(a) A particle of mass $m$ moves in a spherical potential $V(r)$. Show that according to classical mechanics
$$
\frac{\mathrm{d}}{\mathrm{d} t}\left(\mathbf{p} \times \mathbf{L}_{\mathrm{c}}\right)=m r^{2} \frac{\mathrm{d} V}{\mathrm{~d} r} \frac{\mathrm{d} \mathbf{e}_{r}}{\mathrm{~d} t}
$$
where $\mathbf{L}_{\mathrm{c}}=\mathbf{r} \times \mathbf{p}$ is the classical angular-momentum vector and $\mathbf{e}_{r}$ is the unit vector in the radial direction. Hence show that when $V(r)=-K / r$, with $K$ a constant, the Runge-Lenz vector $\mathbf{M}_{\mathrm{c}} \equiv \mathbf{p} \times \mathbf{L}_{\mathrm{c}}-m K \mathbf{e}_{r}$ is a constant of motion. Deduce that $\mathbf{M}_{\mathrm{c}}$ lies in the orbital plane, and that for an elliptical orbit it points from the centre of attraction to the pericentre of the orbit, while it vanishes for a circular orbit.
(b) Show that in quantum mechanics $(\mathbf{p} \times \mathbf{L})^{\dagger}-\mathbf{p} \times \mathbf{L}=-2 \mathbf{i p}$. Hence explain why in quantum mechanics we take the Runge-Lenz vector operator to be
$$
\mathbf{M} \equiv \frac{1}{2} \hbar \mathbf{N}-m K \mathbf{e}_{r} \quad \text { where } \quad \mathbf{N} \equiv \mathbf{p} \times \mathbf{L}-\mathbf{L} \times \mathbf{p}
$$
Explain why we can write down the commutation relation $\left[L_{i}, M_{j}\right]=$ $\mathrm{i} \sum_{k} \epsilon_{i j k} M_{k}$
(c) Explain why $\left[p^{2}, N\right]=0$ and why $[1 / r, \mathbf{p} \times \mathbf{L}]=[1 / r, \mathbf{p}] \times \mathbf{L}$ Hence show that
$$
[1 / r, \mathbf{N}]=\mathrm{i}\left\{\frac{1}{r^{3}}\left(r^{2} \mathbf{p}-\mathbf{x} \mathbf{x} \cdot \mathbf{p}\right)-\left(\mathbf{p} r^{2}-\mathbf{p} \cdot \mathbf{x} \mathbf{x}\right) \frac{1}{r^{3}}\right\}
$$
(d) Show that
$$
\left[p^{2}, \mathbf{e}_{r}\right]=\mathrm{i} \hbar\left\{-\left(\mathbf{p} \frac{1}{r}+\frac{1}{r} \mathbf{p}\right)+\sum_{j}\left(p_{j} \frac{x_{j}}{r^{3}} \mathbf{x}+\mathbf{x} \frac{x_{j}}{r^{3}} p_{j}\right)\right\}
$$
(e) Hence show that $[H, \mathbf{M}]=0 .$ What is the physical significance of this result?
(f) Show that (i) $\left[M_{i}, L^{2}\right]=\mathrm{i} \sum_{j k} \epsilon_{i j k}\left(M_{k} L_{j}+L_{j} M_{k}\right)$, (ii) $\left[L_{i}, M^{2}\right]=$ 0, where $M^{2} \equiv M_{x}^{2}+M_{y}^{2}+M_{z}^{2} .$ What are the physical implications of these results?
(g) Show that
$$
\left[N_{i}, N_{j}\right]=-4 \mathrm{i} \sum_{u} \epsilon_{i j u} p^{2} L_{u}
$$
and that
$$
\left[N_{i},\left(\mathbf{e}_{r}\right)_{j}\right]-\left[N_{j},\left(\mathbf{e}_{r}\right)_{i}\right]=-\frac{4 \mathrm{i} \hbar}{r} \sum_{t} \epsilon_{i j t} L_{t}
$$
and hence that
$$
\left[M_{i}, M_{j}\right]=-2 \mathrm{i} \hbar^{2} m H \sum_{k} \epsilon_{i j k} L_{k}
$$
What physical implication does this equation have?