Chapter 9, Problem 9.17P

(a)

To determine

The WKB wave function in regions (i) $x>x_2$, (ii) $x_1<x<x_2$, and (iii) $0<x<x_1$.

(b)

To determine

Show that $\tan \theta =\pm 2e^{\varphi }$.

(c)

To determine

Show that the quantization condition becomes $\theta \approx \left(n+\frac{1}{2}\right)\pi \mp \frac{1}{2}{e}^{-\varphi }$ under given conditions.

(d)

To determine

Sketch the given potential and find $\theta$ and show that $E_n^{\pm }\approx \left(n+\frac{1}{2}\right)\hslash \omega \mp \frac{\hslash \omega }{2\pi }{e}^{-\varphi }$.

(e)

To determine

Show that particle starts out in the right well oscillates back and forth between the wells, with a period $\tau =\frac{2{\pi }^2}{\omega }{e}^{\varphi }$.

(f)

To determine

The expression for $\varphi$ for the specific potential in part (d) and show that $V\left(0\right)\gg E,\varphi \sim m\omega a^2/\hslash$.