52 Wave–particle duality and its manifestation
From the last inequality for the minimal energy of the electron we obtain
E
min
=
h
2
2m
e
d
2
, (2.135)
which gives E
min
≈ 2.6 × 10
−20
J ≈ 0.16 eV, i.e., an electron cannot have energy
smaller than E
min
. Because E
min
is finite, the electron cannot be in the state of
rest that corresponds to E = 0 and p = 0.
2.5 The world of the nanoscale and the wavefunction
The study of processes and phenomena involving microscopic particles, where
the wave properties are taken into account, is carried out in the framework of the
so-called wave mechanics, which is also frequently called quantum mechanics.
For the description of a particle’s motion well-known classical quantities such as
the position vector r, momentum p, energy E, etc. are used. The state of a particle
at a given instant is defined by a number of physical quantities, i.e., by a set of
variables. The significant peculiarity of the quantum-mechanical description is
that this set includes a smaller number of variables than is necessary for a classical
description.
The quantum-mechanical description uses three variables. These variables
can be either the three projections of r(t) or the three projections of p(t ), but
not both. In classical physics all six variables, r(t) and p(t), are required for
the description of a particle’s state. Usually the coordinates r(t ) are chosen for
the quantum-mechanical description as the total set of variables. Then, such a
description of a particle’s state is called the coordinate representation.
According to the uncertainty principle (2.118), in quantum mechanics it is
impossible to measure simultaneously a particle’s coordinate and its momentum.
From the physical point of view this is equivalent to the absence of a trajectory
for the particle. Indeed, a trajectory is a series of sequential close positions of
a particle, which are defined by the position vectors at close instants in time:
r(t
0
), r(t
1
), r(t
2
), . . .. To define the position of a particle at an instant in time t
k
it is necessary to know its position at the previous instant in time, t
k−1
, and its
momentum at that time, p(t
k−1
), i.e.,
r(t
k
) = r(t
k−1
) + (t
k
− t
k−1
)
p(t
k−1
)
m
. (2.136)
The impossibility of simultaneous definition of position and momentum leads to
the impossibility of indicating the exact position of the particle at any given time.
Therefore, the motion of a particle cannot be described by its position vector r
as a function of time. For any microscopic process it makes sense to talk only
about the probability of finding a certain value of a physical quantity, i.e., the
quantum-mechanical description is a probabilistic description.
Another feature of the description on a microscopic scale lies in the fact that
some values of physical quantities cannot be measured because these values are