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Quantum Theory
I INTRODUCTION Quantum Theory, also quantum mechanics, in physics, a theory based on using the concept of the quantum unit to describe the dynamic properties of subatomic particles and the interactions of matter and radiation. The foundation was laid by the German physicist Max Planck, who postulated in 1900 that energy can be emitted or absorbed by matter only in small, discrete units called quanta. Also fundamental to the development of quantum mechanics was the uncertainty principle, formulated by the German physicist Werner Heisenberg in 1927, which states that the position and momentum of a subatomic particle cannot be specified simultaneously.
II
EARLY HISTORY
In the 18th and 19th centuries, Newtonian, or
classical, mechanics appeared to provide a wholly accurate description of the
motions of bodies—such as, for example, planetary motion. In the late 19th and
early 20th centuries, however, experimental findings raised doubts about the
completeness of Newtonian theory. Among the newer observations were the lines
that appear in the spectra of light emitted by heated gases, or by gases in
which electric discharges take place. From the model of the atom developed in
the early 20th century by the New Zealand-born physicist Ernest Rutherford, in
which negatively charged electrons circle a positive nucleus in orbits
prescribed by Newton's laws of motion, scientists had also expected that the
electrons would emit light over a broad frequency range, rather than in the
narrow frequency ranges that form the lines in a spectrum.
Another puzzle for physicists was the coexistence of two theories of light: the corpuscular theory, which explains light as a stream of particles, and the wave theory, which views light as electromagnetic waves. A third problem was the absence of a molecular basis for thermodynamics. In his book Elementary Principles in Statistical Mechanics (1902), the American mathematical physicist J. Willard Gibbs conceded the impossibility of framing a theory of molecular action that embraced the phenomena of thermodynamics, radiation, and electrical phenomena as they were then understood.
III
PLANCK'S INTRODUCTION OF THE QUANTUM
At the turn of the century, physicists did not
yet clearly recognize that these and other difficulties in physics were in any
way related. The first development that led to the solution of these
difficulties was Planck's introduction of the concept of the quantum, as a
result of physicists' studies of blackbody radiation during the closing years of
the 19th century. (The term blackbody refers to an ideal body or surface
that absorbs all radiant energy without any reflection.) A body at a moderately
high temperature—a “red heat”—gives off most of its radiation in the
low-frequency (red and infrared) regions; a body at a higher
temperature—”white heat”—gives off comparatively more radiation at
higher frequencies (yellow, green, or blue). During the 1890s physicists
conducted detailed quantitative studies of these phenomena and expressed their
results in a series of curves or graphs. The classical, or prequantum, theory
predicted an altogether different set of curves from those actually observed.
What Planck did was to devise a mathematical formula that described the curves
exactly; he then deduced a physical hypothesis that could explain the formula.
His hypothesis was that energy is radiated only in quanta of energy hu,
where u is the frequency and h is the quantum of action, now known as
Planck's constant.
IV
EINSTEIN'S CONTRIBUTION
The next important developments in quantum
mechanics were the work of Albert Einstein. He used Planck's concept of the
quantum to explain certain properties of the photoelectric effect—an
experimentally observed phenomenon in which electrons are emitted from metal
surfaces when radiation falls on these surfaces.
According to classical theory, the energy, as measured by the voltage of the emitted electrons, should be proportional to the intensity of the radiation. Actually, however, the energy of the electrons was found to be independent of the intensity of radiation—which determined only the number of electrons emitted—and to depend solely on the frequency of the radiation. The higher the frequency of the incident radiation, the greater is the electron energy; below a certain critical frequency no electrons are emitted. These facts were explained by Einstein by assuming that a single quantum of radiant energy ejects a single electron from the metal. The energy of the quantum is proportional to the frequency, and so the energy of the electron depends on the frequency.
