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The fundamental commutation relations for angular momentum allow for half-integer (as well as integer) eigenvalues. But for orbital angular momentum only the integer values occur. There must be some extra constraint in the specific form $L = r \times p$ that excludes half-integer values. Let a be some convenient constant with the dimensions of length (the Bohr radius, say, if we're talking about hydrogen), and define the operators $q_1\equiv\frac{1}{\sqrt2}[x+(a^2/\hbar)p_y];\\ p_1\equiv\frac{1}{\sqrt2}[p_x-(\hbar/a^2)y];\\ q_2\equiv\frac{1}{\sqrt2}[x-(a^2/\hbar)p_y];\\ p_2\equiv\frac{1}{\sqrt2}[p_x+(\hbar/a^2)y].$ (a) Verify that $[q_1,q_2] = [p_1,p_2]=0; [q_1,p_1]=[q_2,p_2]=i\hbar.$ Thus the q's and the p's satisfy the canonical commutation relations for position and momentum, and those of index 1 are compatible with those of index 2. (b) Show that $L_z=\frac{\hbar}{2a^2}(q_1^2-q_2^2)+\frac{a^2}{2\hbar}(p_1^2-p_2^2).$ (c) Check that $L_z=H_1-H_2,$ where each H is the Hamiltonain for a harmonic oscillator with mass $m=\habr/a^2$ and frequency $\omega = 1.$ (d) We know that the eigenvalues of the harmonic oscillator Hamiltonian are $(n+1/2)\hbar \omega,$ where n = 0,1,2,.... Use this to conclude that the eigenvalues of $L_z$ must be integers.

(a) Show the following commutation relations:

\begin{align*} \begin{array}{lll} {\left[\hat{Y}, \hat{L}_y\right]=0,} & {\left[\hat{Y}, \hat{L}_z\right]=i \hbar \hat{X},} & {\left[\hat{Y}, \hat{L}_x\right]=-i \hbar \hat{Z},} \\ {\left[\hat{Z}, \hat{L}_z\right]=0,} & {\left[\hat{Z}, \hat{L}_x\right]=i \hbar \hat{Y},} & {\left[\hat{Z}, \hat{L}_y\right]=-i \hbar \hat{X} .} \end{array} \end{align*}

(b) Using a cyclic permutation of $x y z$ , apply the results of (a) to infer expressions for $\left[\hat{X}, \hat{L}_x\right],\left[\hat{X}, \hat{L}_y\right]$ , and $\left[\hat{X}, \hat{L}_z\right]$ .
(c) Use the results of (a) and (b) to calculate $\left[\hat{R}^2, \hat{L}_x\right],\left[\hat{R}^2, \hat{L}_y\right]$ , and $\left[\hat{R}^2, \hat{L}_x\right]$ , where $\hat{R}^2=\hat{X}^2+\hat{Y}^2+\hat{Z}^2$ .

(a) Work out all of the canonical commutation relations for components of the operators r and p: $[x, y], [x, p_y], [x, p_x], [p_y, p_z],$ and so on. (b) Confirm Ehrenfest's theorem for 3-dimensions: $\frac{d}{dt}\left \langle r \right \rangle=\frac{1}{m}\left \langle p \right \rangle$ and $\frac{d}{dt}\left \langle p \right \rangle=\left \langle -\triangledown V \right \rangle$ (Each of these, of course, stands for three equations—one for each component.) (c) Formulate Heisenberg's uncertainty principle in three dimensions.