Page 28 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

1.4 Particles and waves 19

theory, which requires that its rest mass be zero. The magnitude of the
momentum p for a particle with zero rest mass is related to the relativistic
energy E by p ˆ E=c, so that
E hí "ù
p ˆ ˆ ˆ
c c c
Since the velocity c equals ù=k, the momentum is related to the wave number
k for a photon by
p ˆ "k (1:33)
Einstein's postulate was later con®rmed experimentally by A. Compton (1924).
Noting that it had been fruitful to regard light as having a corpuscular nature,
L. de Broglie (1924) suggested that it might be useful to associate wave-like
behavior with the motion of a particle. He postulated that a particle with linear
momentum p be associated with a wave whose wavelength ë is given by
2ð h
ë ˆ ˆ (1:34)
k p
and that expressions (1.32) and (1.33) also apply to particles. The hypothesis of
wave properties for particles and the de Broglie relation (equation (1.34)) have
been con®rmed experimentally for electrons by G. P. Thomson (1927) and by
Davisson and Germer (1927), for neutrons by E. Fermi and L. Marshall (1947),
and by W. H. Zinn (1947), and for helium atoms and hydrogen molecules by I.
Estermann, R. Frisch, and O. Stern (1931).
The classical, non-relativistic energy E for a free particle, i.e., a particle in
the absence of an external force, is expressed as the sum of the kinetic and
potential energies and is given by
1 p 2
2
E ˆ mv ‡ V ˆ ‡ V (1:35)
2 2m
where m is the mass and v the velocity of the particle, the linear momentum p
is
p ˆ mv
and V is a constant potential energy. The force F acting on the particle is given
by
dV
F ˆÿ ˆ 0
dx
and vanishes because V is constant. In classical mechanics the choice of the
zero-level of the potential energy is arbitrary. Since the potential energy for the
free particle is a constant, we may, without loss of generality, take that constant
value to be zero, so that equation (1.35) becomes