91影视

To be considered an electron in the ground state of a hydrogen atom

The law of conservation of momentum states that the sum of kinetic energy and the potential energy for a particle always remains constant.

(a)

The expectation value of potential energy is,

$$\begin{gathered} PE={鈭�}_0^{鈭�}-\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}\mathrm{r}}P\left(r\right)dr \\ =-\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}\mathrm{r}}\frac{4}{{a_{鈭�}}^3}{鈭�}_0^{鈭�}r{e}^{-2r/a_{鈭�}}dr \end{gathered}$$

Substitute x for $\frac{2r}{a_{鈭�}}$in the integration.

role="math" localid="1659615862149" $$\begin{gathered} PE=-\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}}{鈭�}_0^{鈭�}x{e}^{x}dr \\ =-\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}} \end{gathered}$$

Therefore, the expectation value of potential energy is $-\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}}$.

(b)

The total energy is $\left(-\frac{1}{2}\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}}\right)$

The kinetic energy can be calculated by subtracting potential energy from the total energy

KE= E - PE

for role="math" localid="1659615925271" $E=-\frac{1}{2}\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}}$androle="math" localid="1659615868874" $PE=-\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}}$, we have-
role="math" localid="1659615966673" $\begin{array}{rcl}KE& =& \left(-\frac{1}{2}\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}}\right)-\left(-\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}}\right)\\ & =& \frac{1}{2}\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}}\end{array}$

Therefore, the expectation value of kinetic energy is $\begin{array}{rcl}& & \frac{1}{2}\frac{e^2}{4{\mathrm{蟺蔚}}_{鈭�}}\frac{1}{a_{鈭�}}\end{array}$.