The virial theorem is a statement about the relationship between the kinetic and potential energies for a system bound in a potential that is a power of relative distance between particles. For such a system, both total energy and angular momentum are conserved, and the potential energy \(U\) is proportional to the kinetic energy \(K\), with \(U+K=E\), the total energy. In this problem, we will study the virial theorem and its consequences classically and quantum mechanically.

(a) Identify the proportionality relationship between the kinetic and potential energies and prove the virial theorem for a classical system of a pair of masses bound by a potential energy \(U(r)=\frac{n}{|n|} k r^{n}\), where \(r\) is the distance between the masses, \(n\) is a non-zero real number, and \(k\) is a constant that has units of energy/distance \({ }^{n}\).
Hint: Remember that angular momentum is conserved and think about Newton's second law for the system.
(b) Now, let's study the virial theorem in a quantum mechanical system. Consider a two-body system for which the potential is
\[\begin{equation*}V(\hat{r})=\frac{n}{|n|} k \hat{r}^{n} \tag{9.171}\end{equation*}\]
where \(\hat{r}\) is the operator of the distance between the two masses. On an energy eigenstate \(|\psiangle\), argue that the operator
\[\begin{equation*}\hat{\mathcal{O}}=\hat{\vec{r}} \cdot \hat{\vec{p}}=\hat{x} \hat{p}_{x}+\hat{y} \hat{p}_{y}+\hat{z} \hat{p}_{z} \tag{9.172}\end{equation*}\]
has a time-independent expectation value. Using this observation, determine the relationship between the expectation values of the kinetic energy operator \(\langle\hat{K}angle\) and the potential operator \(\langle V(\hat{r})angle\).
(c) Why is the operator \(\hat{\mathcal{O}}=\hat{\vec{r}} \cdot \hat{\vec{p}}\) used to prove the virial theorem quantum mechanically? What does this operator do?
 Consider its action and eigenstates/values in position space.
(d) Explicitly show that the virial theorem holds for the ground state of the hydrogen atom by taking expectation values of the kinetic and potential energy operators.
(e) Explicitly show that the virial theorem holds for every energy eigenstate of the one-dimensional quantum harmonic oscillator. Does the virial theorem provide a new interpretation of the Heisenberg uncertainty principle for the harmonic oscillator?