(a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators:

$$\begin{gathered} \left[L_Z,X\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}\mathbf{y}\mathbf{,}\mathbf{}\left[L_Z,y\right]\mathbf{=}\mathbf{-}\mathbf{i}\sout{\mathbf{h}}\mathbf{x}\mathbf{,}\mathbf{}\left[L_Z,Z\right]\mathbf{=}\mathbf{0} \\ \left[L_Z,p_x\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{p}}_{\mathbf{y}}\mathbf{}\mathbf{,}\left[L_Z,p_y\right]\mathbf{=}\mathbf{-}\mathbf{i}\sout{\mathbf{h}}{\mathbf{p}}_{\mathbf{x}}\mathbf{}\mathbf{,}\mathbf{}\left[L_Z,p_z\right]\mathbf{=}\mathbf{0} \end{gathered}$$

(b) Use these results to obtain $\left[L_Z,L_X\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{L}}_{\mathbf{y}}$directly from Equation 4.96.

(c) Evaluate the commutators $\left[L_z,r^2\right]$and$\left[L_z,p^2\right]$(where, of course, ${\mathbf{r}}^{\mathbf{2}}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{p}}^{\mathbf{2}}\mathbf{=}{\mathbf{p}}_{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{p}}_{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{p}}_{\mathbf{z}}^{\mathbf{2}}\mathbf{}\mathbf{)}$

(d) Show that the Hamiltonian $\mathbf{H}\mathbf{=}\left(p^2/2m\right)\mathbf{+}\mathbf{V}$commutes with all three components of L, provided that V depends only on r . (Thus $\mathbf{H}\mathbf{,}{\mathbf{L}}^{\mathbf{2}}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{L}}_{\mathbf{Z}}$and are mutually compatible observables.)

The basic relation in canonical conjugate quantities is termed as canonical commutation relation. It is there in quantum mechanics. It explains the algebra of quantities.

The coordinate representations of the orbital angular momentum in Quantum Mechanics which is similar to its classical forms are as follows,

$$\begin{gathered} \left(1\right){\hat{\mathrm{L}}}_{\mathrm{x}}=\hat{\mathrm{y}}{\mathrm{p}}_{\mathrm{z}}-\hat{\mathrm{z}}{\mathrm{p}}_{\mathrm{y}} \\ \left(2\right){\mathrm{L}}_{\mathrm{y}}=\hat{\mathrm{z}}{\hat{\mathrm{p}}}_{\mathrm{x}}-\hat{\mathrm{X}}{\hat{\mathrm{P}}}_{\mathrm{z}} \\ \left(3\right){\mathrm{L}}_{\mathrm{z}}=\hat{\mathrm{x}}{\mathrm{p}}_{\mathrm{y}}-\hat{\mathrm{y}}{\mathrm{p}}_{\mathrm{x}} \end{gathered}$$

Solve the given commutator,$\left[L_z,x\right]$.

$$\begin{gathered} \left[{\mathrm{L}}_{\mathrm{z}},\mathrm{x}\right]=\left[{\mathrm{xp}}_{\mathrm{y}}-{\mathrm{yp}}_{\mathrm{x}},\mathrm{x}\right] \\ =\left[{\mathrm{xp}}_{\mathrm{y}},\mathrm{x}\right]-\left[{\mathrm{yp}}_{\mathrm{x}},\mathrm{x}\right] \\ =\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},\mathrm{x}\right]+\left[\mathrm{x},\mathrm{x}\right]{\mathrm{p}}_{\mathrm{y}}-\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},\mathrm{x}\right]-\left[\mathrm{y},\mathrm{x}\right]{\mathrm{p}}_{\mathrm{x}} \\ =-\mathrm{y}\left[\mathrm{p}-\mathrm{x},\mathrm{x}\right] \\ =\mathrm{i}\sout{\mathrm{h}}\mathrm{y} \end{gathered}$$

Solve the given commutator,$\left[{\mathrm{L}}_{\mathrm{z}},\mathrm{y}\right]$.

