© E. F. Schubert
1
Johannes Kepler (1571–1630)
Sir Isaac Newton (1642–1727)
Founder of celestial mechanics
Founder of modern mechanics
1
Classical mechanics and the advent of quantum mechanics
1.1 Newtonian mechanics
The principles of classical mechanics do not provide the correct description of physical
processes if very small length or energy scales are involved.
Classical
or
newtonian
mechanics
allows a
continuous
spectrum of energies and allows
continuous
spatial distribution of matter.
For example, coffee is distributed homogeneously within a cup. In contrast, quantum mechanical
distributions are not continuous but
discrete
with respect to energy, angular momentum, and
position. For example, the bound electrons of an atom have discrete energies and the spatial
distribution of the electrons has distinct maxima and minima, that is, they are not
homogeneously distributed.
Quantum-mechanics does not contradict newtonian mechanics. As will be seen, quantum-
mechanics merges with classical mechanics as the energies involved in a physical process
increase. In the classical limit, the results obtained with quantum mechanics are identical to the
results obtained with classical mechanics. This fact is known as the
correspondence principle
.
In classical or newtonian mechanics the instantaneous state of a particle with mass
m
is fully
described by the particle’s position [
x
(
t
),
y
(
t
),
z
(
t
)] and its
momentum
[
p
x
(
t
),
p
y
(
t
),
p
z
(
t
)]. For the
sake of simplicity, we consider a particle whose motion is restricted to the
x
-axis of a cartesian
coordinate system. The position and momentum of the particle are then described by
x
(
t
) and
p
(
t
) =
p
x
(
t
)
.
The
momentum
p
(
t
)
is
related
to
the
particle’s
velocity
v
(
t
)
by
p
(
t
) =
m v
(
t
) =
m
[d
x
(
t
) / d
t
] . It is desirable to know not only the instantaneous state-variables
x
(
t
) and
p
(
t
), but also their functional evolution with time. Newton’s first and second law enable
us to determine this functional dependence.
Newton’s first law
states that the momentum is a
constant, if there are
no forces
acting on the particle,
i. e.

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