Chapter 8, Problem 8.17P

(a)

To determine

A bound on the ground state of the one-dimensional harmonic oscillator using the function $\psi \left(x\right)=\{\begin{array}{cc}A\cos \left(\pi x/a\right)& -a/2<x<a/2\\ 0& \text{otherwise}\end{array}$ and the “best” value of $a$. Also compare ${\langle H\rangle }_{\min }$ with the exact energy.

(b)

To determine

A bound on the first excited state using $\psi \left(x\right)=B\sin \left(\pi x/a\right)$ on the interval $\left(-a,a\right)$.