The Science Book

230 ERWIN SCHRÖDINGER

only certain distances, and certain
energy levels, would be stable.
However, de Broglie’s solution relied
on treating the matter wave as a
one-dimensional wave trapped in
orbit around the nucleus—a full
description would need to describe
the wave in three dimensions.

The wave equation
In 1925, three German physicists,
Werner Heisenberg, Max Born, and
Pascual Jordan, tried to explain the
quantum jumps that occurred in
Bohr’s model of the atom with a
method called matrix mechanics,
in which the properties of an atom
were treated as a mathematical
system that could change over
time. However, the method could
not explain what was actually
happening inside the atom, and
its obscure mathematical language
did not make it very popular.
A year later, an Austrian
physicist working in Zurich, Erwin
Schrödinger, hit upon a better
approach. He took de Broglie’s
wave-particle duality a step further
and began to consider whether
there was a mathematical equation
of wave motion that would describe
how a subatomic particle might
move. To formulate his wave
equation, he began with the laws
governing energy and momentum
in ordinary mechanics, then
amended them to include the
Planck constant and de Broglie’s
law connecting the momentum of
a particle to its wavelength.
When he applied the resulting
equation to the hydrogen atom,
it predicted exactly the specific
energy levels for the atom that had
been observed in experiments. The
equation was a success. But one
awkward issue remained, because
no one, not even Schrödinger, knew
exactly what the wave equation
really described. Schrödinger tried

to interpret it as the density of

electric charge, but this was not

entirely successful. It was Max

Born who eventually suggested

what it really was—it was a

probability amplitude. In other

words, it expressed the likelihood

of a measurement finding the

electron in that particular place.

Unlike matrix mechanics, the

Schrödinger wave equation or

“wave function” was embraced by

physicists, although it threw open

a whole range of wider questions

about its proper interpretation.

Pauli’s exclusion principle

Another important piece of the

puzzle fell into place in 1925

courtesy of another Austrian,

Wolfgang Pauli. In order to describe

why the electrons within an atom

did not all automatically fall directly

into the lowest possible energy

state, Pauli developed the

exclusion principle. Reasoning

that a particle’s overall quantum

state could be defined by a certain

number of properties, each with a

fixed number of possible discrete

values, his principle stated

that it was impossible for two

particles within the same system

to have the same quantum

state simultaneously.

In order to explain the pattern

of electron shells that was apparent

from the periodic table, Pauli

calculated that electrons must be

described by four distinct quantum

numbers. Three of these—the

principal, azimuthal, and magnetic

quantum numbers—define the

electron’s precise place within

the available orbital shells and

subshells, with the values of the

A classic illustration of wave-particle

duality involves firing electrons from a

“gun” through a barrier with two slits in

it. If electrons are allowed to build up over

time, an interference pattern forms, just

as it would for light waves.

Interference

pattern

Narrow slits

Electrons

Gun