Interpreting "momentum" in WKB approximation

[deep838]

According to WKB approximation, the wave function $$\psi (x) \propto \frac{1}{\sqrt{p(x)}}$$
This implies that the probability of finding a particle in between x and x+dx is inversely proportional to the momentum of the particle in the given potential.

According to the book, R. Shankar, this is "familiar" to us since higher momentum corresponds to higher velocities and consequently, higher velocity implies lesser chances of finding the particle at some position.

However, I'm confused regarding the fact that the momentum, being inversely proportional to the de Broglie wavelength implies that the probability is proportional to λ. This seems unreasonable, since higher wavelengths imply the particle is more "spread out in space" corresponding to less chances of finding it.

This is an honest doubt I've had ever since I came across the de Broglie relation and it just keeps building up, which leads me to question the physical meaning of the term p(x). Any insights will be very much appreciated.

 

[Vanadium 50]

If you see a stop sign, you have two options: a) stop, or b) go as fast as you can through the intersection, minimizing the time you spend there, right?

If you are traveling with velocity v, the time you spend between x and x + dx is inversely proportional to your velocity, and thus momentum. WKB is simply using this classical fact.

 

[deep838]

Yeah I understand the interpretation regarding velocities... What's bothering me is the one regarding wavelengths.

 

[DrDu]

WKB is a high momentum aoproximation. So you are assuming that the wavelength is short compared to the legth over which you are averaging to get the mean probability to find the particle.