V
THE BOHR ATOM
In 1911 Rutherford established the existence of
the atomic nucleus. He assumed, on the basis of experimental evidence obtained
from the scattering of alpha particles by the nuclei of gold atoms, that every
atom consists of a dense, positively charged nucleus, surrounded by negatively
charged electrons revolving around the nucleus as planets revolve around the
Sun. The classical electromagnetic theory developed by the British physicist
James Clerk Maxwell unequivocally predicted that an electron revolving around a
nucleus will continuously radiate electromagnetic energy until it has lost all
its energy, and eventually will fall into the nucleus. Thus, according to
classical theory, an atom, as described by Rutherford, would be unstable. This
difficulty led the Danish physicist Niels Bohr, in 1913, to postulate that in an
atom the classical theory does not hold, and that electrons move in fixed
orbits. Every change in orbit by the electron corresponds to the absorption or
emission of a quantum of radiation.
The application of Bohr's theory to atoms with more than one electron proved difficult. The mathematical equations for the next simplest atom, the helium atom, were solved during the second and third decade of the century, but the results were not entirely in accordance with experiment. For more complex atoms, only approximate solutions of the equations are possible, and these are only partly concordant with observations.
VI
WAVE MECHANICS
The French physicist Louis Victor de Broglie
suggested in 1924 that because electromagnetic waves show particle
characteristics, particles should, in some cases, also exhibit wave properties.
This prediction was verified experimentally within a few years by the American
physicists Clinton Joseph Davisson and Lester Halbert Germer and the British
physicist George Paget Thomson. They showed that a beam of electrons scattered
by a crystal produces a diffraction pattern characteristic of a wave. The wave
concept of a particle led the Austrian physicist Erwin Schrِdinger to
develop a so-called wave equation to describe the wave properties of a particle
and, more specifically, the wave behaviour of the electron in the hydrogen atom.
Although this differential equation was
continuous and gave solutions for all points in space, the permissible solutions
of the equation were restricted by certain conditions expressed by mathematical
equations called eigenfunctions (German eigen, “own”). The Schrِdinger
wave equation thus had only certain discrete solutions; these solutions were
mathematical expressions in which quantum numbers appeared as parameters.
(Quantum numbers are integers developed in particle physics to give the
magnitudes of certain characteristic quantities of particles or systems.) The
Schrِdinger equation was solved for the hydrogen atom and gave conclusions
in substantial agreement with earlier quantum theory. Moreover, it was solvable
for the helium atom, which earlier theory had failed to explain adequately, and
here also it was in agreement with experimental evidence. The solutions of the
Schrِdinger equation also indicated that no two electrons could have the
same four quantum numbers—that is, be in the same energy state. This rule,
which had already been established empirically by Wolfgang Pauli in 1925, is
called the exclusion principle.
VII MATRIX MECHANICS Simultaneously with the development of wave mechanics, Heisenberg evolved a different mathematical analysis known as matrix mechanics. According to Heisenberg's theory, which was developed in collaboration with the German physicists Max Born and Ernst Pascual Jordan, the formula was not a differential equation but a matrix: an array consisting of an infinite number of rows, each row consisting of an infinite number of quantities.. Matrix mechanics introduced infinite matrices to represent the position and momentum of an electron inside an atom. Different matrices exist, one for each of the other observable physical properties associated with the motion of an electron, such as energy, and angular momentum. These matrices, like Schrِdinger's differential equations, could be solved; in other words, they could be manipulated to produce predictions as to the frequencies of the lines in the hydrogen spectrum and other observable quantities. Like wave mechanics, matrix mechanics was in agreement with the earlier quantum theory for processes in which the earlier quantum theory agreed with experiment; it was also useful in explaining phenomena that earlier quantum theory could not explain.
VIII THE MEANING OF QUANTUM MECHANICS Schrِdinger subsequently succeeded in showing that wave mechanics and matrix mechanics are different mathematical versions of the same theory, now called quantum mechanics. Even for the simple hydrogen atom, which consists of two particles, both mathematical interpretations are extremely complex. The next simplest atom, helium, has three particles, and even in the relatively simple mathematics of classical dynamics, the three-body problem (that of describing the mutual interactions of three separate bodies) is not entirely soluble. The energy levels can be calculated, however. In applying quantum-mechanical mathematics to relatively complex situations, a physicist can use one of a number of mathematical formulations. The choice depends on the convenience of the formulation for obtaining suitable approximate solutions.
