Dirac Equation

Type of physical science: Atomic physics

Field of study: Relativistic quantum mechanics

In 1928, Paul Adrien Maurice Dirac postulated a quantum-mechanical wave equation which was consistent with the requirements of special relativity. The resulting theory has become one of the most spectacular single achievements in the history of science. The Dirac equation correctly accounts for the spin of the electron and its magnetic interactions, which could be treated only empirically in nonrelativistic quantum mechanics. It also predicted the existence of the positron, the electron's antiparticle, in advance of its experimental discovery.

Overview

In 1926, Erwin Schrodinger developed the fundamental equation of quantum mechanics to describe the behavior of matter on the atomic scale which have dimensions on the order of 0.0000000001, or 10^-10 meter. The Schrodinger equation is expressed as follows: iħ·∂ψ/∂t = (ħ²/-2m)(∂²ψ/∂x² + ∂²ψ/∂y² + ∂²ψ/∂z²)

Here, m is the mass of the particle (usually an electron); ħ = h/2π, where h is Planck's constant; x, y, and z are the coordinates of the particle; t is the time; i is an imaginary unit (the square root of minus one); and PSI represents the wavefunction, which describes the state of a quantum-mechanical particle. The wavefunction depends on x, y, z, and t, and its square PSI² measures the particle's probability of being found at the point x, y, z, at the time t. The symbol Δ in the Schrodinger equation represents a partial derivative, the basic operation in differential calculus for functions of several variables. Schrodinger's wave mechanics, as this formulation of quantum theory is sometimes called, has since made possible revolutionary advances in a vast domain of phenomena in physics, chemistry, material science, and molecular biology.

The Schrodinger equation is a nonrelativistic theory, however, meaning that it becomes inaccurate when applied to particles approaching the speed of light, c (approximately 3 x 10⁸ meters per second). Albert Einstein had developed his special theory of relativity in 1905. According to this theory, space and time coordinates, such as x, y, z, and t, are inextricably interwoven in the fundamental laws of physics and must appear in a symmetrical way in its equations. This condition is not satisfied, however, in the Schrodinger equation. It can be seen that the Schrodinger equation contains second derivatives with respect to x, y, and z but only a first derivative with respect to t, hence violating a prime mathematical requisite of special relativity. Nonrelativistic quantum mechanics also suffers from a serious error of omission. In order to account for the observed behavior of atoms, molecules, and solids, it is necessary to postulate ad hoc the existence of electron spin, a mysterious internal characteristic of the electron with two possible states.

The first attempt to write down a relativistic generalization of the Schrodinger equation made use of the second derivative with respect to t to match those of x, y, and z. This generalization is known as the Klein-Gordon equation (after Oskar Klein and Walter Gordon) and is expressed as follows: ∂²ψ/∂x² + ∂²ψ/∂y² + ∂²ψ/∂z² - (1/c²)( ∂²ψ/∂t)² = (m²c²/ħ²)ψ

Although consistent with special relativity, the Klein-Gordon equation gave incorrect predictions for certain small energy differences in the hydrogen atom known as fine structure.

Moreover, the equation still appeared to describe a spinless particle, not the electron. It is now understood that the Klein-Gordon equation is actually valid for a class of massive spinless particles including pions, but that it is not the correct equation for the electron.

In 1926, Paul Adrien Maurice Dirac received his Ph.D. in theoretical physics from the University of Cambridge at the age of twenty-four, after receiving an undergraduate degree in electrical engineering. That same year, he created a unified formulation of quantum mechanics, known as transformation theory. This theory showed the equivalence of wave mechanics with an alternative formalism, called matrix mechanics, that was developed by Werner Heisenberg and coworkers. In 1928, Dirac turned his attention to the unification of the two revolutionary achievements of twentieth century physics: quantum theory and relativity. He wrote a relativistic wave equation containing first derivatives in both space and time variables. The result is known as the Dirac equation: iħ·∂ψ/∂t = -iħc [(α1)(∂ψ/∂x) + (α2)(∂ψ/∂y) + (α3)(∂ψ/∂z)] + βmc2ψ

The complication arose that coefficients which were ordinary numbers in the Schrodinger equation had to be replaced by the quantities α and β in the Dirac equation.

