Chapt. 39 Photons and Matter Waves 293
for the stationary states
ψ
.
b) For
V
= 0 (i.e., the free particle system), verify that
ψ
=
e
ikx
are the unnormalized stationary states, where
k
=
±
r
2
mE
h
−
2
.
Why can’t these states be normalized?
c) The in±nite square well has
V
=
b
0
,
for 0
≤
x
≤
a
;
∞
,
for
x <
0 and
x > a
.
Verify that
ψ
=
R
2
a
sin(
kx
)
are the normalized solutions with boundary conditions
ψ
(0) =
ψ
(
a
) = 0. Also show that
k
=
nπ
a
and
E
=
h
−
2
2
m
p
π
a
P
2
n
2
.
039 qfull 02000 3 5 0 tough thinking: complete set of states
Extra keywords:
A good challenge question.
5. It has been mathematically shown that the set of stationary state wave functions for a system
constitute a
COMPLETE SET
: i.e., any function that obeys the same boundary conditions as
the stationary state wave functions can be expanded as a linear combination of those stationary
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 Fall '11
 Dr.Acost
 Physics, Photon, ai φi, state wave functions

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