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A quantum theory primer
In the first half of the twentieth century, our understanding of matter underwent a profound revolu-
tion with the advent of quantum theory. Although a deep understanding is not needed for a reader's com-
prehension of this topic, this section summarizes the essence of quantum theory that now underpins all of
modern physics. This primer is not intended as a substitute for learning more about quantum theory but will
be helpful in our understanding of the semiconductor materials that are central to the modern computer
industry and of attempts to develop a new type of “quantum computer.”
Although it is only about one hundred years old, quantum theory helped settle a scientific debate about
the nature of light that began in the seventeenth century. Was light best described as a stream of particles,
as Isaac Newton claimed, or was light some form of wave motion, as the Dutch physicist Christiaan Huygens
had proposed? In 1801, the English physicist Thomas Young demonstrated that when two rays of light meet,
they form a series of bright and dark bands called an
interference pattern
. Since these patterns are characteristic
of waves - like ripples on a pond - the nature of light seemed to be settled. Then in 1921, Albert Einstein won
the Nobel Prize for his explanation of the “photoelectric effect,” the emission of electrons by a metal when
exposed to light. Einstein found that light is made up of particle-like packets of energy called photons.
The theory of quantum mechanics emerged in the 1920s, pioneered by physicists such as Werner
Heisenberg, Erwin Schrödinger, and Paul Dirac. Quantum mechanics provided successful “explanations” -
in terms of its predictions - of the behavior of light, electrons, atoms, and nuclei in the microscopic world
of atoms and nuclei (see Appendix 1). But there is a price to pay for this success: objects like photons
and electrons behave in an essentially quantum mechanical way. All we can know about their motion is
described by the evolution of a “probability wave.” The wave equation discovered by Schrödinger describes
how the probability wave for a quantum object - usually represented by the Greek letter ψ - evolves with
time. We can only observe probabilities and, according to quantum theory, it is the square of this wave
amplitude ψ that gives us the probability that we will observe the object at any given place and time.
Despite this emphasis on probability and uncertainty - epitomized by Heisenberg's famous “uncer-
tainty principle” - quantum mechanics is the only theory capable of making accurate predictions for
systems of atomic sizes or smaller. In addition, it is the very certainties of quantum mechanics that are
responsible for the existence of the different chemical elements we see around us! According to quantum
theory, electrons bound to an atom can only have
certain energies. We can see how this comes about
as follows. The problem of finding the allowed
energies of an electron in an atom is analogous
to finding the allowed energy levels of a charged
particle in a potential well. In real life, we have
to solve the Schrödinger wave equation in three-
dimensional space but we can get some idea of
the quantum solution for an atom by considering
the problem of finding the allowed energy levels
of an electron confined to a one-dimensional box.
In the classical world, the electron in a box could
have any energy; in the quantum world, the wave
patterns must match the dimensions of the well -
like finding the allowed wavelengths for a violin
string. This means that only certain energies are
allowed for the electron in a box (
Fig. 7.21
).
Energy levels
in a box
n=4
n=4
n=3
n=3
Energy
n=2
n=2
n=1
Zero energy
n=1
Fig. 7.21. Energy levels and wave functions for electrons confined to
a box. (a) Energy levels for a quantum particle in a box. The energy
levels are labeled by the quantum number “n.” (b) This shows how
the corresponding wave forms fit into the box. The wave function
has to be zero at the edges.
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