These quantities did not obey the rules of ordinary arithmetic and had to be interpreted as matrices, in this case 4 x 4 square arrays of numbers. To be consistent, the wavefunction PSI also had to be interpreted as a four-component entity called a spinor.

The components of the spinor could be identified with the internal states of the electron, which is what was needed to express spin. Yet, two, rather than four, components would have sufficed. Dirac found that two of the components did indeed represent the spin-up and spin-down states of a particle of rest energy mc². The concept of rest energy is another consequence of special relativity, contained in Einstein's famous formula E = mc². Rest energy implies that, in a way, matter is frozen energy. The remaining two components of Dirac's spinors, however, presented an enigma. They were again spin-up and spin-down states, but those of a hypothetical particle with negative rest energy, -mc², which has no meaning in relativity theory. Another unfortunate consequence of such negative energy states would be that ordinary positive-energy electrons should have a natural tendency to fall into them, which is entirely contradictory to physical reality.

Dirac resolved these last two difficulties in a bold and imaginative way. Electrons are subject to the Pauli exclusion principle, which requires that no two electrons in a system can exist in the same state. Dirac proposed that the negative-energy states were entirely filled by electrons. While this proposition would require an infinite number of electrons (because negative energies extend down to minus infinity), the exclusion principle would at least guarantee the stability of ordinary electrons, since they could not fall into already occupied states. This infinite sea of negative-energy electrons was supposed to represent the background against which all observable phenomena occur.

If one of the negative-energy electrons were somehow excited to a positive energy value, then one would observe an electron materializing out of the vacuum. At the same time, a hole would appear in the negative energy sea which would have the appearance of a positively charged particle of rest energy +mc². Although a missing negative charge would resemble a positive charge and missing negative energy would resemble positive energy, no one had ever seen a positive electron. Dirac tried to avoid the condition of negative energy states but was unable to do so. With reluctance, he thereby predicted the existence of antimatter.

Confirmation of this theory came in 1932 when Carl David Anderson identified positrons in the cloud-chamber tracks created by cosmic rays.

In the process of pair production, an electron and a positron materialize out of pure energy. In the inverse process, an electron-positron pair mutually annihilate. The resultant energy is usually taken up by high-energy photons, such as γ rays. This interconvertibility of matter and energy also gives the most explicit confirmation of E = mc².

Since electrons permeate ordinary matter, positrons have a rather short lifetime before annihilation, at most a fraction of a microsecond (millionths of a second). It is now known that all elementary particles have their corresponding antiparticles.

The impermanence of matter implied by the Dirac theory led to the development of quantum field theory, with seminal contributions by Dirac himself. According to quantum field theory, particles and the forces between them can be conceptualized as excitations in various types of fields. For example, photons are the particles of the electromagnetic field. The theory embracing electrons and photons, known as quantum electrodynamics, is the most successful scientific theory, measured by the quantitative agreement of prediction with experiment. On the basis of quantum field theory, particles and antiparticles can be treated in a more symmetrical fashion, thereby eliminating the infinite sea of negative energy electrons.

In the consideration of ordinary matter, further analysis of the Dirac equation confirms that electrons possess angular momentum associated with their spin, which is in addition to the angular momentum that is produced by the orbital motion of electrons around the nucleus.

Angular momentum is a characteristic of circular motion, and the spin angular momentum can be pictured as the rotation of an electron about its axis, somewhat analogous to the daily rotation of the earth. The electron is said to have a spin of one-half and can exist in two internal states, usually designated as "spin-up" and "spin-down." In classical electromagnetic theory, a circulating electric charge behaves like a magnet. Such a system is said to possess a magnetic moment. Experiments suggest that spin angular momentum is twice as effective as orbital angular momentum in producing a magnetic moment. Thus, it is stated that the electron spin g-factor equals two. Spin-orbit interaction between these respective magnetic moments is also correctly accounted for by the Dirac theory, as is the fine structure in the spectrum of atomic hydrogen.

