ELECTRONS IN ATOMS - Chemical Fundamentals of Geology and Environmental Geoscience - page 88

where the lateral displacement of the string, y , remains
zero). These points are called nodes , denoted by the
symbol 'N' in Figure 5.2.
The fundamental and the harmonics are members of
a restricted series of stationary waveforms that can
reside on a string of length L . They can all be repres-
ented by one general equation:

does not lend itself to such simple experimentation,
and our understanding of how it works comes mainly
from theoretical physics. In 1926 the Austrian physicist
Erwin Schrödinger formulated an elegant and highly
successful mathematical theory of particle mechanics
incorporating the wave-like properties of the electron.
His method of analysis is known as wave mechanics ,
and it provides the foundation of modern atomic (and
nuclear) physics. Although the mathematics is diff-
icult, the underlying physical concepts are straightfor-
ward, and have a lot in common with the analysis of
the vibrating string.
Schrödinger set up a general wave equation (anal-
ogous to Equation 5.1) describing the physical circum-
stances of the electron in an atom: the nature of the
electrostatic force attracting it toward the nucleus (the
inverse-square law of classical physics) and the (then)
newly recognized wave properties of the electron
itself. Schrödinger's work suggested that the electron
trapped in an atom behaves in much the same way as
any stationary wave, including the one on a stringed
instrument (Table 5.2).
The Schrödinger wave equation is a differential
equation , which offers a number of possible mathe-
matical 'solutions' called stationary states . Each
simply describes a different stationary 'waveform' a
trapped electron can adopt, analogous to those
shown in Figure 5.2. Each distinct electron waveform
with its own specific three-dimensional geometry
is called an orbital . One speaks of an electron
'occupying' a particular orbital, reminding us that
an orbital is the wave-mechanical equivalent of a
planetary orbit.
Equation 5.1 expresses the way the displacement y
varies along the length of the vibrating string (the x
dimension). In the vocabulary of wave theory, the
mathematical function y is called the wave function .
Each solution of the Schrödinger equation expresses
how a wave function ψ (the Greek letter 'psi') varies in
three-dimensional space ( x , y , z ) around the nucleus.
To understand the physical significance of ψ , we need
to square it. It is true of most types of wave that the
intensity of the physical sensation is proportional to
the square of the wave function. For example, the loud-
ness of the sound emanating from the guitar string is
proportional to y 2 , not y .
The magnitude of ψ 2 at each point in the atom tells us
the probability of finding the electron at that point in
space. In keeping with the Uncertainty Principle, this

yA nx
L

= sin ⋅ ⋅ 360 ° (5.1)

n

2

where y represents the lateral displacement of the string
(from its equilibrium position) at distance x from the end
of the string. A n measures the maximum displacement,
and is called the amplitude . The integer n has a different
value for each of the possible harmonics. Entering ' n = 1'
into Equation 5.2 yields the equation describing the
fundamental; ' n = 2' gives the equation for the first
harmonic (Figure 5.2), and so on. Note that the value of n
defines several important features of a harmonic:

(a) n indicates the number of maxima and minima
(the number of 'lobes') of the waveform;
(b) the number of nodes is equal to n − 1;
(c) the wavelength of the stationary wave is equal to
2 L / n (Figure 5.2);
(d) the frequency of vibration of each harmonic is
proportional to n .

The important conclusion is that a string fixed at
both ends can accommodate only a restricted number
of stationary waves compatible with its length L , so the
vibration frequency can adopt only a limited number
of values (the musical pitches of the fundamental and
harmonics). A variable that is allowed by the proper-
ties of the system to have only certain discrete values is
said to be quantized. The integer n which enumerates
these permitted values is called a quantum number.
These terms are used chiefly in the context of atomic
physics, but they identify characteristics common to all
types of stationary waves, regardless of scale.

Electron waves in atoms

The harmonics of a vibrating string can be investigated
using simple apparatus, so a thorough mathematical
treatment is not called for. Because of its size, the atom