Computational Chemistry

and because the product of mass and velocity is momentum, Eq.4.18can be written

pp¼h=l (4.19)

relating the momentum of a photon (in its particle aspect) to its wavelength (in its
wave aspect). If Eq.4.19can be generalized to any particle, then we have

p¼h=l '(4.20)

relating the momentum of a particle to its wavelength; this is the de Broglie
equation.
If a particle has wave properties it should be describable by somehow combining
the de Broglie equation and a classical wave equation. A highly developed nine-
teenth century mathematical theory of waves was at Schr€odinger’s disposal, and the
union of a classical wave equation with Eq.4.20was one of the ways that he derived
his wave equation. Actually, it is said that the Schr€odinger equation cannot actually
be derived, but is rather a postulate of quantum mechanics that can only be justified
by the fact that it works [ 15 ]; this fine philosophical point will not be pursued here.
Of his three approaches [ 15 ], Schr€odinger’s simplest is outlined here. A standing
wave (one with fixed ends like a vibrating string or a sound wave in a flute) whose
amplitude varies with time and with the distance from the ends is described by

d^2 fðxÞ

dx^2

¼"

4 p^2

l^2

fðxÞ (4.21)

f(x)¼amplitude of the wave,x¼distance from some chosen origin,l¼wave-
length
From Eq.4.20:

l¼h=mv (4.22)

l¼wavelength of particle of massmand velocityv
Identifying the wave with a particle and substituting forlfrom Eq.4.22into
Eq.4.21:

d^2 fðxÞ

dx^2

¼"

4 p^2 m^2 v^2

h^2

fðxÞ (4.23)

Since the total energy of the particle is the sum of its kinetic and potential
energies:

Ekin¼E"Epot¼E"V (4.24)

4.2 The Development of Quantum Mechanics. The Schr€odinger Equation 99