46
Problems and Solutions in Quantum Mechanics
(b) Compute the probability current density and demonstrate that it is everywhere continuous. (c) Consider the case of attractive coupling (g < 0) and solve the bound-state problem for E < 0. Find the bound-state wave function and compute the value of the bound-state energy. (d) Show that the energy of the uniquely existing bound state corresponds to a pole of the coefficient F, previously calculated in (a).
Solution (a) Starting from the usual type of ansatz, ikx e + Be−ikx , ψ(x) = Ceikx ,
x <0 x >0
we obtain 1+ B =C from the continuity of the wave function at x = 0. The derivative of the wave function is discontinuous, with a finite jump given by −
h¯ 2 ψ (0+) − ψ (0−) + gψ(0) = 0 2m
This can be obtained by integrating Schroedinger’s equation in an infinitesimal domain around the discontinuity point. This relation translates to C + B − 1 = −i
2mg C k¯h 2
Introducing g˜ ≡
2mg h¯ 2
we obtain C= The wave function is
ψ E (x) = (x) e
2k , 2k + i g˜
ikx
−
i g˜ 2k + i g˜
B= e
−ikx
−i g˜ 2k + i g˜
+ (−x)
It can be written in the compact form ψ E (x) = eikx + F(k)eik|x|
2k 2k + i g˜
eikx