Chapter 4

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

(a) Confirm that the Pauli matrices \\[ \sigma_{x}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \quad \sigma_{y}=\left(\begin{array}{rr} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{array}\right) \quad \boldsymbol{\sigma}_{z}=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) \\] satisfy the angular momentum commutation relations when we write \(s_{q}=\frac{1}{2} \hbar \sigma_{q},\) and hence provide a matrix representation of angular momentum. (b) Why does the representation correspond to \(s=1 / 2\) ? Hint. For the second part, form the matrix representing \(s^{2}\) and establish its eigenvalues.

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.

Suppose that in place of the actual angular momentum commutation rules, the operators obeyed \(\left[l_{x}, l_{y}\right]=-\mathrm{i} \hbar l_{z^{*}}\) What would be the roles of \(l_{2}=l_{x} \pm l_{y} ?\)

Calculate the matrix elements (a) \(\left\langle 0,0\left|l_{z}\right| 0,0\right\rangle\) (b) \(\left\langle 2,1\left|l_{+}\right| 2,0\right\rangle\) (c) \(\left\langle 2,2\left|l_{+}^{2}\right| 2,0\right\rangle,\) (d) \(\left\langle 2,0\left|l_{+} l_{-}\right| 2,0\right\rangle\) (e) \(\left\langle 2,0\left|l \downarrow_{+}\right| 2,0\right\rangle,\) and (f) \(\left\langle 2,0\left|l^{2} l_{z} l^{2}+\right| 2,0\right\rangle\)

Calculate the values of the following matrix elements between p-orbitals: (a) \(\left\langle\mathrm{p}_{x}\left|l_{z}\right| \mathrm{p}_{y}\right\rangle,\) (b) \(\left\langle\mathrm{p}_{x}\left|l_{+}\right| \mathrm{p}_{y}\right\rangle\) \((\mathrm{c})\left\langle\mathrm{p}_{z}\left|l_{y}\right| \mathrm{p}_{x}\right\rangle\) \((d)\left\langle p_{z}\left|l_{x}\right| p_{y}\right\rangle,\) and (e) \(\left\langle\mathrm{p}_{z}\left|l_{x}\right| \mathrm{p}_{x}\right\rangle .\) Hint. Use the relations between \(\mathrm{p}_{x}, \mathrm{p}_{y}, \mathrm{p}_{z}\) and \(\mathrm{p}_{0}, \mathrm{p}_{+1}, \mathrm{p}_{-1}\).

(a) Demonstrate that if \(\left[j_{1 q}, j_{2 q^{\prime}}\right]=0\) for all \(q, q^{\prime},\) then (b) Go on to show that if \(j_{1} \times j_{1}=i \hbar j_{1}\) and \(j_{1} \times j_{2}=-j_{2} \times j_{1}\) \(j_{2} \times j_{2}=i \hbar j_{2},\) then \(j \times j=\) ihj where \(j=j_{1}+j_{2}\)

In some cases \(m_{11}\) and \(m_{12}\) may be specified at the same time as \(j\) because although \(\left[f^{2}, j_{1 z}\right]\) is non-zero, the effect of \(\left[j^{2}, j_{12}\right]\) on the state with \(m_{i 1}=j_{1}, m_{i 2}=j_{2}\) is zero. Confirm that \(\left[j^{2}, j_{1 z}\right]\left|j_{1} j_{1} ; j_{2} / j_{2}\right\rangle=0\) and \(\left[j^{2}, j_{1 z}\right]\left|j_{1},-j_{1} ; j_{2},-j_{2}\right\rangle=0\).

Determine what total angular momenta may arise in the following composite systems: (a) \(j_{1}=3, j_{2}=4 ;\) (b) the orbital momenta of two electrons (i) both in p-orbitals, (ii) both in d-orbitals, (iii) the configuration \(\mathrm{p}^{1} \mathrm{d}^{1} ;\) (c) the spin angular momenta of four electrons. Hint. Use the ClebschGordan series, eqn \(4.42 ;\) apply it successively in (c).

Construct the vector coupling coefficients for a system with \(j_{1}=1\) and \(j_{2}=1 / 2\) and evaluate the matrix elements \(\left\langle j^{\prime \prime} m_{i}^{\prime}\left|j_{1 z}\right| j m_{i}\right\rangle .\) Hint. Proceed as in Section 4.12 and check the answer against the values in Resource section \(2 .\) For the matrix element, express the coupled states in the uncoupled representation, and then operate with \(j_{1 x}\)