Conserved quantities

Type of physical science: Atomic physics

Field of study: Nonrelativistic quantum mechanics

According to the laws of nature, certain physical quantities appear to be conserved in all processes that occur in an isolated system. Physicists depend particularly on conservation laws in order to interpret the behavior of atoms and subatomic particles.

Overview

Physicists frequently summarize their understanding of the operation of the physical world by saying that nature obeys certain laws. The laws developed by physicists for this purpose tend to fall into three general categories. Dynamical, or force, laws describe the interactions among particles as the result of a small number of well-defined forces which determine the changes of speed and direction of the particles' motion. Extremum laws state that physical systems behave in such a way that certain quantities are at a maximum or a minimum. Thus, light travels along the shortest distance between two points and an isolated system is in thermal equilibrium in its state of maximum entropy. Conservation laws are statements that certain well-defined quantities are not changed by physical processes, at least under certain circumstances. The different types of physical law are not independent of one another. In macroscopic systems, for example, Newton's laws of motion imply the conservation laws for linear momentum and angular momentum in general, the conservation law for energy (with some restrictions) and the validity of the extremum principle of least action. The different types of law have different degrees of usefulness, however, in different areas of physics. Conservation laws are particularly important in the atomic and subatomic realm.

The conserved quantities of importance in atomic physics include the classical quantities of linear and angular momentum, energy, and electric charge, as well as at least six quantities only encountered in connection with subatomic particles and several additional quantities that are conserved by the strong nuclear force but not by the weak nuclear force. In classical mechanics, the linear momentum of a particle is the product of its mass and velocity.

Linear momentum is a vector quantity, described by a magnitude and a direction. For an isolated physical system--that is, a system far away from any other particles that might exert a force on it--the total linear momentum or sum of the linear momenta for all the particles is conserved. For particles moving at high speed, the linear momentum must be defined in a way which takes into account the dependence of mass on speed but still obeys the momentum conservation principle.

Angular momentum is a measure of the rate at which the particles of a system are revolving about one another. For one particle revolving about another in a circle with constant speed, the angular momentum is equal to the product of the orbit radius and the linear momentum of the particle in its orbit. Angular momentum is treated as a vector quantity with a direction along the axis of the orbit. For an isolated collection of particles, the total angular momentum (the sum of all the individual particles' angular momenta) is conserved.

In classical mechanics, the energy of an isolated system of particles can be defined as the sum of the kinetic energy of the particles and the potential energy that is associated with the arrangement in space of the objects that make up the system (rest mass). The classical energy thus defined is not necessarily conserved, as forces such as friction can transform kinetic energy into heat and processes such as radioactive decay can convert the rest mass into kinetic energy or electromagnetic radiation. By generalizing the definition of mechanical energy to include heat energy, chemical energy, the energy of any electromagnetic fields, and mass energy, a quantity which is conserved in all known circumstances is obtained.

Physicists' discussions of conservation laws are often focused on the notion of symmetry in nature. An object is said to possess symmetry if something can be done to it that results in a state which is indistinguishable from the original state. A square piece of blank paper has symmetry because it can be rotated by 90 degrees, 180 degrees, or 270 degrees and still appear the same to an observer. A circle has a higher degree of symmetry because it can be rotated about its center at any angle and maintain the same appearance. Each of the classical conservation principles can be related to one of the fundamental assumptions made about the symmetry of space or time in classical physics. The conservation of linear momentum is related to the assumed homogeneity of space--the fact that a physical system ought to behave in the same way no matter where it is located. The conservation of angular momentum is a necessary consequence of the assumed isotropy of space--the fact that rotating all the parts of a physical system through the same angle about a fixed axis should not have any effect on that system. The conservation of energy, in the absence of forces such as friction, is a consequence of the uniform nature of time--the fact that the behavior of a physical system prepared in a given well-defined state depends only on the time elapsed.

Compared to the classical conservation laws, the conservation of electrical charge has a special character. From the classical standpoint, whenever electrical charge is created or destroyed, equal amounts of positive and negative charge either appear or disappear. From an atomic standpoint, the conservation of charge is a consequence of the fact that charged subatomic particles carry electrical charges of the same size and that the individual particles engage in processes only if the total amount of electrical charge is conserved. Quantum mechanics suggests a way in which the conservation of charge is related to symmetry in nature.

