Nonrelativistic Quantum Mechanics

Type of physical science: Atomic physics

Field of study: Nonrelativistic quantum mechanics

The atomic and subatomic worlds do not obey the familiar principles of classical physics such as Newton's laws of motion. In their place, a whole new paradigm called quantum mechanics has been established. Quantum mechanics has become the cornerstone of the modern theory of matter, accounting for a vast range of phenomena in many branches of science.

Overview

Quantum mechanics is the system of laws that govern the behavior of matter on the atomic scale. It is the most successful theory in the history of science, having withstood thousands of experimental tests without a single verifiable exception. It has correctly predicted or explained phenomena in fields as diverse as particle physics, chemistry, solid-state electronics, molecular biology, and cosmology. A host of modern technological marvels, including transistors, lasers, computer chips, and nuclear reactors, are all offspring of the quantum theory.

To understand the modern theory of matter, conceptual hurdles of both psychological and mathematical variety must be overcome.

The prequantum, prerelativistic picture of the physical world that held sway until the end of the nineteenth century is known as classical physics. Classical physics was built on the formidable foundations of Newtonian mechanics, electromagnetic theory, thermodynamics, and statistical mechanics. Classical physics deals with particles and waves. Particles are the subject of classical mechanics; their behavior can be characterized by a trajectory, a path which a particle traces out in the course of its motion. Some physical variables that describe particles are position, velocity, momentum, and energy. Waves are the subject of electromagnetic theory, as well as optics, fluid mechanics, and acoustics. They represent excitations that are spread out in space and time and are described by such variables as frequency, wavelength, and propagation velocity.

Waves exhibit interference phenomena; that is, one wave can be either augmented or reduced, or even canceled out, by another wave interacting with it.

The laws of classical physics are determinate. Therefore, if one knows the initial configuration of a system of particles, one can unambiguously determine its state at any future time. This is impressively illustrated in celestial mechanics, which is Newtonian mechanics applied to astronomical phenomena. Eclipses of the sun and Moon, positions of the planets, reappearances of Halley's comet, and the like can be predicted with astounding accuracy many thousands of years in advance. According to this classical picture, humans live in what has been described as a "clockwork universe."

The variables that describe a classical system, be it particle or wave, can be measured as accurately as instruments permit. Also, any number of such variables can be measured simultaneously. Moreover, the results of such measurements will cover a continuous range of possible values.

In the seventeenth century, two competing theories on the nature of light arose. Sir Isaac Newton promoted the picture of light as a stream of particles. This seemed to support his experiments on breaking up white light into a spectrum of different colors. Christiaan Huygens, on the other hand, favored a wave model for light. The issue was settled in the early years of the nineteenth century, when interference phenomena involving light were demonstrated by Thomas Young and Augustin-Jean Fresnel. These phenomena could be explained only on the basis of the wave theory. In 1864, James Clerk Maxwell's magnificent unification of electricity, magnetism, and optics in his theory of electromagnetism (Maxwell's equations) appeared to add the finishing touches. Yet, in the last few years of the nineteenth century, phenomena were discovered that could not be satisfactorily accounted for by the wave theory. One of these phenomena was the photoelectric effect, whereby electrons are ejected from the surface of a metal by the action of light of a sufficiently high frequency. Evidently, light usually behaves as a wave phenomenon, but occasionally it betrays a particle-like aspect, a schizoid tendency that became known as the wave-particle duality. This paradox provided one of the stimuli that led to development of the quantum theory.

Another phenomenon that seemed to contradict classical ideas in both mechanics and electromagnetic theory was black body radiation, involving the emission of radiation from a hot object. In particular, classical theory predicted an "ultraviolet catastrophe," whereby radiation at higher frequencies (shorter wavelengths) ought to increase without limit.

