as simple variables rather than as vectors that are needed to describe
three-dimensional motion. The kinetic energy of the particle, KE, and the
total energy, E, are related to these parameters and the potential energy,
V, by:
E=KE+V
(9.1)
According to classical mechanics, the energy of the system is always con-
served. Also conserved is the linear momentum, p, which is equal to:
p=m 9 (9.2)
Since the linear momentum is proportional to the velocity, the kinetic
energy and total energy can be rewritten as:
(9.3)
Various possible interactions between particles, such as gravitation or
electrostatic interactions, are described by well-defined relationships
involving the parameters that describe the particles. For example, the
electrostatic force, F, between two charged particles, q 1 and q 2 , separated
by the distance, r 12 , is given by:
(9.4)
where εois a constant, the vacuum permittivity. According to classical
mechanics, once all of the interactions and the initial conditions have
been established, then the time evolution of the system can be predicted
for all times using the principle of the conservation of energy, which states
that the total energy of the system always remains the same, and using
the laws of mechanics, such as the relationship between force, F, mass,
m, and position, x:
(9.5)
where ais the acceleration of the particle and tis the time.
Fmam
x
t
==
d
d
2
2
F
qq
o r
=
1
4
12
12
πε^2
E
p
m
=+V
2
2
KE m
m
m
p
m
===
1
222
2
22 2
9
9
KE= m
1
2
92
176 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY
