Chapter 38, Problem 18P

(a)

To determine

The proof that function $\Psi \left(x,t\right)=A{e}^{i\left(kx-\omega t\right)}$ is a solution to the time dependent Schrödinger equation but $\Psi \left(x,t\right)=A\cos \left(kx-\omega t\right)$ and ${\Psi }_1\left(x,t\right)=A\sin \left(kx-\omega t\right)$ is not the solution.

(b)

To determine

The proof that conservation of energy gives the result $\frac{{\hslash }^2{k}^2}{2m}+U_0=\hslash \omega$.