De Broglie wavelength in non-constant potential?

[greypilgrim]

Hi.

Does the de Broglie wavelength have any significance for the wavefunctions of particles in a potential that is non-constant in no region of space? As far as I can see, the solutions of the time-independent Schrödinger equation are only sinusoidal if $E>V=const$.

This is enough to derive diffraction and the double-slit-behaviour of electrons and even the energy levels of a particle in a box "the old q.m. way", i.e. without the Schrödinger equation. But is it relevant in a wider context?

 

[jtbell]

If a particle's potential energy varies with position, then so does its kinetic energy, in order to keep the total energy constant. If the kinetic energy varies with position, then so do the momentum and the wavelength. You can see this in the energy eigenstates for the quantum harmonic oscillator: the wavelength is longer at locations further from the center.

(from Wikipedia)