In introducing the Laplace-Runge-Lenz vector and the canonically quantized hydrogen atom, there were a number of Poisson brackets for which we just stated the result, without explicit calculation. It's time to address that here.

(a) Evaluate the Poisson bracket of the angular momentum vector \(\vec{L}\) and the hydrogen atom's Hamiltonian \(H\) and show explicitly that angular momentum is conserved (i.e., \(\{H, \vec{L}\}=0\) ).

(b) Evaluate the Poisson bracket of the Laplace-Runge-Lenz vector \(\vec{A}\) and the hydrogen atom's Hamiltonian \(H\) and show explicitly that it is conserved (i.e., \(\{H, \vec{A}\}=0)\).

(c) Evaluate the Poisson bracket of two components of the Laplace-RungeLenz vector and show that

\[\begin{equation*}\left\{A_{i}, A_{j}\right\}=-\sum_{k=1}^{3} \epsilon_{i j k} L_{k} \frac{2}{m_{e}} H \tag{9.170}\end{equation*}\]