Wave-particle Duality

Type of physical science: Atomic physics

Field of study: Nonrelativistic quantum mechanics

Energy transport is accomplished via particles or waves. The realization, based upon symmetry considerations, that particle behavior can be manifested by a wave and wave behavior can be manifested by a particle represented a philosophical milestone that led to the development of quantum mechanical thought.

Overview

At the beginning of the twentieth century, physics began a revolution that would forever change the way in which scientists viewed the nature of space and time, energy and matter, and particles and waves. That revolution led to the development and acceptance of both quantum mechanics and relativity theory, both special relativity and general relativity. Prior to this revolution, particle behavior in energy and momentum transfer was considered separate from wave behavior in energy and momentum transfer. A number of phenomena studied in the latter part of the nineteenth century could not be explained in terms of classical electromagnetism as expressed concisely by James Clerk Maxwell. Both the Lord Rayleigh-Sir James Hopwood Jeans and the Wilhelm Wien description of black body radiation were inadequate to explain the experimentally observed data of spectral radiancy as a function of wavelength across the entire electromagnetic spectrum. A number of observed aspects of the photoelectric effect--the release of electrons from a metal surface when illuminated properly with electromagnetic radiation--were in total variance with predictions of classical electromagnetism. Classical theory did not predict the existence of a cutoff frequency, below which the photoelectric effect ceased regardless of illumination intensity, and the nearly instantaneous release of photoelectrons from a surface regardless of wave intensity.

One of the biggest philosophical steps that led to the explanation of both black body radiation and the photoelectric effect was the idea that an electromagnetic wave does not consist of a continuous flow of energy. The wave nature of electromagnetism was tied to the propagation of mutually perpendicular spatially and temporally dependent oscillating electric and magnetic fields, which did represent a continuous flow of energy. A wave has a well-defined wavelength, the spatial distance over which it completes one full cycle of oscillation, and a well-defined period, which is the time span during which a complete oscillation is made. A wave's frequency is the inverse of the period of oscillation and, as such, is the number of complete oscillations per second. To explain the noncontinuous flow of energy in an electromagnetic wave, a corpuscular (or particle) nature was attributed to it. A wave of a given frequency carries its energy in identical, massless packets of an energy that is proportional to the wave's frequency. The constant of proportionality between the energy and the frequency is called Planck's constant, h, which has a value of 6.63 x 10^-34 joule-seconds. (In the limit that h approaches zero, quantum behavior reverts to classical behavior as espoused by Newtonian mechanics and classical electromagnetism.) The packet of energy is called a photon and is considered the quantum of the electromagnetic field. Energy transfer is accomplished via the exchange of photons.

This ambiguity begs the question as to whether electromagnetic radiation consists of particles or is a wave. The resolution of this ambiguity involves wave intensity and a probabilistic interpretation. The intensity of an electromagnetic wave at any location in space is not the wave's energy itself, but the probability of finding the wave's energy at that point in space. Where a wave intensity is high, large numbers of photons are likely to be detected, and where the wave intensity is low, fewer photons are likely to be observed.

Electromagnetic radiation has both a particle behavior, exhibited by the photon, and a wave behavior. Nevertheless, both behaviors cannot be exhibited at the same time. The equivalence of particle and wave behavior for electromagnetic radiation can be seen most easily in the energy equation. For a particle of zero rest mass, its energy is given by the product of its momentum and the speed of light c. For the wave nature of electromagnetic radiation, the energy is the product of Planck's constant and the wave's frequency. The equivalence of these two energy equations shows the ability of a wave to have momentum given by Planck's constant, multiplied by the wave frequency and divided by the speed of light. Since the dispersion relation for electromagnetic radiation gives the speed of light as the product of wavelength and frequency, the momentum of electromagnetic radiation can be expressed as Planck's constant divided by the wavelength.

One of the beauties of physics is its symmetries. If a wave could have a particle nature, Louis de Broglie postulated, then material (nonzero rest mass) particles (possessing energy and momentum) should have an associated matter wavelength given by Planck's constant divided by the momentum, the product of the particle's mass and velocity. The hypothesis of a wave nature for particles restored the symmetry broken by the acceptance of a particle nature of electromagnetic radiation. Whenever particles have a momentum p, associated with the motion of the particle is a wave of wavelength given as Planck's constant divided by the momentum p. Whenever there is a wave of given wavelength, the square of the wave's amplitude is proportional to the probability of observation of a particle with momentum p given by Planck's constant divided by the wavelength.

Therefore, waves are specified by wavelength, frequency, and a dispersion relation that gives the speed of propagation. Particles are specified by momentum, mass, and a particle speed.

Applying this to electromagnetic radiation, Albert Einstein's relation between energy and mass (E = mc²) allows one to calculate a relativistic mass for a photon in terms of the speed of light, Planck's constant, and the frequency. Applying this to material particles, a frequency of the associated matter wave can be calculated in terms of Planck's constant and the sum of the particle's kinetic and potential energies.

