Physics 471 Problem Set 7 Fall 2003
23. Evaluate the following integrals
(a)
∞
−∞
dx (x2−4x+1)δ(x2−4) .
(b)
π
0
dx sin3(x)δcos(x)−1
2.
24. 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
AIIe−ikx +BII eikx for x>0.
(a) Apply the boundary conditions at x= 0 and construct the Smatrix which relates
(BI,B
II)to(AI,A
II).
(b) Show that Sis unitary, i.e. that SS†=1.
(c) Show that det Shas the form eiα.
25. 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?
26. Griffiths Problem 3.9.