localid="1658205793073" $$\begin{gathered} \left[{\mathrm{L}}_{\mathrm{z}},\mathrm{y}\right]=\left[{\mathrm{xp}}_{\mathrm{y}}-{\mathrm{yp}}_{\mathrm{x}},\mathrm{y}\right] \\ =\left[{\mathrm{xp}}_{\mathrm{y}},\mathrm{y}\right]-\left[{\mathrm{yp}}_{\mathrm{x}},\mathrm{y}\right] \\ =\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},\mathrm{y}\right]+\left[\mathrm{x},\mathrm{y}\right]{\mathrm{p}}_{\mathrm{y}} \\ =\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},\mathrm{y}\right] \\ =\mathrm{i}\sout{\mathrm{h}}\mathrm{x} \end{gathered}$$

Solve the given commutator,$\left[{\mathrm{L}}_{\mathrm{z}},\mathrm{z}\right]$.

$$\begin{gathered} \left[{\mathrm{L}}_{\mathrm{z}},\mathrm{z}\right]=\left[{\mathrm{xp}}_{\mathrm{y}}-{\mathrm{yp}}_{\mathrm{x}},\mathrm{z}\right] \\ =\left[{\mathrm{xp}}_{\mathrm{y}},\mathrm{z}\right]-\left[{\mathrm{yp}}_{\mathrm{x}},\mathrm{z}\right] \\ =\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},\mathrm{z}\right]+\left[\mathrm{x},\mathrm{z}\right]{\mathrm{p}}_{\mathrm{y}}-\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},\mathrm{z}\right]-\left[\mathrm{y},\mathrm{z}\right]{\mathrm{p}}_{\mathrm{x}} \\ =0 \end{gathered}$$

Solve the given commutator,$\left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{p}}_{\mathrm{x}}\right]$.

$$\begin{gathered} \left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{p}}_{\mathrm{x}}\right]=\left[{\mathrm{xp}}_{\mathrm{y}}-{\mathrm{yp}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{x}}\right] \\ =\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{p}}_{\mathrm{x}}\right]-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{x}}\right] \\ =\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},{\mathrm{p}}_{\mathrm{x}}\right]+\left[\mathrm{x},{\mathrm{p}}_{\mathrm{x}}\right]{\mathrm{p}}_{\mathrm{y}}-\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{x}}\right]-\left[\mathrm{y},{\mathrm{p}}_{\mathrm{x}}\right]{\mathrm{p}}_{\mathrm{x}} \\ =\mathrm{i}\sout{\mathrm{h}}\mathrm{y} \end{gathered}$$

Solve the given commutator, $\left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{p}}_{\mathrm{y}}\right]$.

$$\begin{gathered} \left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{p}}_{\mathrm{y}}\right]=\left[{\mathrm{xp}}_{\mathrm{y}}-{\mathrm{yp}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{y}}\right] \\ =\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{p}}_{\mathrm{y}}\right]-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{y}}\right] \\ =\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},{\mathrm{p}}_{\mathrm{y}}\right]+\left[\mathrm{x},{\mathrm{p}}_{\mathrm{y}}\right]{\mathrm{p}}_{\mathrm{y}}-\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{y}}\right]-\left[\mathrm{y},{\mathrm{p}}_{\mathrm{y}}\right]{\mathrm{p}}_{\mathrm{x}} \\ =\mathrm{i}\sout{\mathrm{h}}{\mathrm{p}}_{\mathrm{x}} \end{gathered}$$

Solve the given commutator, $\left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{p}}_{\mathrm{z}}\right]$.

$$\begin{gathered} \left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{p}}_{\mathrm{z}}\right]=\left[{\mathrm{xp}}_{\mathrm{y}}-{\mathrm{yp}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{z}}\right] \\ =\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{p}}_{\mathrm{z}}\right]-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{z}}\right] \\ =\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},{\mathrm{p}}_{\mathrm{z}}\right]+\left[\mathrm{x},{\mathrm{p}}_{\mathrm{z}}\right]{\mathrm{p}}_{\mathrm{y}}-\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{z}}\right]-\left[\mathrm{y},{\mathrm{p}}_{\mathrm{z}}\right]{\mathrm{p}}_{\mathrm{x}} \\ =0 \end{gathered}$$

Thus, all the commutators are verified.

