(a) Prove the three-dimensional virial theorem

$\mathbf{2}\mathbf{⟨}\mathbf{T}\mathbf{⟩}\mathbf{=}\mathbf{⟨}\mathbf{r}\mathbf{\cdot }\mathbf{\nabla }\mathbf{V}\mathbf{⟩}$

(for stationary states). Hint: Refer to problem 3.31,

(b) Apply the virial theorem to the case of hydrogen, and show that

$\mathbf{⟨}\mathbf{T}\mathbf{⟩}\mathbf{=}\mathbf{-}{\mathbf{E}}_{\mathbf{n}}\mathbf{;}\mathbf{⟨}\mathbf{V}\mathbf{⟩}\mathbf{=}\mathbf{2}{\mathbf{E}}_{\mathbf{n}}$

(c) Apply the virial theorem to the three-dimensional harmonic oscillator and show that in this case

$\mathbf{⟨}\mathbf{T}\mathbf{⟩}\mathbf{=}\mathbf{⟨}\mathbf{V}\mathbf{⟩}\mathbf{=}{\mathbf{E}}_{\mathbf{n}}\mathbf{/}\mathbf{2}$

The virial theorem connects the gravitational potential energy, U, of a self-gravitating entity to its total kinetic energy, T, which results from the motions of its individual pieces.

The virial theorem connects a quantum system's expected kinetic energy to its potential. This is both theoretically interesting and crucial for computational methods such as "density functional theory."

The three-dimensional virial theorem described as follows,

It is known that role="math" localid="1658128835661" $[\mathrm{H},\hspace{0.33em}\mathrm{r}\cdot \mathrm{p}]=\sum ^{\begin{array}{c}\mathrm{i}=1\\ 3\end{array}}[{\mathrm{H}}_{\mathrm{j}},\hspace{0.33em}{\mathrm{r}}_{\mathrm{j}}\cdot {\mathrm{p}}_{\mathrm{j}}]$. Apply it to the above expression and then solve it.

Further evaluate the expression.

Further simplify the expression.

Thus, in case of stationary states

Write the expression in case of hydrogen atom.

$$\begin{gathered} \mathrm{V}\left(\mathrm{r}\right)=-\frac{\mathrm{e}}{4\mathrm{\pi }0{\text{'}}_0\mathrm{r}} \\ \nabla \mathrm{V}=\frac{\mathrm{e}}{4\mathrm{\pi }0{\text{'}}_0{\mathrm{r}}^2}\hat{\mathrm{r}} \\ \mathrm{r}.\nabla \mathrm{V}=\frac{\mathrm{e}}{4\mathrm{\pi }0{\text{'}}_0\mathrm{r}} \\ =-\mathrm{V} \end{gathered}$$

So, it can be observed the equation is true but $2⟨\mathrm{T}⟩=-⟨\mathrm{V}⟩$.

It is known that $⟨\mathrm{T}⟩+⟨\mathrm{V}⟩={\mathrm{E}}_{\mathrm{n}}^{\text{'}}$. Substitute the values in this expression.

Thus, the given expression is verified.

Write the expression for Harmonic oscillator.

$$\begin{gathered} \mathrm{V}=\frac{1}{2}{\mathrm{m\omega }}^2{\mathrm{r}}^2 \\ \nabla \mathrm{V}={\mathrm{m\omega }}^2\mathrm{r}\hat{\mathrm{r}} \\ \mathrm{r}.\nabla \mathrm{V}={\mathrm{m\omega }}^2{\mathrm{r}}^2 \\ =2\mathrm{V} \end{gathered}$$

Further solve the expression.

It is known that as $⟨\mathrm{T}⟩+⟨\mathrm{V}⟩={\mathrm{E}}_{\mathrm{n}}$ . Substitute the values in this expression.

Thus, the given expression is verified.