We can quantitatively study this claim, that for the hydrogen atom, we do not need to invert the Hamiltonian to use the power method to estimate its groundstate energy. In this exercise, we'll just consider the power method applied to spherically symmetric energy eigenstates of hydrogen; that is, those states that can be expressed as \(|n, 0,0angle \equiv|nangle\) in the \(n, l, m\) quantum numbers, where \(n \geq 1\). As established in Chap. 9, the state for which \(n=1\) is the ground state and the energy eigenvalues are

\[\begin{equation*}\hat{H}|nangle=-\frac{m_{e} e^{4}}{2\left(4 \pi \epsilon_{0}\right)^{2} \hbar^{2}} \frac{1}{n^{2}}|nangle \tag{10.130}\end{equation*}\]

(a) Consider the generic, spherically symmetric state \(|\chiangle\) of the hydrogen atom, which can be expressed as

\[\begin{equation*}|\chiangle=\sum_{n=1}^{\infty} \beta_{n}|nangle \tag{10.131}\end{equation*}\]

for some coefficients \(\beta_{n}\). Estimate the ground-state energy of hydrogen after \(N\) applications of the power method applied to this state.

(b) For concreteness, let's take the coefficients in the expansion of the state \(|\chiangle\) to be

\[\begin{equation*}\beta_{n} \propto \frac{1}{n} . \tag{10.132}\end{equation*}\]

What is the expectation value of the Hamiltonian on this state? Is it greater than the known, exact value of the ground-state energy?

(c) Determine the estimate of the ground-state energy of hydrogen after \(N\) applications of the power method on the state defined in part (b). Express the result in terms of the Riemann zeta function, which is defined to be

\[\begin{equation*}\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}} \tag{10.133}\end{equation*}\]

for \(s>1\).

(d) Show that this estimate converges to the true ground-state energy as \(N \rightarrow \infty\). Plot the value of the \(N\) th application power method estimate as a function of \(N\). For what \(N\) does the power method produce a result within \(1 \%\) of the true ground-state energy?