Question

Probably still from our high-school days, we all remember the rules of reflection and refraction of light at a plane interface, say between vacuum and a medium with refractive index n:

α = α ′, sin α/ sin β = n. (3)

Here we denoted the incident angle as α, the reflection angle α ′ and the refraction angle β. It is interesting to see if we have a similar situation for material quantum-mechanical waves. For this purpose, let us consider a particle of mass m moving in three-dimensional space with the potential energy step

U (x, y, z) =         0, x < 0,

U0, x > 0. (4)

The yz-plane is thus the interface separating two half-spaces. U0 can be either positive or negative. Consider a plane wave corresponding to the particle with energy E > U0 incident from x = −∞ towards the separating interface at angle α with the normal to the interface, say in the xy-plane.

(a) Establish if reflection and refraction of the incident wave take place according to the “optical rules” (3).

Answer 1