University of Colorado, Department of Physics
PHYS3220, Fall 09, HW#5
due Wed, Sep 23, 2PM at start of class
1. A few more properties of Hermitian operators (total: 10pts)
a) Show that the sum of two Hermitian operators is a Hermitian operator.
b) Suppose that
ˆ
A
is a Hermitian operator and
α
is a number. Under what condition
on
α
is
α
ˆ
A
a Hermitian operator?
2. A few useful properties of the Dirac delta function (total: 10pts)
a) By multipying both sides of the following equations by a differentiable function
f
(
x
),
and integrating over
x
, verify the following equations:
δ
(
x
)
=
δ
(

x
)
(1)
d
dx
δ
(
x
)
=

d
dx
δ
(

x
)
(2)
xδ
(
x
)
=
0
(3)
x
d
dx
δ
(
x
)
=

δ
(
x
)
(4)
b) Prove the following relations
Z
∞
∞
δ
(
a

x
)
δ
(
x

b
)
dx
=
δ
(
a

b
)
(5)
f
(
x
)
δ
(
x

a
)
=
f
(
a
)
δ
(
x

a
)
(6)
3. A few useful properties of Fourier transforms (total: 10pts)
a) Proove Parseval’s theorem
Z
∞
∞

f
(
x
)

2
dx
=
Z
∞
∞

g
(
k
)

2
dk,
(7)
for any regular function
f
(
x
) =
1
√
2
π
Z
∞
∞
g
(
k
) exp(
ikx
)
dx
with
g
(
k
) =
1
√
2
π
Z
∞
∞
f
(
x
) exp(

ikx
)
dx
b) The convolution of two functions
f
1
and
f
2
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 Fall '08
 STEVEPOLLOCK
 Physics, Fourier Series, Hilbert space, Dirac delta function, Hermitian, Hermitian Operator

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