91Ó°ÊÓ

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.

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

…(¾±)

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.

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

Write the expression for the Hermit ion function again.

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

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.

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.

$\overline{V}V=\frac{∂V}{∂r}\hat{r}$

Substitute the above value in.

Thus, it is proved that .