In everyday life, the wave nature of matter is not observed. Tennis balls get caught in a court net; they do not diffract through the way a light wave will diffract through a slit aperture or a grating. If a particle's matter-wave wavelength is comparable to an aperture, then diffraction effects would be seen for that particle.

Assignment of a speed v to a particle of given energy and momentum implies that E and p are localized in space and are transported through space at speed v. How does the particle speed compare to its associated matter-wave phase velocity?

Phase velocity is the ratio of energy to momentum. If E = mc² and p = mv, then the phase velocity is greater than the speed of light, since v less than c for nonzero rest mass particles. The significance of this is simply that a monochromatic wave associated with a nonzero rest-mass particle is unobservable. Calculation of the phase velocity for electromagnetic radiation yields simply c. Here, the phase velocity and the wave speed are identical.

To describe a material particle, a group or packet of individual monochromatic waves differing in frequency and phase velocity must be used. Here, the group velocity--the rate of change of energy with momentum--will always be less than c and the group velocity is identical to the particle's speed v.

Wave and particle description for the same phenomenon are mutually incompatible. If a wave is to have its wavelength given with infinite precision, then it must have an infinite extent in space. Also, if it is confined to a limited region of space so its energy is confined to a localized position at any one time, then it resembles a particle by virtue of this localizability. Thus, one sees that wave-particle duality results from basic uncertainties inherent in nature itself. This principle is clearly demonstrated by the Heisenberg uncertainty principle.

Applications

Consider a 150-gram baseball thrown by a pitcher at 40 meters per second. What does the de Broglie equation predict for the baseball's associated matter wavelength? The baseball's momentum would be 6 kilogram-meters per second. Dividing that into Planck's constant yields a wavelength of only 1.11 x 10^-34 meters, a distance eighteen orders of magnitude below the smallest distance measurable by twentieth century physics. Thus, the wave character of this baseball is indiscernible.

If one considers a molecule of hydrogen at typical room temperature (300 Kelvins), its root-mean-square velocity is 1.84 kilometers per second. A calculation of the associated matter wavelength yields 1.08 angstroms, a distance easily detected. Indeed, this wavelength is typical of the wavelengths of X rays and the interatomic spacings in crystalline lattices. Wave effects can and have been observed for such molecules of hydrogen.

In 1927, Clinton Joseph Davisson and Lester Germer, working at Bell Telephone Laboratories, accidentally verified the existence of matter waves associated with particles. While working on secondary electron emission (a process in which a beam of electrons falls on a solid surface and liberates electrons originally bound to the solid) in nickel, they discovered what looked like a diffraction peak in the angular distribution of electrons emitted from the sample.

This peak was observed after they had heated their nickel sample to high temperature in an attempt to clean the surface. The thermal energy of the heat treatment was sufficient to turn the original high-purity polycrystalline sample into single-crystal nickel, a nearly perfect face-centered cubic lattice structure throughout the sample. The interatomic spacing of the crystal was of the size of angstroms. (Crystallographers often use X rays of several angstroms wavelength to ascertain interatomic spacings in crystalline structures via diffraction techniques.)

When Davisson and Germer illuminated the nickel sample with the electron beam (54 electronvolts of energy), they observed the superposition of a secondary emission spectrum and a diffraction peak (at an angle of 50 degrees).

A calculation of the matter wave wavelength for the electron beam indicated that, based upon the known interatomic spacing for face-centered cubic nickel, a first-order diffraction peak should be observed at the experimentally obtained angle for an X ray of equal wavelength. For the first time, diffraction with electrons, a material particle, had been experimentally demonstrated, and the results precisely followed the Bragg law for wave diffraction and the de Broglie relation for matter waves.

Since that accidental verification of particle diffraction, numerous other nonzero rest mass particles have shown diffraction effects. Consider a thermal neutron, a neutron at room temperature (300 Kelvins). The kinetic energy of this particle, by equipartition of energy (an energy of one-half the Boltzmann constant per unit Kelvins for each degree of freedom) is 6.2 x 10^-21 joules. Equating this energy to 1/2 mv², the speed of such a neutron is found to be 2.7 kilometers per second. The matter wavelength would then be 1.4 angstroms. Therefore, diffraction peaks would be observed if thermal neutrons were incident on many varieties of naturally occurring crystalline solids.

Applications of wave-particle duality could be somewhat confusing. Both the wave and particle models were used to explain various phenomena for electromagnetic radiation and material particles. Nevertheless, the wave model and the particle model were never applied simultaneously. If one considers electromagnetic radiation, the wave model can be used to explain diffraction and interference. The photoelectric effect, Compton effect, pair production, and pair annihilation require use of the particle model. Thus, electromagnetic radiation exhibits both wave and particle aspects, but never at the same time.

This phenomenon is a result of the complementarity principle enunciated in 1928 by Niels Bohr. This principle states that wave and particle aspects are complementary rather than antagonistic. A full description of the nature of electromagnetic radiation, for example, requires a knowledge of the wave aspect and also the particle aspect. A knowledge of only one aspect would represent only partial understanding.

