Consider the scaled one-dimensional Schrödinger equation with a potential $V(x)$ such that there is a complete set of real, normalized bound states $\psi_{n}(x), n=0,1,2, \ldots$, with discrete energies $E_{0}<E_{1}<E_{2}<\ldots$, satisfying

$$-\frac{d^{2} \psi_{n}}{d x^{2}}+V(x) \psi_{n}=E_{n} \psi_{n}$$

Show that the quantity

$$\langle E\rangle=\int_{-\infty}^{\infty}\left(\left(\frac{d \psi}{d x}\right)^{2}+V(x) \psi^{2}\right) d x$$

where $\psi(x)$ is a real, normalized trial function depending on one or more parameters $\alpha$, can be used to estimate $E_{0}$, and show that $\langle E\rangle \geqslant E_{0}$.

Let the potential be $V(x)=|x|$. Using a suitable one-parameter family of either Gaussian or piecewise polynomial trial functions, find a good estimate for $E_{0}$ in this case.

How could you obtain a good estimate for $E_{1}$ ? [ You should suggest suitable trial functions, but DO NOT carry out any further integration.]