Consider the time-independent Schrödinger equation

$$\frac{d^{2} \psi}{d x^{2}}+\lambda^{2} q(x) \psi(x)=0$$

where $\lambda \gg 1$ denotes $\hbar^{-1}$ and $q(x)$ denotes $2 m[E-V(x)]$. Suppose that

and consider a bound state $\psi(x)$. Write down the possible Liouville-Green approximate solutions for $\psi(x)$ in each region, given that $\psi \rightarrow 0$ as $|x| \rightarrow \infty$.

Assume that $q(x)$ may be approximated by $q^{\prime}(a)(x-a)$ near $x=a$, where $q^{\prime}(a)>0$, and by $q^{\prime}(b)(x-b)$ near $x=b$, where $q^{\prime}(b)<0$. The Airy function $\operatorname{Ai}(z)$ satisfies

$$\frac{d^{2}(\mathrm{Ai})}{d z^{2}}-z(\mathrm{Ai})=0$$

and has the asymptotic expansions

$$\operatorname{Ai}(z) \sim \frac{1}{2} \pi^{-1 / 2} z^{-1 / 4} \exp \left(-\frac{2}{3} z^{3 / 2}\right) \quad \text { as } \quad z \rightarrow+\infty$$

and

$$\operatorname{Ai}(z) \sim \pi^{-1 / 2}|z|^{-1 / 4} \cos \left[\left(\frac{2}{3}|z|^{3 / 2}\right)-\frac{\pi}{4}\right] \quad \text { as } \quad z \rightarrow-\infty .$$

Deduce that the energies $E$ of bound states are given approximately by the WKB condition:

$$\lambda \int_{a}^{b} q^{1 / 2}(x) d x=\left(n+\frac{1}{2}\right) \pi \quad(n=0,1,2, \ldots)$$

$$\begin{aligned} & q(x)>0 \quad \text { for } \quad a<x<b \\ & \text { and } q(x)<0 \text { for }-\infty<x<a \text { and } b<x<\infty \end{aligned}$$