There remain minute experimental energy differences between certain atomic energy levels which are not given correctly by the Dirac theory. The best known case is the energy difference between two low-lying excited states of the hydrogen atom, specifically the states designated 2S½ and 2P½. According to the Dirac equation, these states should have exactly the same energy. Very accurate measurements by Willis Lamb and coworkers, however, showed that the two states are separated by an energy corresponding to the radio frequency 1,058 megahertz, which is known as the Lamb shift. In addition, an accurate measurement of the electron's g-factor gives the value 2.0023, rather than exactly two. These and other small effects are within the range of quantum electrodynamics, but beyond the scope of the Dirac equation.

The Dirac equation also applies to the massless particle with a spin of 1/2 known as the neutrino. Neutrinos are produced in radioactive decay and, very copiously, by the nuclear reactions that take place in the interior of stars. Because m = 0, the spinors describing neutrinos have only two components. As a consequence, every neutrino has a spin in the opposite direction to its linear motion; it is said to be a left-handed particle. Conversely, all corresponding antiparticles, called antineutrinos, are right-handed.

In 1931, Dirac predicted the existence of magnetic monopoles, unattached north or south magnetic poles. This prediction follows from the fact that both electric charge and angular momentum are quantized; that is, they occur only as multiples of a fundamental unit. Magnetic monopoles have not yet been observed in nature. They are, however, a speculative feature of some grand unified theories that have been proposed.

Applications

The Dirac theory is such a fundamental cornerstone of modern physical theory that almost every modern development in particle physics, theoretical or experimental, can be considered one of its applications. The relativistic quantum theory of electrons and photons, for which the Dirac equation was the starting point, had for many years served as a model for developing theories of strong and weak interactions. These theories have culminated in the so-called standard model for strong, weak, and electromagnetic interactions.

The most successful experimental technique of high-energy physics is based on colliding beams of electrons and positrons, such as at the Stanford Linear Accelerator Center (SLAC) at Stanford University and the Large Electron-Positron (LEP) Collider at the Centre Europeen de Recherche Nucleaire (CERN) in Geneva, Switzerland. The energy that is made available by the mutual annihilation of ultrahigh energy electrons and positrons can rematerialize into more exotic particle-antiparticle pairs and their decay products. The charmed and bottom quarks were discovered in this way, in addition to the W and Z bosons, the carriers of weak interaction. These latter particles are of the order of one hundred times as massive as the proton, which is itself 1,860 times heavier than the electron.

The chemical behavior of certain elements in the latter part of the periodic table can be attributed to relativistic effects involving their electrons. Relativistic corrections to atomic structure vary approximately as the fourth power of the atomic number Z. The most spectacular example of the effect of relativity concerns the chemical nature of gold (Z = 79). Gold, being exceptionally stable and unreactive, is highly prized as one of the "noble metals." The Finnish quantum chemist Pekka Pyykko has attributed these features to a rearrangement of some of gold's atomic energy levels as a consequence of relativity. In other words, the energy levels of a gold atom in a hypothetical nonrelativistic world would make the atom closer chemically to copper or silver. Therefore, in a nonrelativistic world, gold might tarnish like silver or turn green like copper after long exposures to air. The fact that the metallic element mercury (Z = 80) is a liquid at room temperature can likewise be attributed to a relativistic effect on the interatomic attractive forces.

Context

Classical mechanics, the laws of motion first enunciated by Sir Isaac Newton in the seventeenth century, were adequate to understand almost all known physical phenomena until the end of the nineteenth century. In the latter part of the nineteenth century, Newtonian mechanics was augmented by the laws of electromagnetism formulated by James Clerk Maxwell and by an understanding of statistical mechanics, the work of Maxwell and Ludwig Boltzmann. It appeared to many physicists that a complete comprehensive theory of their subject was nearly at hand.

This belief was highly premature, however, because developments in the twentieth century were to completely undermine the foundations of classical physics. The quantum theory had its origin in 1900 with the work of Max Planck on black body radiation. Further advances came with Einstein's explanation of the photoelectric effect and Neils Bohr's theory of the hydrogen atom in 1913. More complete formulations of quantum mechanics, the successor to Newtonian mechanics, were developed from 1925 to 1926, principally by Heisenberg, Schrodinger, and Dirac.

