PHY 6645 Fall 2002 – Homework 2
Due by 5 p.m. on Friday, September 13. Partial credit will be available for
solutions submitted by 5 p.m. on Monday, September 16.
Answer all questions. To gain maximum credit you should explain your reasoning and show
all working. Please write neatly and include your name on the front page of your answers.
1. A useful and important property of Hermitian operators is the following: If Ω and Λ
are two commuting Hermitian operators, there exists a basis of common eigenvectors
that diagonalizes both operators.
Read Shankar pp. 43–46, then do Sakurai Problem 1.23, which is reproduced below:
Consider a three-dimensional ket space. If a certain set of orthonormal kets—say, |1i,
|2i, and |3i—are used as the base kets, the operators A and B are represented by


a
0
0
0
A=
 0 −a

0



,
0 

−a


b
0
0



B=
 0 0 −ib  ,


0 ib 0
with a and b both real.
(a) Obviously A exhibits a degenerate spectrum. Does B also exhibit a degenerate
spectrum?
(b) Show that A and B commute.
(c) Find a new set of orthonormal kets which are simultaneous eigenkets of both A
and B. Specify the eigenvalues of A and B for each of the three eigenkets. Does
your specification of eigenvalues completely characterize each eigenket?
2. Read Shankar Section 1.9, then do the following problem:
Simplify eaΛ Ω e−aΛ (where Ω and Λ are linear operators, and a is a complex scalar) for
the following special cases:
(a) [Λ, Ω] = h̄,
(b) [Λ, Ω] = h̄Λ,
(c) [Λ, Ω] = h̄Ω,
(d) [Λ, Ω] = 0.
3. Adapted from Ballentine Problem 1.10: Consider the Hilbert space H consisting of all
complex-valued functions f (x) such that
Z
∞
−∞
dx|f (x)|2 < ∞,
and the reduced Hilbert space HR containing those elements of H that vanish faster
∗
than any power of |x| for |x| → ∞. Let H∗ and HR
be the duals of these two vector
spaces.
∗
State which (if any) of the four vector spaces (H, H∗ , HR , and HR
) each of the
following functions belongs
to: (a) cos ax; (b) (cos x)/x; (c) cosh ax; (d) (cosh ax)/x;
√
(e) exp(−a2 x2 ); (f) 1/ a2 + x2 ; (g) arctanh ax. You may assume that a is a real scalar.
4. The Dirac delta function δ(x) is defined in terms of the limit α → 0 of a set of
functions δα (x) that have the following properties when integrated with any “wellbehaved” function f (x):
Z
∞
−∞
Z
−
−∞
Z ∞
dx f (x) δ(x) ≡ lim
Z
∞
α→0 −∞
dx f (x) δ(x) ≡ lim
Z
−
α→0 −∞
Z ∞
dx f (x) δ(x) ≡
lim
dx f (x) δα (x) = f (0),
dx f (x) δα (x) = 0,
dx f (x) δα (x) = 0,
α→0 for any > 0. We can loosely write
δ(x) = lim δα (x),
α→0
provided it is realized that this definition may only be used when integrated against a
well-behaved function.
Consider the following sets of functions that give rise to representations of the delta
function (α is assumed to be a positive real):
(a) The “top hat” function
δα (x) =


 1/(2α),
|x| ≤ α,

 0
|x| > α.
(b) The Gaussian
δα (x) = √
1
exp(−x2 /α2 ).
πα
(c) The Lorentzian
δα (x) =
1
α
.
π x2 + α 2
δα (x) =
1
πx
sin
.
πx
α
(d) The sinc function
For each case, examine limα→0 δα (x) for x 6= 0, and explain any apparent inconsistency
with the standard notion that “δ(x) = 0” for any x 6= 0.