Total Reflection of a Classical Particle off a Two-Dimensional Potential

[klg0218]

Homework Statement

This question is motivated by Problem 8, Ch 1 of Cohen-Tannoudji's "Quantum Mechanics". Suppose that you are given a two dimensional potential of the form V(x,y) = V if x>0 and V(x,y)=0 if x<0. A particle travels toward the the potential obliquely, i.e. neither parallel to the x-axis nor to the y-axis, and its kinetic energy is less than V. Describe the classical trajectory of the particle upon reflection.

Homework Equations

Presumably, conservation of the y-component of momentum and conservation of energy.

The Attempt at a Solution

The classical trajectory is described as a "lateral shift upon reflection." I assume that the answer will somehow depend on the angle \theta between the trajectory and the normal to the surface. Thanks!

 

[mark726]

Hello there,

Thank you for bringing this problem to our attention. I can offer you some insights into this problem.

First, let's consider the classical trajectory of a particle traveling towards the potential obliquely. As you mentioned, the conservation of the y-component of momentum and conservation of energy must be taken into account. This means that the particle's y-component of momentum and energy remain constant throughout its motion.

Now, when the particle reaches the potential barrier, it will experience a change in direction due to the potential. The angle of reflection will depend on the angle of incidence and the potential barrier's shape. In this case, the potential barrier is a step function, so the angle of reflection will depend on the angle of incidence and the height of the potential barrier.

If the particle's kinetic energy is less than the potential barrier, it will be reflected back in the opposite direction. However, due to the oblique angle of incidence, the reflected particle's trajectory will be shifted laterally, as you mentioned. The amount of lateral shift will depend on the angle of incidence and the height of the potential barrier.

I hope this helps you understand the classical trajectory of the particle upon reflection in this scenario. Let me know if you have any further questions or need clarification.