Verify the given equation by proofing theleft-hand side equal to the right-hand side.

$$\begin{gathered} \left({\mathrm{L}}_{\mathrm{z}},{\mathrm{L}}_{\mathrm{x}}\right)=\left[{\mathrm{xp}}_{\mathrm{y}}-{\mathrm{yp}}_{\mathrm{x}},{\mathrm{yp}}_{\mathrm{z}}-{\mathrm{zp}}_{\mathrm{y}}\right] \\ =\left[{\mathrm{xp}}_{\mathrm{y}}{\mathrm{yp}}_{\mathrm{z}}\right]-\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{zp}}_{\mathrm{y}}\right]-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{yp}}_{\mathrm{z}}\right]+\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{zp}}_{\mathrm{y}}\right] \\ =\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},\mathrm{y}\right]{\mathrm{p}}_{\mathrm{z}}+\left[\mathrm{x},\mathrm{y}\right]{\mathrm{p}}_{\mathrm{y}}{\mathrm{p}}_{\mathrm{z}}+\mathrm{y}\left[\mathrm{x},{\mathrm{p}}_{\mathrm{z}}\right]{\mathrm{p}}_{\mathrm{y}}+\mathrm{yx}\left[{\mathrm{p}}_{\mathrm{y}},{\mathrm{p}}_{\mathrm{z}}\right] \\ =-\left[\mathrm{x},\mathrm{z}\right]{\mathrm{p}}_{\mathrm{y}}{\mathrm{p}}_{\mathrm{y}}-\mathrm{z}\left[\mathrm{x},{\mathrm{p}}_{\mathrm{y}}\right]{\mathrm{p}}_{\mathrm{y}}-\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},\mathrm{z}\right]{\mathrm{p}}_{\mathrm{y}}-\mathrm{zx}\left[{\mathrm{p}}_{\mathrm{y}},{\mathrm{p}}_{\mathrm{y}}\right]-\left[\mathrm{y},\mathrm{y}\right]{\mathrm{p}}_{\mathrm{x}}{\mathrm{p}}_{\mathrm{z}} \end{gathered}$$

Further solve the expression.

$$\begin{gathered} \left({\mathrm{L}}_{\mathrm{z}},{\mathrm{L}}_{\mathrm{x}}\right)=-\mathrm{y}\left[\mathrm{y},{\mathrm{p}}_{\mathrm{z}}\right]{\mathrm{p}}_{\mathrm{x}}-\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},\mathrm{y}\right]{\mathrm{p}}_{\mathrm{z}}-\mathrm{yy}\left[{\mathrm{p}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{z}}\right]+\left[\mathrm{y},\mathrm{z}\right]{\mathrm{p}}_{\mathrm{x}}{\mathrm{p}}_{\mathrm{y}}+\mathrm{z}\left[\mathrm{y},{\mathrm{p}}_{\mathrm{y}}\right]\mathrm{px}+\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},\mathrm{z}\right]{\mathrm{p}}_{\mathrm{y}}+\mathrm{zy}\left[{\mathrm{p}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{y}}\right] \\ =-\mathrm{i}\sout{\mathrm{h}}{\mathrm{xp}}_{\mathrm{z}}+\mathrm{i}\sout{\mathrm{h}}{\mathrm{zp}}_{\mathrm{x}} \\ =\mathrm{i}\sout{\mathrm{h}}\left({\mathrm{zp}}_{\mathrm{x}}-{\mathrm{xp}}_{\mathrm{z}}\right) \\ =\mathrm{i}\sout{\mathrm{h}}\mathrm{Ly} \end{gathered}$$

Hence, the given equation is verified.

