=1, s=1 space, explain the error in the 3+3 reasoning, and confirm your dimension using the coupled-basis decomposition 1⊗1=2⊕1⊕0.
((b)) Test the claim about (L^x2,L^y2,L^z2) directly, using the commutator definition of a rank-1 tensor operator, [L^z,T^q(1)]=qℏT^q(1) and [L^±,T^q(1)]=ℏ(1∓q)(2±q)T^q±1(1), taking T^0(1)=L^z2 as the natural q=0 candidate (it commutes with L^z, as required for q=0). Show whether the operator that the raising relation forces you to build as the q=+1 partner can possibly be one of L^x2, L^y2 (or a real linear combination of them), and state the verdict on the original claim.