Step-Up Potential

Consider a particle scattering in one dimension from a potential V(x) which is a simple step at x = 0:

(i)

Figure 1.1

where V₀ > 0. A particle with kinetic energy E > V₀ is incident from the left (see Figure 1.1).

a) Find the intensity of the reflected (R) and transmitted (T) waves.

b) Find the currents (J_R, J_T) and the sum J_R + J_T, of the reflected and transmitted waves.

SOLUTION

Write the energy as E = h²k²/2m, where k is the wave vector on the left of zero. Since E > V₀, define a wave vector k' on the right as k'² = k² – 2mV₀/h².

a) The wave functions on the left and right of the origin are

(1)

where r and t are the amplitudes of the reflected and transmitted waves. Matching the wave function and its slope at x = 0 gives two equations:

(2)

(3)

These two equations are solved to obtain r and t:

(4)

(5)

(6)

(7)

b) The particle currents are the velocities times the intensities. The velocities are hk/m on the left and hk'/m on the right:

(8)

(9)

(10)

The last expression equals the current of the incoming particle.