Chapter 11, Problem 11.36P

(a)

To determine

The classical position of the oscillator, assuming it started from rest at the origin $\left(x_c\left(0\right)={\dot{x}}_c\left(0\right)=0\right)$.

(b)

To determine

Show that the solution to the time-dependent Schrodinger equation for this oscillator can be written as $\Psi \left(x,t\right)={\psi }_n\left(x-x_c\right)\exp \left(\frac{i}{\hslash }\left[-\left(n+\frac{1}{2}\right)\hslash \omega t+m{\dot{x}}_c\left(x-\frac{x_c}{2}\right)+\frac{m{\omega }^2}{2}{\displaystyle \underset{0}{\overset{t}{\int }}f\left(t\text{'}\right)x_c\left(t\text{'}\right)dt}\right]\right)$.

(c)

To determine

Show that the eigenfunctions and eigenvalues of $H\left(t\right)$ are ${\psi }_n\left(x,t\right)={\psi }_n\left(x-f\right)$ and $E_n\left(t\right)=\left(n+\frac{1}{2}\right)\hslash \omega -\frac{1}{2}m{\omega }^2{f}^2$.

(d)

To determine

Show that in the adiabatic approximation the classical position reduces to $x_c\left(t\right)\approx f\left(t\right)$.

(e)

To determine

Show that $\Psi \left(x,t\right)\approx {\psi }_n\left(x,t\right)e^{i{\theta }_n\left(t\right)}{e}^{i{\gamma }_n\left(t\right)}$ and check whether the dynamic phase has the correct form. And determine the geometric phase.