BioPHYSICAL chemistry

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 mathematic-
ally 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 expec-
tation, value. Every physical observable phas an associated operator
(eqn 9.17) and the average, or expectation, value of the observable is
given by:

(9.52)

where Vis 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 xcomponent 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 probabil-
ity 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

pri

x

x =−() ()r

⎛

⎝

⎜⎜

⎞

⎠

∫ψ ⎟⎟

∂

∂

*dZ ψτ

xxxxx^22 =∫ψψ*d() ()

rrrr=∫ψψτ*d() ()

prr=∫ψψτ*d() ()V

1

0

= () ()

∞

∫ψψ τ*drr

CHAPTER 9 QUANTUM THEORY 189

Wavefunction

Probability

density

Figure 9.7The sign

of a wavefunction

may be either

positive or negative,

but the probability is

always zero or

positive.