Physics 471 Problem Set 4 Fall 2006
16. A particle of mass min an infinite potential well of width ahas the initial wave function
Ψ(x, 0) = Asin3(πx/a).
(a) Normalize Ψ(x, 0).
(b) Express Ψ(x, 0) in terms of the energy eigenstates
ψn(x) = s2
asin(nπx/a), n = 1,2,3,··· ,
as
Ψ(x, 0) = X
n
Cnψn(x).
Hint: There are only two terms in the expansion and they can be found by expressing
sin3(x) as a combination of sin(x) and sin(3x) using
e3ix = (cos(x) + isin(x))3.
(c) Obtain Ψ(x, t) and find hxias a function of time.
17. The wave function for a particle of mass mmoving in the potential
V=
∞for x= 0
0 for 0 < x ≤a
V0for x≥a
is
ψ(x) = (Asin(kx) for 0 < x ≤a
Ce−κx for x≥a
with
k=s2mE
¯h2κ=s2m(V0−E)
¯h2.
(a) Apply the boundary conditions at x=aand obtain the transcendental equation
which determines the bound state energies E.
(b) If
s2mV0a2
¯h2= 3π ,
determine the allowed bound state energies. Express your answers in the form of a
numerical factor multiplying the dimensional factor (¯h2/2ma2). (Note: The
FindRoot[ ] command in Mathematica is one way to locate solutions to the transcen-
dental equation.)
