2 The wavefunction may be a complex function but the probability is
always real, since:
ψ(x) = A(x) + iB(x)
ψ*(x) = A(x) − iB(x)
(9.50)
so ψ*(x)ψ(x) = [A(x) + iB(x)][A(x) − iB(x)] = A
2
(x) + B
2
(x) ≥ 0
3 The probability is always a positive and real number (Figure 9.7).
The total probability of finding an object anywhere in space must
be equal to one. The sum of all of the probabilities is mathematically written as the integral of the probability. Therefore, the integral
of the probability over all space must be equal to one:
(9.51)
4 Particles do not have a specific position or momentum but rather
there is a distribution of values that reflect the distribution of the
particle. The physically relevant quantity is the average, or expectation, value. Every physical observable p has an associated operator
(eqn 9.17) and the average, or expectation, value of the observable is
given by:
(9.52)
where V is the operator.
For example, the average values of the position and momentum of
a particle can be calculated by substituting the operators for position r
and x component of the momentum:
(9.53)
(9.54)
5 To be physically reasonable, some restrictions can be placed on the
allowed values of the wavefunction. Since the probability must be less
than one everywhere, the wavefunction must be finite. The probability must have a single, unique solution at every position (Figure 9.8).
Solutions of Schrödinger’s equation that yield more than one solution at
a certain position are not allowed. When comparing the solutions of the
wavefunction at two close locations, there cannot be any discontinuities
p
r i
x
r
x
( )
( )
=
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
∫ ψ
∂
∂
ψ
τ
*
d
Z
x
xx x x
2
2
( ) ( )
= ∫ ψ
ψ
*
d
r
rr r
( ) ( )
= ∫ ψ ψ τ
*
d
p
r
r
( ) ( )
= ∫ ψ ψ τ
*
d
V
1
0
( ) ( )
=
∞
∫ ψ ψ τ
*
d
r r
CHAPTER 9
QUANTUM THEORY
189
Wavefunction
Probability
density
Figure 9.7 The sign
of a wavefunction
may be either
positive or negative,
but the probability is
always zero or
positive.
9781405124362_4_009.qxd 4/29/08 10:42 Page 189