In quantum mechanics, the state of a charged particle is described by a mathematical function, called the wave function, which assigns a complex number to every point in space. The wave function is a solution of the Schrodinger wave equation, a differential equation which describes the forces acting on the particle as the result of electric or magnetic fields. In the Schrodinger equation, the electric and magnetic fields appear not directly but in the form of scalar and vector potential functions V and A, respectively, from which the fields may be derived. The electric and magnetic fields are themselves solutions to Maxwell's equations, a set of differential equations with an important but subtle symmetry property called gauge invariance.

In essence, a gauge transformation is a change in the scalar and vector potential functions (such as from V₁ and A₁ to V₂ and A₂) which does not affect the electric and magnetic fields derived from them, and gauge invariance refers to the mathematical properties of Maxwell's equations that allow gauge transformations to be possible.

The solution to the Schrodinger equation, which contains the potentials V₁ and A₁, can be converted into the corresponding solution of the equation that contains the potential V₂ and A₂ by multiplying the original solution by a function which involves the difference in the potentials and the elementary unit of charge. The two different Schrodinger equations describe exactly the same physical situation--that is, the same electric and magnetic fields--and must describe the same physical behavior and unit of electric charge. These requirements are related to the ability to make a gauge transformation without changing the electric or magnetic forces acting in a system.

Applications

Conservation laws are used by molecular and atomic physicists as a guide to the interpretation of experiments. One of the most useful conservation laws in atomic and molecular spectroscopy is that of the conservation of parity. Parity refers to the behavior of a wave function under an inversion of the coordinate system, that is, turning the x, y, and z coordinates of each point into -x, -y, and -z. Parity conservation is related to an assumed symmetry of space under reflection and, as a result, the mirror image of any allowed physical process is also considered to be an allowed physical process. Wave functions for the quantum mechanical states of the nucleus and the electrons in an atom can be classified as being either of even parity or of odd parity, Even-parity wave functions are unchanged under an inversion of the coordinate system (taking the center of the nucleus as the origin), while odd-parity wave functions are multiplied by -1 under the same transformation.

The parity of a multi-particle wave function is found by multiplying the parities of the wave functions that are associated with the individual particles. Therefore, the wave function for a two-particle system would have even parity if both of the particles were in states of odd parity or even parity. The wave function would have odd parity, however, if one particle were in a state of even parity and the other in a state of odd parity. When an atom or nucleus emits or absorbs energy in the form of a single photon, the photon can be treated as being in a state of odd parity, so that the atom or nucleus must make a transition to a state of the opposite parity. Thus, the conservation of parity leads to a selection rule which allows one to rule out certain transitions when interpreting the spectrum of an atom or nucleus.

Physicists make use of conserved quantities to organize the experimental information on families of subatomic particles. It is generally agreed that the known subatomic particles fall into four natural groupings. The photon, the quantum of light or electromagnetic energy, is in a group by itself and interacts by the electromagnetic force only. The number of photons is not conserved, and photons can be created or destroyed by a wide range of processes in which the energy and momentum of the photon are transferred to other particles. The electron is a member of the lepton family, which includes the muon, the τ particle, the positron (or antielectron), the antimuon, the antitau, and three pairs of massless particles called neutrinos (the electron neutrino and electron antineutrino, the muon neutrino and muon antineutrino, and the τ neutrino and tau antineutrino). Leptons interact with other particles through the electromagnetic interaction and through the short-range weak nuclear force. In order to explain the reactions that occur within the lepton family, physicists have proposed the existence of three separate conserved quantities: the electron number, the muon number, and the τ particle number. The electron and electron neutrino have an electron number of 1, the antielectron and electron antineutrino have an electron number of -1, and all other particles have an electron number of 0. Thus, an electron and positron together can be transformed entirely into γ radiation since their total electron number is 0.

Likewise, a neutron cannot decompose into an electron and a proton without the simultaneous emission of an electron antineutrino. A muon decays to form an electron, a muon neutrino, and an electron antineutrino, thus conserving a muon number of 1 and an electron number of 0.

