Chapter 11, Problem 11.35P

(a)

To determine

To check that the equation ${\Phi }_n\left(x,t\right)=\sqrt{\frac{2}{\omega }}\sin \left(\frac{n\pi }{\omega }x\right)e^{i\left(mv{x}^2-2E_n^{i}at\right)/2\hslash \omega }$ satisfies the time-independent Schrodinger equation, with the appropriate boundary conditions.

(b)

To determine

To show that expansion coefficients can be written in the form $c_n=\frac{2}{\pi }{\displaystyle \underset{0}{\overset{\pi }{\int }}{e}^{-ia{z}^2}\sin \left(nz\right)\sin \left(z\right)dz}$ .

(c)

To determine

The value of $T_e$ , $T_i$ , $c_n$ and the expression for $\Psi \left(x,t\right)$ .

(d)

To determine

To show that the phase factor in $\Psi \left(x,t\right)$ can be written in the form $\theta \left(t\right)=-\frac{1}{\hslash }{\displaystyle \underset{0}{\overset{t}{\int }}{E}_1\left(t^{\prime }\right)d{t}^{\prime }}$ .