Physics 471 Problem Set 7 Fall 2006
28. In the attractive δ-function potential problem, let mα/¯h2= 1/a, which means that the
bound state eigenfunction and energy are
ψ(x) = 1
√ae−|x|/a, E =−¯h2
2ma2.
If a particle moving in this potential is in the initial state
Ψ(x, 0) = (A(a2−x2) for −a≤x≤a
0 for |x|> a
what is the probability that a measurement of the particle’s energy will give the result
different from −¯h2/2ma2?
29. For the repulsive δ-function potential V(x) = αδ(x), with α > 0, consider a wave function
of the form
ψ(x) = (AIeikx +BIe−ikx for x < 0
AII e−ikx +BI I eikx for x > 0.
(a) Apply the boundary conditions at x= 0 and construct a matrix Swhich relates
(BI, BII ) to (AI, AI I ) using the parameter β=mα/k¯h2, i.e.
ÃBI
BII !=ÃS11 S12
S21 S22 !Ã AI
AII !,
where the Sij depend on β.
(b) Show that Sis unitary, i.e. that SS†= 1.
(c) Show that det Shas the form eiφ.
30. A particle of mass mmoves in the double δ-function potential
V(x) = −¯h2λ
2ma (δ(x−a) + δ(x+a)) ,
where aand λare positive constants. Because V(x) = V(−x), the wave functions can be
chosen to be even or odd functions of x.
(a) Find the transcendental equation that determines the bound state energy in the case
when the wave function is even, i.e. ψ(−x) = ψ(x). Hint: Analyze the solutions with
E < 0 in the regions x < −a,−a<x<aand x > a. Use the finiteness at x→ ± ∞
and the symmetry under x→ −xto reduce the number of coefficients to two, and
then apply the boundary conditions at x=a. Is there always a bound state?
(b) Find the corresponding transcendental equation for the case when the solution satis-
fies ψ(−x) = −ψ(x). Is there always a bound state in this case?
