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Physics 401  Homework #1
1) (3 points) Complex numbers are commonly expressed in two forms: cartesian (
iy
x
z
)
and polar (
). Convert
i
Ae
z
i
1
1
to both of these forms, and state explicitly the values of x,
y, A, and
2) For any complex number,
1
z
z
.
a) (3 points) Show that this is true using the cartesian form (
iy
x
z
).
b) (3 points) Show that this is true using the polar form (
).
i
Ae
z
Which method is easier?
3) (1 point each, 4 points total) Prove the following identities for complex numbers:
a) Re(z) = (z +z*)/2
b) Im(z) = (z  z*)/2i
c) cos(
) = [exp(i
) + exp(i
)]/2
d) sin(
) = [exp(i
)  exp(i
)]/2i
4) (1 point each, 4 points total)
a) What is the phase of the following wavefunction?:
)
(
)
,
(
t
kx
i
Ae
t
x
b) Show that the phase is shifted by
/2 when the function is multiplied by i
c) Show that the phase is shifted by
when the function is multiplied by 1.
d) Suppose we multiply the wavefunction by a phase factor
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 Fall '11
 HALL
 Work, Quantum Physics

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