2.5 The world of the nanoscale and the wavefunction 53
forbidden by the quantum-mechanical laws. The set of all values that a physical
quantity is allowed to have is called the spectrum. Depending on the character of
this physical quantity and the specific physical conditions, the spectrum can be
discrete, continuous,ormixed. A discrete spectrum consists of a set of individual
values of the physical quantity, between which intervals of forbidden values
occur. A continuous spectrum occurs if the physical quantity can have any values
from the given interval. Finding the spectra of physical quantities is one of the
main problems of quantum mechanics. Examples of finding the spectra of specific
systems will be considered later.
Summarizing the above, it follows that according to the quantum-mechanical
description we may talk only about the probability of a particle being at a given
point at a given instant – namely in an infinitesimal volume dV = dx dy dz,
which includes this point. Accordingly, the so-called wavefunction (r, t), which
depends on time and coordinates, can be introduced. With the help of the wave-
function the above-mentioned probability, P, is defined by the expression
dP =
∗
dV =||
2
dV. (2.137)
In general, the wavefunction can be a complex function, i.e., it consists of
real and imaginary parts. Therefore, the wavefunction itself does not have any
physical meaning, but the square of its modulus, ||
2
, which is defined as a
product of the wavefunction, , and its complex conjugate,
∗
, has. According
to Eq. (2.137), the physical quantity
ρ(r, t ) =
dP
dV
=||
2
(2.138)
defines the so-called probability density (the probability of finding the particle
per volume unit). The quantity ρ(r, t) can be experimentally observed, but the
function (r, t) cannot. From Eq. (2.137) we can find the finite probability of the
particle, described by the wavefunction (r, t), being within an arbitrary volume
V :
P =
V
ρ(r, t )dV =
V
||
2
dV. (2.139)
Since the probability of finding a particle somewhere must equal unity, its wave-
function, , is chosen in such a way that it satisfies the so-called normalization
condition:
V
||
2
dV = 1, (2.140)
where the integral is taken over the entire space or over the region where the
wavefunction, , is non-zero. A wavefunction that satisfies condition (2.140)is
called normalized.
In electromagnetism the superposition principle for the wave fields follows
from the linearity of Maxwell’s equations. The wavefunctions that describe the