9-21(1) Consider the following potential, which is a one-dimensional analog of the
hydrogen atom:
$\begin{equation*}
V(x) = \begin{cases}
-\frac{K}{x}, & x \ge 0 \\
\infty, & x \le 0.
\end{cases}
\end{equation*}$
Carry out a variational analysis for a particle of mass $m$ moving in this
potential, taking as the trial wavefunction
$\phi(x) = Cxe^{-\beta x}$ ($x \ge 0$; $\phi(x) = 0$ otherwise),
where $C$ is the normalization constant and $\beta$ is the variational parameter. If
$\kappa = e^2/4\pi\epsilon_0$, as in the Coulomb potential, how does your estimate of the
ground-state energy (for an electron) compare with that for the usual
Coulomb potential?
