Question

Question: Use the virial theorem (Problem 4.40) to prove Equation 6.55.

Short answer

It is proved that .

Step-by-step solution

Expression of thepotential energy and Bohr energy

The expression for potential energy is as follows,

$\mathit{V}\mathbf{=}\frac{\mathbf{-}{\mathbf{e}}^{\mathbf{2}}}{\mathbf{4}\mathbf{\pi }{\dot{\mathbf{o}}}_{\mathbf{0}}}\frac{\mathbf{1}}{\mathbf{r}}$

The expression for Bohrenergy is as follows,

localid="1658214362022" ${\mathit{E}}_{\mathbf{n}}\mathbf{=}\mathbf{-}\mathbf{\left[}\frac{\mathbf{m}}{\mathbf{2}{\sout{\mathbf{h}}}^{\mathbf{2}}}{\left(\frac{-e^2}{4\pi {\dot{o}}_0}\right)}^{\mathbf{2}}\mathbf{\right]}\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{2}}}$

Verification of <1r>=1an2

Write the expression for the relation between potential energy and Bohr energy.

Substitute all the values in the above expression.

localid="1658214341492"

Consider $a=\frac{4\mathrm{\pi }{\dot{\mathrm{o}}}_0{\sout{\mathrm{h}}}^2}{{\mathrm{me}}^2}$, where, a is the Bohr radius.

Substitute$\frac{4\mathrm{\pi }{\dot{\mathrm{o}}}_0{\sout{\mathrm{h}}}^2}{{\mathrm{me}}^2}$ for a in the above expression.

Thus, the equation 6.55 is verified.