Question

# Particles of mass m are incident from the positive x axis (moving to the left) onto...

Particles of mass m are incident from the positive x axis (moving to the left) onto a potential energy step at x=0. At the step the potential energy drops from the positive value U_0 for all x>0 to the value 0 for all x<0. The energy of the particles is greater than U_0.

A) Sketch the potential energy U(x) for this system.

B) How would the wavelength of a particle change in the x<0 region compared to the x>0 region? Explain your answer.

C) Write down the general form of the particle wavefunction for x<0 and also for x>0. Assume the amplitude of the incident wave is A. Clearly define all other constants used.

D) What are the boundary conditions on the particle wavefunction at x=0.

E) Use your particle wavefunction and boundary conditions to determine the values of the amplitude of the transmitted and reflected waves in terms of the energy E, the potential energy U_0, the amplitude of the incident wave A, and the mass of the particle m (as well as fundamental constants).

F) Sketch the incoming, reflected, and transmitted waves, being sure to pay attention to your answers above.