Physical Chemistry , 1st ed.

The wavefunctionin the above example has not changed. It is still a sine

function. However, it is now multiplied by a constant so that the normaliza-

tion condition is satisfied. The normalization constant does not affect the

shape of the function. It only imposes a scaling factor on the amplitude—a

very convenient scaling factor, as we will find. For the remainder of this text,

all wavefunctions must be or will be normalized unless stated otherwise.

Example 10.8

The wavefunction  2 sin xis valid for the range x0 to 1. Verify

that an electron has a 100% probability of existing in this range, thus verify-

ing that this wavefunction is normalized.

Solution

Evaluate the expression

P 2 

1

0

sin^2 x dx

and show that it is identically equal to 1. This integral has a known solution,

and substituting that solution, we get

P (^2) 
2
x

4

1

sin (2 x)^10 

 (^2) 

1

2

 0 

0

2

 (^0) 
where the limits have been substituted into the expression for the integral.
Solving:
P 1
which verifies that the wavefunction is normalized. Thus, from the Born in-
terpretation, the probability of finding the particle in the range x0 to 1 is
exactly 100%.

10.7 The Schrödinger Equation

One of the most important ideas in quantum mechanics is the Schrödinger

equation, which deals with the most important observable: energy. A change

in the energy of an atomic or molecular system is usually one of the easiest

things to measure (usually by spectroscopic methods, as discussed in the pre-

vious chapter), so it is important that quantum mechanics be able to predict

energies.

In 1925 and 1926, Erwin Schrödinger (Figure 10.4) brought together many

of the ideas presented in Chapter 9 as well as in earlier sections of this chap-

ter, ideas like operators and wavefunctions. The Schrödinger equation is based

on the Hamiltonian function (section 9.2), since these equations naturally pro-

duce the total energy of the system:

EtotKV

where Krepresents the kinetic energy and Vis the potential energy. We will

start with a one-dimensional system. Kinetic energy, energy of motion, has a

10.7 The Schrödinger Equation 285

Figure 10.4 Erwin Schrödinger (1887–1961).

Schrödinger proposed an expression of quan-

tum mechanics that was different from but

equivalent to Heisenberg’s. His expression is use-

ful because it expresses the behavior of electrons

in terms of something we understand—waves.

The Schrödinger equation is the central equa-

tion of quantum mechanics.

CORBIS-Bettmann