Understanding WKB Approximation for E-V(x)

[curious.cat]

Consider E>V(x). WKB states the wavefunction will remain sinusoidal with a slow variation of wavelength $ \lambda $ and amplitude given that V(x) varies slowly. From the equation \begin{equation}
k(x)=\frac{\sqrt{2m(E-V(x))}}{\hbar}
\end{equation}, I can see that the k(x) is directly proportional to E-V(x), implying that $ \lambda $ is inversely proportional to E-V(x). I cannot understand why the amplitude should change with a variation in V(x). I do know that if E-V(x) becomes negative then the wavefunction becomes exponentially decaying in which case the wavefunction is no longer a periodic function (hence, amplitudes and wavelengths no longer apply). What is the equation connecting amplitude with V(x)? I am stuck and unable to proceed any further. Sorry about the equation numbering. I am rather new to LATEX

 

[DrDu]

k(x) is only the first term in an expansion of the wavefunction into orders of $\hbar$, the next order therm, called the eikonal, gives you the variation of the amplitude.
It should be obvious why the amplitude changes with k: $p=\hbar k$ is momentum and proportional to v. Already classically, the probability to find, say, a planet at a given point of its orbit when averaging over time is inversely proportional to v. But the amplitude of the wavefunction is the square root of this probability. Everything you ever want to know about the WKB approximation can be found in the review by Berry and Mount: http://iopscience.iop.org/article/1...678CE73A943EED846ECCF5898519449.ip-10-40-1-74

 

[curious.cat]

Thank you!