Question

(a) Prove that for a particle in a potential V(r)the rate of change of the expectation value of the orbital angular momentum L is equal to the expectation value of the torque:

$\frac{\mathbf{d}}{\mathbf{dt}}<L>\mathbf{=}<N>$

Where,

$\mathbf{N}\mathbf{=}\mathbf{r}\mathbf{\times }\left(\overline{V}V\right)$

(This is the rotational analog to Ehrenfest's theorem.)

(b) Show that $\mathbf{d}<L>\mathbf{/}\mathbf{dt}\mathbf{=}\mathbf{0}$for any spherically symmetric potential. (This is one form of the quantum statement of conservation of angular momentum.)

Short answer

(a) The given expression has been verified.

(b) It is proved that .

Step-by-step solution

Definition of Ehrenfest's theorem

Ehrenfest’stheorem connects the time derivative of the position and momentum operators to the expected value of the force on a heavy particle traveling in a scalar potential.

Step 2: (a) Verification of the given expression

Write equation 3.71 (Also, according to the energy-time uncertainty principle for the case of $L_x$).

…(i)

Write the expression for theHermition function.

$\begin{array}{rcl}\left[\mathrm{H},{\mathrm{L}}_{\mathrm{x}}\right]& =& \left[\frac{{\mathrm{p}}^2}{2\mathrm{m}}+\mathrm{V},{\mathrm{L}}_{\mathrm{x}}\right]\\ & =& \left[\frac{{\mathrm{p}}^2}{2\mathrm{m}},{\mathrm{L}}_{\mathrm{x}}\right]+\left[\mathrm{V},{\mathrm{L}}_{\mathrm{x}}\right]\\ & =& \frac{1}{2\mathrm{m}}\left[{\mathrm{p}}^2,{\mathrm{L}}_{\mathrm{x}}\right]+\left[\mathrm{V},{\mathrm{yp}}_{\mathrm{z}}-{\mathrm{zp}}_{\mathrm{y}}\right]\\ & =& \frac{1}{2\mathrm{m}}\left(0\right)+\left[\mathrm{V},{\mathrm{yp}}_{\mathrm{z}}\right]-\left[\mathrm{V},{\mathrm{zp}}_{\mathrm{y}}\right]\end{array}$

Further simplify the above expression.

$\begin{array}{rcl}\left[\mathrm{H},{\mathrm{L}}_{\mathrm{x}}\right]& =& \mathrm{y}\left[\mathrm{V},{\mathrm{p}}_{\mathrm{z}}\right]+\left[\mathrm{V},\mathrm{y}\right]{\mathrm{p}}_{\mathrm{z}}-\mathrm{z}\left[\mathrm{V},{\mathrm{p}}_{\mathrm{y}}\right]-\left[\mathrm{V},\mathrm{z}\right]{\mathrm{p}}_{\mathrm{y}}\\ & =& \mathrm{y}\left[\mathrm{V},{\mathrm{p}}_{\mathrm{z}}\right]-\mathrm{z}\left[\mathrm{V},{\mathrm{p}}_{\mathrm{y}}\right]\\ & =& \mathrm{y}\left[\mathrm{V},\mathrm{i}\sout{\mathrm{h}}\frac{\partial }{\partial \mathrm{z}}\right]-\mathrm{z}\left[\mathrm{V},\mathrm{i}\sout{\mathrm{h}}\frac{\partial }{\partial \mathrm{y}}\right]\\ & =& \mathrm{yi}\sout{\mathrm{h}}\frac{\partial \mathrm{V}}{\partial \mathrm{z}}-\mathrm{zi}\sout{\mathrm{h}}\frac{\partial \mathrm{V}}{\partial \mathrm{z}}\left[\mathrm{H},{\mathrm{L}}_{\mathrm{x}}\right]\\ & =& \mathrm{i}\sout{\mathrm{h}}\left[\mathrm{y}\frac{\partial \mathrm{V}}{\partial \mathrm{z}}-\mathrm{z}\frac{\partial \mathrm{V}}{\partial \mathrm{y}}\right]\end{array}$

Write the expression for the Hermit ion function again.

$\left[\mathrm{H},{\mathrm{L}}_{\mathrm{x}}\right]=\mathrm{i}\sout{\mathrm{h}}{\left(\overline{\mathrm{r}}\times \overline{\mathrm{V}}\mathrm{V}\right)}_{\mathrm{x}}$

Substitute the above value in equation (i).

Obtain similar results for and in the same way.

Apply and infer.

Thus, the given expression has been verified.

Step 3: (b) Verification of the given relation

Write the expression for the potential when it is spherically symmetric.

$V\left(\overline{r}\right)=V\left(r\right)$

Write the expression for the potentialin spherical coordinates.

$\overline{V}V=\frac{\partial V}{\partial r}\hat{r}$

Substitute the above value in.

Thus, it is proved that .