Chapter 13.7, Problem 20P

Write the Schrödinger equation (3.22) if $\psi$ is a function of x, and $V=\frac{1}{2}m{\omega }^2{x}^2$ (this is a one-dimensional harmonic oscillator). Find the solutions ${\psi }_n(x)$ and the energy eigenvalues $E_n.$ Hints: In Chapter 12, equation (22.1) and the first equationin (22.11), replace $x$ by $\alpha x$ where $\alpha =\sqrt{m\omega /h}$.

(Don’t forget appropriate factorsof $\alpha$ for the $x^{\prime }$ s in the denominators of $D=d/dx$ an ${\psi }^{\prime \prime }=d^2\psi /d{x}^2,$ ) Compare your results for equation (22.1) with the Schrödinger equation you wroteabove to see that they are identical if $E_n=\left(n+\frac{1}{2}\right)\hslash \omega .$ Write the solutions ${\psi }_n(x)$ of the Schrödinger equation using Chapter 12, equations (22.11) and (22.12).