Context

An overriding concern in physics is symmetry. Symmetry principles are what provide physics with its intrinsic beauty of formulation. In the early development of quantum physics, particle behavior was attributed to what had previously been considered a wave phenomena. An explanation of black body radiation and the photoelectric effect was not within the capabilities of classical electromagnetic theory as summarized by Maxwell's equations. In 1900, Max Planck introduced the notion of quantization in order to interpret successfully experimental observations of black body radiation. In 1905, Albert Einstein introduced the concept of the photon--a particle quantum of electromagnetic energy with zero rest mass--to interpret successfully experimental observations of the photoelectric effect. (Actually, Einstein was not the first to attribute a particle nature to light. Sir Isaac Newton had espoused a corpuscular theory of light that failed to achieve universal acceptance.) Several other interactions between matter and energy were successfully explained, combining the notion of quantization with a particle nature attributed to electromagnetic radiation.

In 1924, a French graduate student in physics at the University of Paris proposed, principally on the grounds of symmetry considerations, that the movement of material particles is accomplished by a complementary matter wave whose wavelength was determined by the momentum of the material particle. Louis de Broglie restored symmetry in early quantum theory by attributing a wave nature to material particles, where the particle nature of electromagnetic waves had been established. The wave nature of the electron orbiting the proton in the nucleus of a hydrogen atom confirmed Bohr's postulates of atomic structure in a theoretical framework derivable from first principles. The electron could orbit the nucleus only in such a way that its matter wavelength, determined by its momentum, could be only integral multiples of the electron's orbital circumference. This condition mimicked a standing wave that has no net transfer of energy, thus explaining the stability of atomic structure. Only when the orbit was changed could the atom emit electromagnetic radiation, carrying away the energy and momentum. Two years after de Broglie's hypothesis of matter waves, Davisson and Germer experimentally verified diffraction of an electron beam in a single crystal of nickel.

Today, physicists believe in a wave-particle duality of existence. That which carries energy and has momentum has both a wave nature and a particle nature. When the wave nature is manifested, the particle nature is totally suppressed. When the particle nature is manifested, the wave nature is totally suppressed. Wave-particle duality has become the cornerstone upon which modern quantum mechanical theory is firmly constructed.

Principal terms

MATERIAL PARTICLE: a particle having nonzero rest mass

MATTER WAVE: wavelike characteristics of a material particle

MOMENTUM: a quantity of motion of a particle, usually expressed as the product of mass and velocity

PHOTON: a quantum of electromagnetic energy possessing no rest mass

QUANTIZATION: limitation of certain physical parameters to discrete values rather than a continuum of values

RELATIVISTIC MASS: the mass of an object when measured under conditions of motion relative to an inertial reference frame

REST MASS: the mass of an object when measured at rest with respect to an inertial reference frame

WAVELENGTH: the spatial repetition of a waveform; the distance covered by one complete wave oscillation

Bibliography

Beiser, Arthur. CONCEPTS OF MODERN PHYSICS. New York: McGraw-Hill, 1987. Excellent text on the development of nonclassical understanding of space and time, matter and energy. Easily accessible to the general reader. Illustrations and examples abound.

Burns, Marshall L. MODERN PHYSICS FOR SCIENCE AND ENGINEERING. San Diego: Harcourt Brace Jovanovich, 1988. Well-written account of modern physics. Although calculus-based and aimed at a college physics audience, qualitative descriptions of basic quantum phenomena are understandable to the layperson. Illustrations are few, but well chosen.

Halliday, David, and Robert Resnick. FUNDAMENTALS OF PHYSICS. 3d rev. ed. New York: John Wiley & Sons, 1988. This edition of the classic undergraduate physics text contains highly descriptive chapters on modern physics including a thorough elementary description of quantum physics concepts, such as wave-particle duality. Illustrations and sample problems abound. Excellent questions for further thought.

Jammer, Max. THE PHILOSOPHY OF QUANTUM MECHANICS. New York: John Wiley & Sons, 1974. High-powered, but readable account of the philosophical implications of modern nonclassical thought about matter and energy.

Ohanian, Hans C. PHYSICS. New York: W. W. Norton, 1985. Although calculus-based, the text is not mathematically rigorous. Accessible to those with modest mathematics skills. Excellent descriptions of difficult physical concepts. Well illustrated; provides practical examples.

Weidner, Richard T., and Robert L. Sells. ELEMENTARY MODERN PHYSICS. Boston: Allyn & Bacon, 1985. Beautifully written explanation of modern physics concepts such as the development of nonclassical thought. Wave-particle duality is demonstrated through discourse of matter and energy interactions. Well illustrated. Mathematics level is adequate for the layperson as well as the physics student.

Wilson, Jerry D., and John Kinard. COLLEGE PHYSICS. Boston: Allyn & Bacon, 1990. Basic text excellent for the reader unfamiliar with calculus. More qualitative than rigorous. Well-illustrated and thorough description of basic concepts without resorting to advanced mathematics. Excellent for high school physics instruction. Young's double-slit experiment displayed the wave nature of light

Electrons and Atoms

The Interpretation of Quantum Mechanics