the wavefunctions, verify this probability by considering the area
under the plot of ψ(x)
2 versus x between x = 0 and x = L/2.
Solution This probability is given by the square of the normalized
wavefunction integrated between 0 and L/2.
ð
L=2
0
ψ∗ x
ð Þψ x
ð Þ =
2
L
ð
L=2
0
sin
2 nπx
L
dx
Using the trigonometric identity sin
2 x =
1
2
= (1 − cos 2x) we can
write the above expression as
2
L
ð
L=2
0
1
2
1 − cos 2
nπx
L
dx =
2
L
ð
L=2
0
1
2
−
1
2
cos 2
nπx
L
dx
Integrating gives
2
L
1
2
x −
1
2
L
2nπ
sin 2
nπx
L
L=2
0
Subtracting the integration limits and realizing that sin(nπx) is zero
gives
2
L
L
4
−
L
4nπ
sin nπ − 0 −
L
4nπ
sin 0
ð Þ
=
2
L
Â
L
4
=
1
2
Figure 4.6 shows a plot of ψ(x)
2 versus x between x = 0 and x = L for
two representative states (n = 1 and n = 2). The total area under
these curves represents the probability of finding the electron
between x = 0 and x = L. For normalized wavefunctions this is equal
to one. Also indicated by the shaded region in the figure is the area
under the curve between x = 0 and x = L/2. In both cases, we see
that this is exactly 50% (or 0.5). This result is consistent with the
calculation in Example 4.5, which shows that the probability of
finding the electron between x = 0 and x = L/2 is always 50%,
regardless of the value of n.
There are a few important features in Figure 4.5 that are in stark contrast
to the classical picture. The energy is quantized (it increases with n) as
opposed to a continuously varying function. Also, the probability of finding the electron at x = 0 and x = L is zero for all values of n. For instance, in
CONFINEMENT OF ELECTRONS IN BOXES 109
under the plot of ψ(x)
2 versus x between x = 0 and x = L/2.
Solution This probability is given by the square of the normalized
wavefunction integrated between 0 and L/2.
ð
L=2
0
ψ∗ x
ð Þψ x
ð Þ =
2
L
ð
L=2
0
sin
2 nπx
L
dx
Using the trigonometric identity sin
2 x =
1
2
= (1 − cos 2x) we can
write the above expression as
2
L
ð
L=2
0
1
2
1 − cos 2
nπx
L
dx =
2
L
ð
L=2
0
1
2
−
1
2
cos 2
nπx
L
dx
Integrating gives
2
L
1
2
x −
1
2
L
2nπ
sin 2
nπx
L
L=2
0
Subtracting the integration limits and realizing that sin(nπx) is zero
gives
2
L
L
4
−
L
4nπ
sin nπ − 0 −
L
4nπ
sin 0
ð Þ
=
2
L
Â
L
4
=
1
2
Figure 4.6 shows a plot of ψ(x)
2 versus x between x = 0 and x = L for
two representative states (n = 1 and n = 2). The total area under
these curves represents the probability of finding the electron
between x = 0 and x = L. For normalized wavefunctions this is equal
to one. Also indicated by the shaded region in the figure is the area
under the curve between x = 0 and x = L/2. In both cases, we see
that this is exactly 50% (or 0.5). This result is consistent with the
calculation in Example 4.5, which shows that the probability of
finding the electron between x = 0 and x = L/2 is always 50%,
regardless of the value of n.
There are a few important features in Figure 4.5 that are in stark contrast
to the classical picture. The energy is quantized (it increases with n) as
opposed to a continuously varying function. Also, the probability of finding the electron at x = 0 and x = L is zero for all values of n. For instance, in
CONFINEMENT OF ELECTRONS IN BOXES 109