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Physics 3220 – Quantum Mechanics 1 – Fall 2008
Problem Set #3
Due Wednesday, September 10 at 2pm
Problem 3.1
: Practice with complex numbers. (20 points)
Every complex number
z
can be written in the form
z
=
x
+
iy
where
x
and
y
are real; we
call
x
the
real part
of
z
, written
x
= Re
z
, and likewise
y
is the
imaginary part
of
z
,
y
= Im
z
.
We further deFne the
complex conjugate
of
z
as
z
*
≡
x

iy
.
a) Prove the following relations that hold for any complex numbers
z
,
z
1
and
z
2
:
Re
z
=
1
2
(
z
+
z
*
)
,
(1)
Im
z
=
1
2
i
(
z

z
*
)
,
(2)
Re (
z
1
z
2
) = (Re
z
1
)(Re
z
2
)

(Im
z
1
)(Im
z
2
)
,
(3)
Im (
z
1
z
2
) = (Re
z
1
)(Im
z
2
) + (Im
z
1
)(Re
z
2
)
.
(4)
b) The modulussquared of
z
is deFned as

z

2
≡
z
*
z
. What is Im

z

2
, and what is Im
z
2
?
In doing quantum mechanics confusing
z
2
and

z

2
is very common; be careful!
c) Any complex number can also be written in the form
z
=
Ae
iθ
, where
A
and
θ
are real
and
θ
is usually taken to be in the range [0
,
2
π
);
A
and
θ
are called the
modulus
and the
phase
of
z
, respectively. Use Euler’s relation (which is provable using a Taylor expansion),
e
ix
= cos
x
+
i
sin
x,
(5)
to Fnd Re
z
, Im
z
,
z
*
and

z

in terms of
A
and
θ
.
d) Use the above relations on
e
i
(
α
+
β
)
=
e
iα
e
iβ
to derive trigonometric identities for sin(
α
+
β
)
and cos(
α
+
β
).
e) The secondorder di±erential equation,
d
2
dx
2
f
(
x
)=

k
2
f
(
x
)
,
(6)
has two linearly independent solutions. These can be written in more than one way, and two
convenient forms are
f
(
x
Ae
ikx
+
Be

ikx
,f
(
x
a
sin(
kx
)+
b
cos(
)
.
(7)
Verify that both are solutions of (6). Since both are equally good solutions, we must be able
determine
a
and
b
in terms of
A
and
B
; do so.
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This note was uploaded on 02/27/2012 for the course PHYSICS 3220 taught by Professor Stevepollock during the Fall '08 term at Colorado.
 Fall '08
 STEVEPOLLOCK
 mechanics

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