Evaluate the commutator$\left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{r}}^2\right]$.

localid="1658209407417" $$\begin{gathered} \left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{r}}^2\right]=\left[{\mathrm{xp}}_{\mathrm{y}}-{\mathrm{yp}}_{\mathrm{x}},{\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2\right] \\ =\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{x}}^2\right]-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{x}}^2\right]+\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{y}}^2\right]-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{y}}^2\right]-\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{z}}^2\right]-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{z}}^2\right] \\ =0+0-\mathrm{y}\left(\left[{\mathrm{p}}_{\mathrm{x}},\mathrm{x}\right]\mathrm{x}+\mathrm{x}\left[{\mathrm{p}}_{\mathrm{x}},\mathrm{x}\right]\right)-0+\mathrm{x}\left(\left[{\mathrm{p}}_{\mathrm{y}},\mathrm{y}\right]\mathrm{y}+\mathrm{y}\left[{\mathrm{p}}_{\mathrm{y}},\mathrm{y}\right]\right)+0-0-0+0+0-0-0-0 \\ =-\mathrm{y}\left(-\mathrm{i}\sout{\mathrm{h}}\mathrm{x}-\mathrm{i}\sout{\mathrm{h}}\mathrm{x}\right)+\mathrm{x}\left(-\mathrm{i}\sout{\mathrm{h}}\mathrm{y}-\mathrm{i}\sout{\mathrm{h}}\mathrm{y}\right) \\ =2\mathrm{i}\sout{\mathrm{h}}\mathrm{yx}-2\mathrm{i}\sout{\mathrm{h}}\mathrm{xy} \\ =0 \end{gathered}$$

Evaluate the commutators $\left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{p}}^2\right]$

$$\begin{gathered} \left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{p}}^2\right]=\left[{\mathrm{xp}}_{\mathrm{y}}-{\mathrm{yp}}_{\mathrm{x}},{\mathrm{px}}^2+{\mathrm{py}}^2+{\mathrm{pz}}^2\right] \\ =\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{p}}^2\mathrm{x}\right]-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{p}}^2\mathrm{x}\right]+\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{p}}^2\mathrm{y}\right]-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{p}}^2\mathrm{y}\right]-\left[{\mathrm{xp}}_{\mathrm{y}},{\mathrm{p}}^2\mathrm{z}\right] \\ =-\left[{\mathrm{yp}}_{\mathrm{x}},{\mathrm{p}}^2\mathrm{z}\right]-\left[\mathrm{x},{\mathrm{p}}^2\mathrm{x}\right]{\mathrm{p}}_{\mathrm{y}}+\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},{\mathrm{p}}^2\mathrm{x}\right]-\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},{\mathrm{p}}^2\mathrm{x}\right]-\left[\mathrm{y},{\mathrm{p}}^2\mathrm{x}\right]{\mathrm{p}}_{\mathrm{x}}+\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},{\mathrm{p}}^2\mathrm{y}\right]+\left[\mathrm{x},,{\mathrm{p}}^2\mathrm{y}\right]{\mathrm{p}}_{\mathrm{y}} \\ =-\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},{\mathrm{y}}^2\right]-\left[\mathrm{y},{\mathrm{p}}^2\mathrm{y}\right]{\mathrm{p}}_{\mathrm{x}}+\mathrm{x}\left[{\mathrm{p}}_{\mathrm{y}},{\mathrm{p}}^2\mathrm{z}\right]+\left[\mathrm{x},{\mathrm{p}}^2\mathrm{z}\right]{\mathrm{p}}_{\mathrm{y}}-\mathrm{y}\left[{\mathrm{p}}_{\mathrm{x}},{\mathrm{p}}^2\mathrm{z}\right]-\left(\mathrm{y},{\mathrm{p}}^2\mathrm{z}\right){\mathrm{p}}_{\mathrm{x}} \end{gathered}$$

Further solve the expression.

