Step-Down Potential

A particle of mass m obeys a Schrodinger equation with a potential

(1)

Since V₀ > 0, the potential is higher on the left of zero than on the right. Find the reflection coefficient R(p) for a particle coming in from the left with momentum p (see Figure 1.1).

Figure 1.1

SOLUTION

Denote by p' the momentum of the particle to the right of the origin, and ±p is momentum on the left. Since energy is conserved, we have

(1)

(2)

Now we set up the most general form for the wave function, assuming the incoming wave has unit amplitude:

(3)

Matching the wave function and its derivative at the origin gives two equations for the unknowns R and T which are solved to find R:

(4)

(5)

(6)