Concise Physical Chemistry

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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)
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