localid="1658210515968" $$\begin{gathered} \left[{\mathrm{L}}_{\mathrm{Z}},{\mathrm{p}}^2\right]=\left(\left[\mathrm{x},{\mathrm{p}}_{\mathrm{x}}\right]{\mathrm{p}}_{\mathrm{x}}+{\mathrm{p}}_{\mathrm{x}}\left[\mathrm{x},{\mathrm{p}}_{\mathrm{x}}\right]\right){\mathrm{p}}_{\mathrm{y}}+0-0-0+0+0-0-\left(\left[\mathrm{y},{\mathrm{p}}_{\mathrm{y}}\right]{\mathrm{p}}_{\mathrm{y}}+{\mathrm{p}}_{\mathrm{y}}\left[\mathrm{y},{\mathrm{p}}_{\mathrm{y}}\right]\right){\mathrm{p}}_{\mathrm{x}}+0+0-0-0 \\ =\left(\mathrm{i}\sout{\mathrm{h}}{\mathrm{p}}_{\mathrm{x}}+\mathrm{i}\sout{\mathrm{h}}{\mathrm{p}}_{\mathrm{x}}\right){\mathrm{p}}_{\mathrm{y}}-\left(\mathrm{i}\sout{\mathrm{h}}{\mathrm{p}}_{\mathrm{y}}+\mathrm{i}\sout{\mathrm{h}}{\mathrm{p}}_{\mathrm{y}}\right){\mathrm{p}}_{\mathrm{x}} \\ =2\mathrm{i}\sout{\mathrm{h}}{\mathrm{p}}_{\mathrm{x}}{\mathrm{p}}_{\mathrm{y}}-2\mathrm{i}\sout{\mathrm{h}}{\mathrm{p}}_{\mathrm{y}}{\mathrm{p}}_{\mathrm{x}} \\ =2\mathrm{i}\sout{\mathrm{h}}\left[{\mathrm{p}}_{\mathrm{x}},{\mathrm{p}}_{\mathrm{y}}\right] \\ =0 \end{gathered}$$

$\mathrm{Thus},\mathrm{the}\mathrm{value}\mathrm{of}\left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{r}}^2\right]\mathrm{is}\mathrm{o}\mathrm{and}\mathrm{the}\mathrm{value}\mathrm{of}\left[{\mathrm{L}}_{\mathrm{z}},{\mathrm{p}}^2\right]\mathrm{is}\mathrm{also}0.$

It is known that $\left[{\mathrm{L}}_{\mathrm{Z}},{\mathrm{r}}^2\right]=\left[{\mathrm{L}}_{\mathrm{Z}},{\mathrm{p}}^2\right]=0$and by the symmetry of x,y and z, $\left[{\mathrm{L}}_{\mathrm{x}},{\mathrm{r}}^2\right]=\left[{\mathrm{L}}_{\mathrm{x}},{\mathrm{p}}^2\right]=0$, and $\left[{\mathrm{L}}_{\mathrm{y}},{\mathrm{r}}^2\right]=\left[{\mathrm{L}}_{\mathrm{y}},{\mathrm{p}}^2\right]$. So, $\left[{\mathrm{L}}_{\mathrm{y}},{\mathrm{r}}^2\right]=\left[{\mathrm{L}}_{\mathrm{y}},{\mathrm{p}}^2\right]=0$.

It can be observed that all the components commute with ${\mathrm{p}}^2\mathrm{and}{\mathrm{r}}^2$

Write the expression forthe Hamiltonian function.

$$\begin{gathered} \mathrm{H}=\frac{{\mathrm{p}}^2}{2\mathrm{m}}+\mathrm{V}\left(\sqrt{{\mathrm{r}}^2}\right) \\ \left[\mathrm{H},\mathrm{L}\right]=0 \end{gathered}$$

Thus, the Hamiltonian $\mathrm{H}=\left({\mathrm{p}}^2/2\mathrm{m}\right)+\mathrm{V}$commutes with all three components of L.