Physics 486 Homework 3 Due February 11, 2005
1) This is Griffiths, problem 2.29 (page 82) [10 points]
Analyze the odd bound state wave functions for the finite square well. Derive the
transcendental equation for the allowed energies, and solve it graphically. Examine the two
limiting cases (Wide, deep well and shallow, narrow well, see p. 80). Is there always an odd
bound state.
2) This is Griffiths, problem 2.41 (page 86) [10 points]
A particle of mass, m, in the potential, V(x) = ½m
ω
2x2, starts out (at t = 0) in the state
!(x,0) =A1"2m
#
!
x
$
%
&'
(
)
2
e"m
#
2!
x2
, for some constant, A.
a: What is the expectation value, <E>,of the energy?
b: At some later time, T, the wave function is
!(x,T)=B1+2m
"
!
x
#
$
%&
'
(
2
e)m
"
2!
x2
, for some constant, B.
What is the smallest possible value of T?
3) This is Griffiths, problem 2.42 (page 86) [5 points]
Find the allowed energies of the half harmonic oscillator
V(x)=
1
2m
!
2x2, for x>0,
"!!!!!, for x<0.
#
$
%
(This represents, for example, a spring that can be stretched, but not compressed.)
Hint: This requires careful thought, but little actual computation.
