Chapter 40, Problem 40.58P

(a)

To determine

To show: That for one dimensional Schrodinger equation $-\frac{{\hslash }^2}{2m}\frac{d^2\psi \left(x\right)}{d{x}^2}+U\left(x\right)\psi \left(x\right)=E\psi \left(x\right)$ the wave function $\psi \left(x\right)=A_{1}x{e}^{-\frac{{\alpha }^2{x}^2}{2}}$ is a solution with energy corresponding to $n=1$.

(b)

To determine

The normalization constant $A_1$ for the wave function $\psi \left(x\right)=A_{1}x{e}^{-\frac{{\alpha }^2{x}^2}{2}}$.

(c)

To determine

To show: That probability density for the wave function The wave function is $\psi \left(x\right)=A_{1}x{e}^{-\frac{{\alpha }^2{x}^2}{2}}$. has its maximum at $x=\pm \frac{1}{\alpha }$ and minimum at $x=0$.