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 argues as follows.
((a)) "A d electron has orbital angular momentum l=2, giving 5 orbital states, and spin s=21, giving 2 spin states. Combining the two systems means combining their degrees of freedom, so the total number of states is 5+2=7." Say whether this is correct, and if not, give the correct total number of states and the values of j that appear when l=2 is coupled to s=21.
((b)) The same person then argues: "the three operators J^x2,J^y2,J^z2 form a rank-1 tensor operator, because a rank-1 tensor is just any object with three components, exactly like J^x,J^y,J^z themselves." Test this claim by computing [J^z,J^x2] and comparing it with what the commutator definition of a rank-1 tensor would require for the x-component, then state whether (J^x2,J^y2,J^z2) is a rank-1 tensor operator.