The stationary Schrödinger equation in one dimension has the form

$$\epsilon^{2} \frac{d^{2} \psi}{d x^{2}}=-(E-V(x)) \psi$$

where $\epsilon$ can be assumed to be small. Using the Liouville-Green method, show that two approximate solutions in a region where $V(x)<E$ are

$$\psi(x) \sim \frac{1}{(E-V(x))^{1 / 4}} \exp \left\{\pm \frac{i}{\epsilon} \int_{c}^{x}\left(E-V\left(x^{\prime}\right)\right)^{1 / 2} d x^{\prime}\right\}$$

where $c$ is suitably chosen.

Without deriving connection formulae in detail, describe how one obtains the condition

$$\frac{1}{\epsilon} \int_{a}^{b}\left(E-V\left(x^{\prime}\right)\right)^{1 / 2} d x^{\prime}=\left(n+\frac{1}{2}\right) \pi$$

for the approximate energies $E$ of bound states in a smooth potential well. State the appropriate values of $a, b$ and $n$.

Estimate the range of $n$ for which $(*)$ gives a good approximation to the true bound state energies in the cases

(i) $V(x)=|x|$,

(ii) $V(x)=x^{2}+\lambda x^{6}$ with $\lambda$ small and positive,

(iii) $V(x)=x^{2}-\lambda x^{6}$ with $\lambda$ small and positive.