91Ó°ÊÓ

Evaluate the commutator \(\left[l_{x}, l_{y}\right]\) in (a) the position representation, (b) the momentum representation.

Using the Pauli matrix representation, reduce each of the operators (a) \(s_{x} s_{y}\) (b) \(s_{x} s_{y}^{2} s_{z}^{2},\) and (c) \(s_{x}^{2} s_{y}^{2} s_{z}^{2},\) to a single spin operator.

What is the expectation value of the \(z\) -component of orbital angular momentum of electron 1 in the \(\left|G, M_{L}\right\rangle\) state of the configuration \(\mathrm{d}^{2}\) ? Hint. Express the coupled state in terms of the uncoupled states, find \(\left\langle\mathrm{G}, M_{L}\left|l_{1 z}\right| \mathrm{G}, M_{L}\right\rangle\) in terms of the vector coupling coefficients, and evaluate it for \(M_{L}=+4,+3, \ldots,-4\)