Quantum theory was born in 1900, when Max Planck proposed his theory of black body radiation. He was led to the conclusion that light could be absorbed or emitted from a black body only in discrete packets or "quanta" of energy. Albert Einstein explained the photoelectric effect in 1905 by proposing further that these quanta were carried by light in the form of particles, which he named "photons." The Planck-Einstein formula relating the energy of a photon to the frequency of the corresponding light is given by E = hv, where E is the energy in joules, v is the frequency in hertz (cycles per second), and h is Planck's constant, a new fundamental constant of nature with the value 6.626 x 10^-34 joules per second. This explicit connection between energy, a particle-like property, and frequency, a wavelike property, lucidly portrays the wave-particle duality. The fact that Planck's constant is so minuscule by macroscopic standards accounts for the unobservability of quantum effects in everyday phenomena.

Concurrent with the development of quantum theory, the understanding of the structure of the atom also emerged. Antoine-Henri Becquerel discovered radioactivity in 1896, showing that the atom was not indestructible. Sir Joseph John Thomson identified the first subatomic particle, the electron, in 1897. Ernest Rutherford's experiments on the scattering of α particles from metal foils led him to the nuclear model of the atom in 1911. According to this model, almost all the mass of the atom is concentrated in a tiny, positively charged "nucleus" orbited by much lighter, negatively charged electrons, much like a miniature solar system. The diameter of the electron orbits, on the order of 10^-10 meters, establishes the typical size of an atom. The nucleus is ten thousand times smaller--on the order of 10^-15 meters in diameter--and is effectively a point mass from an atomic viewpoint.

In 1913, Niels Bohr developed his theory of the hydrogen atom. Hydrogen is the simplest element, its atoms consisting of one negatively charged electron orbiting one positively charged proton. The spectrum of atomic hydrogen consists of a series of discrete lines at certain frequencies, rather than a continuum over the entire range of frequency, as would be expected on the basis of classical physics. Bohr theorized that a hydrogen atom could have only one of a discrete set of allowed energy levels, which he pictured as orbits of the electron around the proton; that spectral lines originated from a jump of the atom from one orbit to another with the emission or absorption of a photon; and that the energy of the photon, equal to the energy difference between the electron orbits, was given by the Planck-Einstein formula. Bohr's pioneering achievement was the recognition that the energy levels of an atomic system were quantized. The reason they were quantized had to await later theoretical developments.

In 1924, Louis de Broglie speculated that matter might also exhibit a wave-particle duality. A wavelike aspect of the electron might, for example, be responsible for the discrete nature of Bohr orbits. According to de Broglie's hypothesis, the "matter waves" associated with a particle have a wavelength given by λ = h/p, where λ stands for wavelength, p is the momentum of the particle (mass times velocity), and h is Planck's constant.

The correctness of de Broglie's conjecture was most dramatically confirmed by the observations in 1927 by Clinton Joseph Davisson and Lester Germer in the United States and by George Paget Thomson (the son of Joseph John Thomson) in England of the diffraction of beams of electrons by metal foils, an interference effect that is characteristic of waves. Detailed measurements showed that de Broglie's formula did indeed give the correct wavelength.

Classical wave phenomena are governed by wave equations, which determine how the wave varies as a function of space and time coordinates. For example, Maxwell's equations govern the behavior of electromagnetic waves. In 1926, Erwin Schrodinger proposed an equation for de Broglie's matter waves. The Schrodinger equation marks the starting point of modern quantum mechanics.

(-ħ²/2m)∙[(∂²ψ/∂x²) + (∂²ψ/∂y²) + (∂²ψ/∂z²)] +Vψ = Eψ The key quantity in the Schrodinger equation is the wave function ψ, which describes the state of the quantum mechanical system in much the same way that the electric and magnetic fields describe the state of a classical electromagnetic system. In the above equation, E represents the energy of the quantum state. In contrast to the classical situation, only certain values of energy are allowed, often in the form of a discrete set of energy levels.

In his original paper, Schrodinger solved the equation for the hydrogen atom and recovered Bohr's formula for the quantized energy levels. These results now followed from a more rational and complete theory, in contrast to Bohr's ad hoc approach. Moreover, whereas the Bohr model failed for the helium atom (containing two electrons) and more complex systems, the Schrodinger equation appeared to be applicable. In 1928, an approximate solution of the Schrodinger equation for helium was worked out by Egils Hylleraas.

The wave function in the Schrodinger equation has no analogue in classical mechanics.

