Download Physics 5002 (Spring 2017) Discussion Problem (4/20) Consider

Physics 5002 (Spring 2017) Discussion Problem (4/20)
Consider one-dimensional scattering of a particle of mass m by the potential barrier
(
V (x) =
V0 , 0 < x < L,
0, elsewhere.
(1)
The energy of the particle is E = h̄2 k 2 /(2m) > V0 . Take the incident wave to be exp(ikx) in
the region of x < 0.
1. Derive the amplitude A of the reflected wave A exp(−ikx) in the region of x < 0.
2. Derive the amplitude D of the transmitted wave D exp[ik(x − L)] in the region of x > L.
3. Show that |A|2 + |D|2 = 1.
4. Show that |A|2 = 0 when the imaginary part of D vanishes.
Due to the geometric differences, the Optical Theorem for three-dimensional scattering does
not have a simple counterpart in one dimension. The results in parts 3 and 4 are close to such
a counterpart.