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\begin{document}

WKB=(Wentzel,Kramers,Brillouin, 1926) Approximation = classical approximation.

A semi-classical method for solving 1-d schrodinger approximation.

valid when de Broglie wavelength changes slowly: \( \frac{\delta \lambda }{\lambda }=\frac{d\lambda }{dx}<<1 \). \( \frac{d\lambda }{dx}=\frac{mh}{p^{3}}\frac{dV}{dx} \) small when h-\(>\)0,
m-\(>\)0, p-\(>\)\( \infty  \)or when \( \frac{dV}{dx} \)-\(>\)0.

WKB doesn't work at the classical turning points.

Formal expansion:

Schrodinger equation:\( -\frac{\hbar ^{2}}{2m}\frac{d^{2}\psi }{dx^{2}}+V\psi =E\psi  \) so \( \psi ''+k^{2}(x)\psi (x)=0 \) for \( k^{2}(x)=\frac{2m}{\hbar ^{2}}(E-V(x)) \).

V= constant -\(>\) k=constant, \( \psi =e^{ikx} \).

if V not a constant, try \( \psi =Ae^{iS(x)/\hbar } \) This gives the equation: \( -i\hbar S''+S'^{2}=\hbar ^{2}k^{2}(x)=p^{2}(x)=2m(E-V(x)) \).

The expansion as \( \hbar \rightarrow 0 \) is \( S(x)=\sum _{i}\hbar ^{i}S_{i}(x)/i! \).

plugging back into the equation, you get: \( (S_{0}')^{2}-p^{2}+2\hbar (S_{0}'S_{1}'-\frac{i}{2}S_{0}'')+O(\hbar ^{2})=0 \).

each coefficient must seperately go to 0 so, 

\begin{itemize}

\item \( (S_{0}')^{2}=\hbar ^{2}k^{2}(x) \) which means \( S_{0}(x)=\pm \hbar \int k(x)dx \)

\item \( S_{0}'S_{1}'-\frac{i}{2}S_{0}''=0 \) which means \( S_{1}(x)=\frac{i}{2}ln(S_{0}')+C=\frac{i}{2}ln(p(x)) \)

\item neglect higher order terms

\end{itemize}

Then \( \psi (x)\sim Ce^{i(\pm \int k(x)dx+\frac{i}{2}ln(p(x))} \) \( =\frac{C}{\sqrt{p(x)}}e^{\pm i\int k(x)dx} \)

works in both classically allowed and forbidden regions as long as
\(|\)E-V\(|\)\(>\)\(>\)0

In forbidden regions: \( \psi (x)\sim  \) \( =\frac{C}{\sqrt{\hbar K(x)}}e^{\pm \int K(x)dx} \)

Example: high energy scattering off of attractive potential

There are applications to 

\begin{itemize}

\item bound states

\item tunnelling

\end{itemize}

\end{document}
