# Wavelengths of Particle passing Potential Barriers

## Homework Statement

Compare the wavelengths of a particle when it passes a barrier of height (a) +V0 and (b) -V0 where E > |V0|. Calculate the momentum and kinetic energy for both cases.

(see below)

## The Attempt at a Solution

I know the wavelength changes in the area of potential inside the barriers, but my textbook doesn't really give me much else information. It just leads on to explain how when passing the negative potential region, reflection and transmission may occur, which isn't what I'm interested in. With regards to the positive potential region, it gives what we 'thought' happened classically, and says this is not the case. Not very helpful either.

My book refers to the regions outside of the barriers as regions I and III, and the region within the barriers as region II. So what I'm looking for is how the wavelength changes in region II.

Equation given for 'wave number' in region II is as follows:

$\kappa_{II} = \frac{\sqrt{2m(E - V_{0})}}{\hbar}$

And the 'wave function' in region II is:

$\psi_{II} = Ce^{\kappa x} + De^{-\kappa x}$

How am I supposed to tie these together with wavelength, momentum, and kinetic energy? Am I even supposed to use these equations? I'm so confused as to how I'm supposed to approach this.