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16
WAVE MECHANICS OF
SIMPLE SYSTEMSBecause of the wave nature of the electron, solution of problems of atomic and
molecular structure requires solvingwave equations. Problems in atomic and molec-
ular structure involve electrons that are tied to a positive nucleus or group of nuclei
by electrostatic attraction. They arebound. Therefore we shall be concerned with
mathematicalboundary conditionsimposed on the solutions of wave equations.16.1 WAVE MOTION
Any mathematical description of a vibrating guitar string has to take into account
the fact that it is tied down at both ends. This puts severe restrictions on the wave
forms the string can take which are not characteristic of a free wave. A free wave
can assume any wavelength, but a bound wave is restricted to waves that come to
zero at either end of the string. The three boundwavelengthsshowninFig.16.1are
allowed by these two boundary conditions, but an infinite number of intermediate
wavelengths are not allowed. The wave of longest wavelength is thefundamental
(one observeshalfa sine wave), and the others are calledovertones. A fundamental
and its overtones are well described by the sine function.
The wavelengthλof the first overtone is one-half the wavelength of the funda-
mentalλ=^12 λfundamental, the next overtone hasλ=^13 λfundamental, and we can imagine
many overtones atλ=^1 nλfundamental. This is the source ofintegers nthat appear for
bound waves. De Broglie’s observation that particles (specifically electrons) have aConcise Physical Chemistry,by Donald W. Rogers
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