The largest grouping of subatomic particles is the hadron group, which includes the meson family and the baryon family. Particles within these families interact with one another through the strong nuclear force as well as through electromagnetic and weak nuclear interactions. Mesons are unstable and decay to form leptons and photons. The lowest mass baryon is the proton, which is either absolutely stable or has a lifetime which is much longer than the current age of the universe. In order to account for the stability of the proton, physicists have assigned a baryon number of 1 to each of the baryons and a baryon number of -1 to each of their antiparticles. Thus, a proton and antiproton can annihilate each other to form photons or pairs of leptons, but the proton, by itself, cannot decay into any smaller particle. Baryons of greater mass than the proton are unstable and, when isolated, will decay to form a proton and groups of lighter particles. In order to explain the variation in baryon lifetimes, physicists have introduced the notion of additional quantities that are conserved under the strong nuclear interaction but not under the weak interaction. The first such quantity was termed "strangeness" and the second, "charm." Baryons may carry up to three units of strangeness or three units of charm. In the quark model for hadron structure, each of these quantities is carried by a separate particle, called a quark, and each baryon is composed of three quarks, thus allowing for up to three units, positive or negative, of strangeness or charm. Most physicists believe that each of these conserved quantities is related to a symmetry of the gauge invariance type.

Context

The conservations of linear momentum and angular momentum are direct consequences of Newton's laws of motion and thus have been understood at least implicitly since the seventeenth century. The idea of energy as a conserved quantity could not be developed until the equivalence of heat energy and mechanical energy was demonstrated by the English physicist James P. Joule in the 1840's. The principle of the conservation of energy is attributed both to Joule and to the German physician Julius von Mayer, who is thought to have been led to the idea by his observations of the metabolism of sailors in different parts of the world. At several times, scientists have thought that newly discovered atomic phenomena might violate the conservation of energy, which is now considered to be one of the most firmly established law of physics. For example, with the discovery of radioactivity by the French physicist Antoine-Henri Becquerel, no one could imagine a process which could account for the energy of the particles emitted.

Albert Einstein later showed, however, that a small decrease in the rest mass of the radioactive atom was sufficient to account for the energy of the emitted particles and the energy of γ rays.

The discovery of β decay presented an even greater challenge. In this process, a neutron within an atomic nucleus breaks into a proton, which remains in the nucleus, and an electron, which is ejected. Instead of being ejected with a fixed kinetic energy, however, as would have been required by the simultaneous conservation of energy and momentum, the electrons emerged with kinetic energies that varied over a considerable range. The solution to this puzzle was proposed by the German physicist Wolfgang Pauli, who suggested in 1931 that a second particle was also emitted, one with a mass smaller than the electron and which interacted with matter only weakly. The Italian physicist Enrico Fermi gave this particle the name "neutrino" to distinguish it from the much heavier nuetron that was discovered in 1932. The neutrino must be a particle with no electrical charge and no rest mass which interacts with other particles only by the weak nuclear force. Because of their extremely weak interactions with ordinary matter, neutrinos were not identified in elementary particle experiments until the 1950's.

The discovery of neutrinos confirmed the principle of energy conservation.

The relationship between the classical conservation laws and symmetry was demonstrated by the German mathematician Emmy Noether, one of the first women to receive an advanced degree, in 1904, through the German university system. The concept of gauge invariance was proposed by Fritz London in 1927. In 1954, theoretical physicists Chen Ning Yang and Robert Mills published a theory of the force between baryons based on a form of gauge invariance. In later years, the idea that all conserved quantities in the subatomic realm are related to symmetries of the gauge invariance type gained widespread acceptance among particle physicists.

In the area of conserved quantities, perhaps the greatest surprise in the second half of the twentieth century was the discovery that parity is in fact not conserved in weak interactions.

In 1956, Yang and Tsung-Dao Lee published a paper stating their reasons for believing that conservation parity might not be obeyed by the weak interaction. They proposed that an experiment involving the β decay of cobalt nuclei with an atomic weight of sixty would reveal the effect. By aligning the direction of spin, and thus the magnetic moments, of the cobalt 60 nuclei, it was possible to determine whether the electrons were emitted more often from the north or the south magnetic pole. Since the mirror image of a spinning object is spinning in the opposite direction, conservation of parity would require an equal probability of emission from either magnetic pole. Their experiment showed that parity was not conserved. Yang and Lee were honored for their remarkable proposal with the 1957 Nobel Prize in Physics.

