We explicitly constructed the ground-state and first excited-state wavefunctions for the hydrogen atom in this chapter, but we can go further. The LaplaceRunge-Lenz operator commutes with the Hamiltonian, but its action changes the angular momentum of a state so can be used to relate known eigenstate

wavefunctions to presently unknown wavefunctions. We know what the energy eigenvalue of the \(n\)th state is, and that will also help us construct the energy eigenstate wavefunctions.
(a) Consider the \(n\) th-energy eigenstate with total angular momentum \(\ell=0\). What is a reasonable ansatz for the form of the eigenstate wavefunction \(\psi_{n, 0,0}(r)\), that is purely a function of the electron's distance from the proton?
 What behavior must you have as \(r \rightarrow 0\) ? How do you get the factor of \(n^{2}\) in the energy eigenvalue? How many nodes should there be in the wavefunction of the \(n\) th-energy eigenstate?
(b) Using the wavefunction of the first excited state of hydrogen, \(\psi_{2,0,0}(\vec{r})\), calculated in Example 9.1, we can then act with components of the LaplaceRunge-Lenz operator to change the value of the angular momentum of the state, while maintaining the energy eigenvalue. Here, we'll construct all states that have the same energy as the \(n=2, \ell=0\) eigenstate. This simplifies our task because the angular momentum factors in the Laplace-RungeLenz operator annihilate this state. With this observation, determine the Laplace-Runge-Lenz operator with zero angular momentum; you should find
\[\begin{equation*}\hat{A}_{k}=-i \frac{\hbar}{m_{e}} \hat{p}_{k}-\frac{e^{2}}{4 \pi \epsilon_{0}} \frac{\hat{r}_{k}}{\hat{r}} \tag{9.174}\end{equation*}\]
(c) Using this result, construct the energy eigenstate wavefunctions \(\psi_{1,1,1}(\vec{r})\), \(\psi_{1,1,0}(\vec{r})\), and \(\psi_{1,1,-1}(\vec{r})\) from the action of \(\hat{A}_{+}, \hat{A}_{z}\), and \(\hat{A}_{-}\)on the wavefunction \(\psi_{2,0,0}(r)\) of Eq. (9.138), respectively.