Although quantum mechanics describes the atom purely in terms of mathematical interpretations of observed phenomena, a rough verbal description can be given of what the atom is now thought to be like. Surrounding the nucleus is a series of stationary waves; these waves have crests at certain points, each complete standing wave representing an orbit. The absolute square of the amplitude of the wave at any point at a given time is a measure of the probability that an electron will be found there. Thus, an electron can no longer be said to be at any precise point at any given time.
IX THE UNCERTAINTY PRINCIPLE The impossibility of pinpointing an electron at any precise time was analysed by Werner Heisenberg, who in 1927 formulated the uncertainty principle. This principle states the impossibility of simultaneously specifying the precise position and momentum of any particle. In other words, physicists cannot measure the position of a particle, for example, without causing a disturbance in the velocity of that particle. Knowledge about position and velocity are said to be complementary, that is, they cannot be precise at the same time. This principle is also fundamental to the understanding of quantum mechanics as it is generally accepted today: the wave and particle characters of electromagnetic radiation can be understood as two complementary properties of radiation.
X RESULTS OF QUANTUM THEORY Quantum mechanics solved all of the great difficulties that troubled physicists in the early years of the 20th century. It gradually enhanced the understanding of the structure of matter, and it provided a theoretical basis for the understanding of atomic structure and the phenomenon of spectral lines: each spectral line corresponds to the energy of a photon transmitted or absorbed when an electron makes a transition from one energy level to another. The understanding of chemical bonding was fundamentally transformed by quantum mechanics and came to be based on Schrِdinger's wave equations. New fields in physics emerged— solid-state physics, condensed-matter physics, superconductivity, nuclear physics, and elementary particle physics—that all found a consistent basis in quantum mechanics.
XI FURTHER DEVELOPMENTS In the years since 1925, no fundamental deficiencies have been found in quantum mechanics, although the question of whether the theory should be accepted as complete has come under discussion . In the 1930s the application of quantum mechanics and special relativity to the theory of the electron allowed the British physicist Paul Dirac to formulate an equation that referred to the existence of the spin of the electron. It further led to the prediction of the existence of the positron, which was experimentally verified by the American physicist Carl David Anderson.
The application of quantum mechanics to the subject of electromagnetic radiation led to explanations of many phenomena, such as bremsstrahlung (German, “braking radiation”, the radiation emitted by electrons slowed down in matter) and pair production (the formation of a positron and an electron when electromagnetic energy interacts with matter). It also led to a grave problem, however, called the divergence difficulty: certain parameters, such as the so-called bare mass and bare charge of electrons, appear to be infinite in Dirac's equations. (The terms bare mass and bare charge refer to hypothetical electrons that do not interact with any matter or radiation; in reality, electrons interact with their own electric field.) This difficulty was partly resolved in 1947-1949 in a programme called renormalization, developed by the Japanese physicist Shin'ichirô Tomonaga, the American physicists Julian S. Schwinger and Richard Feynman, and the British-born American physicist Freeman Dyson. In this programme, the bare mass and charge of the electron are chosen to be infinite in such a way that other infinite physical quantities are cancelled out in the equations. Renormalization greatly increased the accuracy with which the structure of atoms could be calculated from first principles.
XII
FUTURE PROSPECTS
Quantum mechanics underlies current attempts to
account for the strong nuclear force and to develop a unified theory for
all the fundamental interactions of matter (. Nevertheless, doubts exist about
the completeness of quantum theory. The divergence difficulty, for example, is
only partly resolved. Just as Newtonian mechanics was eventually amended by
quantum mechanics and relativity, many scientists—and Einstein was among
them—are convinced that quantum theory will also undergo profound changes in
the future. Great theoretical difficulties exist, for example, between quantum
mechanics and chaos theory, which began to develop rapidly in the 1980s. Ongoing
efforts are being made by theorists such as the British physicist Stephen
Hawking to develop a system that encompasses both relativity and quantum
mechanics.