Principal terms

ANGULAR MOMENTUM: a measurement of the total rotational motion of a system; measured in units of mass multiplied by length squared per unit time and assigned a direction along the axis of rotational motion

BARYON FAMILY: the largest family of subatomic particles, which includes the proton, the neutron, and the majority of heavier unstable particles; baryons participate in the strong and weak nuclear interactions

CHARGE: the property of a particle or object that determines its reaction to other electrically charged particles and to the electromagnetic field

CHARM: a quantum mechanical property of baryons and mesons which is conserved in strong nuclear interactions but not in weak interactions

ENERGY: a measure of the total physical work that can be obtained from a system; most frequently measured in electronvolts when describing atoms and subatomic particles

LEPTON FAMILY: a grouping of the lightest subatomic particles, which includes the electron, the muon, the τ particle, and the associated neutrinos and their antiparticles; leptons do not participate in the strong nuclear interaction

LINEAR MOMENTUM: a measure of the total motion of a system of particles, measured in units of mass times velocity

STRANGENESS: a quantum mechanical property of baryons and mesons which is conserved in strong nuclear interactions but not in weak interactions

STRONG NUCLEAR FORCE: the short range interaction between protons, neutrons, and other baryons resulting from the exchange of mesons; the force responsible for the binding of protons and neutrons in a nucleus

WEAK NUCLEAR FORCE: a short range interaction between subatomic particles, much weaker than the strong or electromagnetic forces, which is responsible for beta decay and related processes

Bibliography

Amaldi, Ginestra. THE NATURE OF MATTER. Chicago: University of Chicago Press, 1961. This relatively brief and very readable volume covers the development of atomic theory from ancient Greek speculation to the subatomic "zoo" of the early 1960's. The development of conservation laws for subatomic particles is one of the major themes in this work.

Asimov, Isaac. ASIMOV'S NEW GUIDE TO SCIENCE. New York: Basic Books, 1984. Chapters 6 through 9 of this comprehensive, readable volume by the prolific science writer describes the development of modern atomic theory and its applications to chemistry and technology.

Boorse, Henry A., and Lloyd Motz. THE WORLD OF THE ATOM. New York: Basic Books, 1966. This two-volume compilation of landmark papers in the history of atomic theory includes portions of the original papers by many of the major contributors to the modern understanding of atomic structure. The last section includes Chen Ning Yang and Tsung-Dao Lee's work on the nonconservation of parity.

Crease, Robert P., and Charles C. Mann. THE SECOND CREATION. New York: Macmillan, 1986. An informal biography of the personalities and ideas of twentieth century physics, presented largely as an attempt to identify the symmetries and conservation laws for subatomic particles.

Davies, P. C. W. THE FORCES OF NATURE. Cambridge, England: Cambridge University Press, 1986. This well-written book provides an overview of the behavior of electrons and other elementary particles in nontechnical language and using a bare minimum of mathematics.

Feynman, Richard P. THE CHARACTER OF PHYSICAL LAW. Cambridge, Mass.: MIT Press, 1965. This compact volume presents a number of lectures which were presented to a general audience by one of the leading physicists of the twentieth century. Feynman discusses the character of physical laws in general and conservation laws in particular.

Feynman, Richard P., Robert B. Leighton, and Matthew Sands. THE FEYNMAN LECTURES ON PHYSICS. 3 vols. Reading, Mass.: Addison-Wesley, 1963-1965. This comprehensive set of lectures by one of the leading theoretical physicists of the twentieth century is an attempt to convey both modern and classical physics to beginning university students. Feynman's introductory lectures provide an easy-to-follow discussion of the role of conservation laws in physics.

Gamow, George. THIRTY YEARS THAT SHOOK PHYSICS. New York: Dover, 1985. Gamow, a distinguished physicist, took time from his professional pursuits to write several books for the general reader. Gamow's personal acquaintance with Wolfgang Pauli, Enrico Fermi, Niels Bohr, and other leading investigators of the subatomic realm adds much to his description of their discoveries.

Pais, Abraham. INWARD BOUND. New York: Oxford University Press, 1986. Also by a prominent physicist, an in-depth but readable history of elementary particle physics from the initial discovery of radioactivity until the mid-1980's.

Electrons and Atoms

Group Theory and Elementary Particles

Leptons and the Weak Interaction

Nuclear Reactions and Scattering

Quarks and the Strong Interaction