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Date: 25-8-2016
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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 (JR, JT) and the sum JR + JT, 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.
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