The Foundations of Chemistry

kind of standing-wave mathematics that is applied to the vibrating guitar string. In this
approach, the electron is characterized by a three-dimensional wave function,. In a given
space around the nucleus, only certain “waves” can exist. Each “allowed wave” corresponds
to a stable energy state for the electron and is described by a particular set of quantum
numbers.
The quantum mechanical treatment of atoms and molecules is highly mathematical.
The important point is that each solution of the Schrödinger wave equation (see the
following Enrichment section) describes a possible energy state for the electrons in the
atom. Each solution is described by a set of quantum numbers.These numbers are in
accord with those deduced from experiment and from empirical equations such as the
Balmer-Rydberg equation. Solutions of the Schrödinger equation also tell us about the
shapes and orientations of the probability distributions of the electrons. (The Heisenberg
Principle implies that this is how we must describe the positions of the electrons.) These
atomic orbitals(which are described in Section 5-16) are deduced from the solutions of the
Schrödinger equation. The orbitals are directly related to the quantum numbers.

Figure 5-18 When a string that is fixed at both ends—such as (a) a guitar string—is
plucked, it has a number of natural patterns of vibration, called normal modes. Because the
string is fixed at both ends, the ends must be stationary. Each different possible vibration is a
standing wave, and can be described by a wave function. The only waves that are possible
are those in which a whole number of half-wavelengths fits into the string length. These
allowed waves constitute a harmonic series. Any total motion of the string is some
combination of these allowed harmonics. (b) Some of the ways in which a plucked guitar
string can vibrate. The position of the string at one extreme of each vibration is shown as a
solid line, and at the other extreme as a dashed line. (c) An example of vibration that is not
possible for a plucked string. In such a vibration, an end of the string would move; this is
not possible because the ends are fixed.

5-14 The Quantum Mechanical Picture of the Atom 207

The Schrödinger Equation

In 1926, Erwin Schrödinger (1887–1961) modified an existing equation that described a
three-dimensional standing wave by imposing wavelength restrictions suggested by
de Broglie’s ideas. The modified equation allowed him to calculate the energy levels in the
hydrogen atom. It is a differential equation that need not be memorized or even under-
stood to read this book. A knowledge of differential calculus would be necessary.

 V E

This equation has been solved exactly only for one-electron species such as the hydrogen
atom and the ions Heand Li^2 . Simplifying assumptions are necessary to solve the equa-
tion for more complex atoms and molecules. Chemists and physicists have used their
intuition and ingenuity (and modern computers), however, to apply this equation to more
complex systems.

∂^2



∂z^2

∂^2



∂y^2

∂^2



∂x^2

h^2



82 m

E

nrichment

In 1928, Paul A. M. Dirac (1902–1984) reformulated electron quantum mechanics to take
into account the effects of relativity. This gave rise to a fourth quantum number.

A stationary string

1 half-wavelength

2 half-wavelengths (or one full wave)

3 half-wavelengths

2– half-wavelengths (not possible)

(a)

(b)

(c)^12

Schrödinger’s equation and wave

functions.