A second revolution in twentieth century physics was the theory of relativity, almost the single-handed creation of Einstein. The special theory of relativity, which dealt with the symmetry between space and time, was incorporated into quantum mechanics by the work of Dirac in 1928. The Dirac equation for the electron has proven to be one of the most spectacular successes in the history of science. It accounted for the spin of the electron and predicted the existence of antimatter. Dirac's relativistic quantum mechanics paved the way for the subsequent development of quantum field theory, a unified formulation governing elementary particles and their interactions. Quantum field theory has evolved into the standard model, a provisional theory of strong, weak, and electromagnetic interactions (but still excluding gravity) which provides a coherent framework for understanding all known physical phenomena. In addition, the mutual annihilation of particle-antiparticle pairs, a possibility inherent in the Dirac theory, provides the experimental basis for modern high-energy physics.

Principal terms

ANTIMATTER: a collective term for antiparticles

ANTIPARTICLE: a "mirror image" twin of an elementary particle having the same mass and spin, but the opposite electric charge; when a particle and its antiparticle collide, they mutually annihilate to produce pure energy

HYDROGEN ATOM: the simplest atom, consisting of a single proton and electron

PLANCK'S CONSTANT: the fundamental unit of action in quantum mechanics, designated h and equal to 6.626 x 10^-34 joules per second

POSITRON: the electron's antiparticle, whose existence was predicted by the Dirac equation

QUANTUM MECHANICS: the theory governing the behavior of matter on the atomic scale

SCHRODINGER EQUATION: the fundamental wave equation of nonrelativistic quantum mechanics

SPECIAL RELATIVITY: Albert Einstein's theory that measurements of space and time are relative to a given observer and that only the mathematical forms of the laws of physics are invariant; as a consequence, matter and energy become equivalent

SPIN: an intrinsic property of elementary particles, a mechanical analogue of which is the rotation of a particle around its axis

SPINOR: a wavefunction with more than one component which describes a particle with spin, such as the electron

WAVEFUNCTION: a solution of a wave equation describing the state of a quantum-mechanical system

Bibliography

Cline, Barbara Lovett. MEN WHO MADE A NEW PHYSICS. Chicago: University of Chicago Press, 1987. A lively biographical account of the creators of the quantum theory. Dirac's contributions are discussed on pages 186-191.

Crease, Robert P., and Charles C. Mann. THE SECOND CREATION. New York: Macmillan, 1986. A lucid account of the quest for the fundamental principles governing matter and energy. Discusses the continuing revolution in twentieth century physics, from relativity and quantum mechanics to black holes and superstrings. Dirac's several contributions are described on pages 76-88.

Dirac, P. A. M. THE DEVELOPMENT OF QUANTUM THEORY. New York: Gordon & Breach, 1971. Contains Dirac's own account of how he developed his equation.

Dirac, P. A. M. "The Evolution of the Physicist's Picture of Nature." SCIENTIFIC AMERICAN 208 (May, 1963): 45-53. Dirac's analysis of how physical theories have evolved in the past and some speculations on how they might develop in the future. Relativistic quantum mechanics is discussed in this context.

Kursunoglu, Behram N., and Eugene P. Wigner, eds. REMINISCENCES ABOUT A GREAT PHYSICIST: PAUL ADRIEN MAURICE DIRAC. Cambridge, England: Cambridge University Press, 1987. Contributions by a number of eminent contemporaries on the man and his work are presented.

Rose, Morris E. RELATIVISTIC ELECTRON THEORY. New York: John Wiley & Sons, 1961. An advanced textbook on relativistic quantum mechanics and its applications.

Sherwin, Chalmers W. INTRODUCTION TO QUANTUM MECHANICS. New York: Henry Holt, 1959. Most standard graduate and undergraduate texts on quantum mechanics contain a chapter on the Dirac equation. Sherwin's treatment is one of the more accessible accounts.

Electrons and Atoms