Before we can interpret the wavefunction, we need to take care of a small
piece of bookkeeping. Specifically, we need to know the value of the
normalization constant—the constant that guarantees that the wavefunction will remain within the bounds we set. In the example below we
illustrate this process.
Example 4.4 Normalizing the Wavefunction
The wavefunction described by Equation 4.13 is not normalized.
Use Equation 4.7 to normalize the wavefunction and determine the
normalization constant A n .
Solution A normalized wavefunction must obey
ð L
0
ψ x
ð Þψ x
ð Þdx = 1
n = 5
ψ 5 (x)
ψ 5
2 (x)
ψ 4
2 (x)
ψ 3
2 (x)
ψ 2
2 (x)
ψ 1
2 (x)
ψ 4 (x)
ψ 3 (x)
ψ 2 (x)
ψ 1 (x)
n = 4
n = 3
n = 2
n = 1
x
+
+
+
+
+
+
+
+
+
–
–
–
–
–
–
E 5 =
25h 2
8mL 2
E 4 = 2h 2
mL 2
E 3 =
9h 2
8mL 2
E 2 =
h 2
2mL 2
E 1 =
h 2
8mL 2
0
(a)
(b)
L
x
0
L
Nodal point
Figure 4.5 The wavefunctions (a) and the wavefunctions squared (b) for a particle confined to a line of length L.
These are plotted as a function of n. Also shown are the energy values. The nodal points represent regions of zero
probability.
CHAPTER 4: Quantum Effects at the Nanoscale
106