Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

WAVE EQUATIONS 249

1

1

1

1

sin(x)

sin(2x)

sin(3x)

0 x

FIGURE 16.1 Graph of sin(x), sin(2x), and sin(3x) shown over the interval [0,π]. The

fundamental mode of oscillation of a vibrating string is^12 of a sine wave, the first overtone is

a full sine wave from 0 to 2π, and the second overtone is^32 ofasinewave.

wave nature leads toquantum numbers nin atomic spectra (Bohr) and to the con-

nection between wave equations and atomic structure (Schrodinger, Hartree). Born’s ̈

observation that the wave equation governs the probability of finding an electron,

which may contribute to or oppose formation of a chemical bond, leads to the con-

nection between wave equations and molecular structure, energy, and reactivity that

is the basis of modernquantum chemistry.

16.2 WAVE EQUATIONS

The function sinφ(x) describes the excursion away fromφ(x)=0ofaninfinite

number of points along the variablex=0tol(Fig. 16.1). For any selected harmonic,

the fractionx/λtells us “where we are” on the sine wave. Ifx=λ, we are at the very

end of the sine function. Ifx=λ/2, we are half way. In order to fully describe a wave,

we need one more piece of information beyond the sine function and wavelengthλ.

The height of the wave is essential; a big sine wave has a largeamplitudeof oscillation

Aand a small sine wave has a small amplitude. The complete description of the wave

isφ(x)=Asin

2 πx

λ

.

We can obtain the second derivative ofφ(x)=Asin

2 πx

λ

:

d^2 φ(x)

dx^2

=−A

4 π^2

λ^2

sin

2 πx

λ

=−

4 π^2

λ^2

Asinφ(x)=−

4 π^2

λ^2

φ(x)

This is typical of the wave equations that Schrodinger used: ̈

d^2 φ(x)

dx^2

=kφ(x)