Max Born's interpretation of its physical significance is now accepted by most scientists.

According to Born, the square of the wave function, the quantity ψ², measures the probability of finding a particle at a given point in space at a given time. Implicit in this interpretation is the recognition that repeated measurements on an identical quantum system will, in general, give different results. In striking contrast to the clockwork universe of classical physics, quantum mechanics is evidently characterized by statistical laws. It has not been easy for scientists to give up habits of thought developed over hundreds of years, but given the perfect agreement of the quantum laws with experiment, there has been no reasonable alternative to acceptance of these aspects of quantum mechanics that are at odds with everyday experience and common sense.

In 1925, the year before the Schrodinger equation was proposed, Werner Heisenberg, along with Born and Ernst Pascual Jordan, had developed another version of quantum mechanics. In this theory, physical observables, such as position, momentum, and energy, were represented by "matrices," square arrays of numbers well known to mathematicians. A remarkable property of matrices is that they do not necessarily commute: This means that the product of P times Q is not necessarily equal to the product of Q times P. As a consequence of this, measurement of one observable, for example, position, will interfere with simultaneous measurement of another, such as momentum. This fundamental limitation on the ability to measure a quantum system is known as the Heisenberg uncertainty principle. (No such restriction applies in classical mechanics.) More specifically, the uncertainty principle can be written Δx Δp ≥ or = /4π, where Δx and Δp represent, respectively, the uncertainties in the measurement position and momentum. The product of these uncertainties is of the order of Planck's constant, which explains why this limitation on measurement plays no role in everyday life.

The two independent formulations of quantum theory, Schrodinger's wave mechanics and Heisenberg's matrix mechanics, were shown to be mathematically equivalent by Paul Adrien Maurice Dirac in 1926. Dirac synthesized an elegant formalism, known as transformation theory, which contained both wave and matrix mechanics as special cases.

The formulation of nonrelativistic quantum mechanics described above is applicable whenever particle velocities are small compared with the speed of light, approximately 3 x 10⁸ meters per second. If this is not the case, it is necessary to reformulate the equations of quantum mechanics to make them consistent with the requirements of special relativity. In 1928, Dirac first proposed the correct relativistic equation for the electron. Certain consequences of relativity are evident even under nonrelativistic conditions, most notably the existence of spin, an internal attribute of the electron and other elementary particles. Dirac's equation is in the field of relativistic quantum mechanics.

Applications

After 1928, quantum mechanics was applied with remarkable success to elucidate a vast range of phenomena involving atoms, molecules, and macroscopic matter. In one type of radioactive decay, a heavy element spontaneously emits α particles, which are helium nuclei, made up of two protons plus two neutrons. The original nucleus is thereby transmuted into a nucleus of the element two steps lower in the periodic table, with a mass number four units lower. Alpha-emitting isotopes generally have half-lives measured in thousands of years. The energy of the emitted α particles is typically in the range of 4 million electronvolts. Yet, in order to reverse the nuclear reaction, that is, to reattach an α particle to the product nucleus, typically requires around 30 million electronvolts. This discrepancy was a mystery for many years. In 1928, George Gamow showed that α particles have a certain probability of "tunneling" out of their containing nucleus even when their energies are apparently insufficient to do so. Tunneling is a uniquely quantum mechanical phenomenon having no analogue in classical physics.

Chemistry had developed as a coherent and productive science for more than two hundred years with the assistance of empirical rules and models that were able to account for a vast array of phenomena. One such example is the periodic table, which reveals regularities in the physical and chemical properties of the chemical elements; another example is the rule that a carbon atom can form four single bonds, with the geometry of a tetrahedron, thus explaining a vast amount of structural organic chemistry. With the advent of quantum mechanics, it became possible to actually "deduce" chemical principles from the fundamental laws of atomic structure.

In fact, Dirac in 1929 could claim that "the underlying physical laws necessary for the mathematical theory of . . . the whole of chemistry are thus completely known. . . ." This turned out to be a bit too optimistic since much of the mathematical theory was not susceptible to exact solution, but it does remain true that a majority of chemistry can be explained by quantum mechanics.

Solution of the Schrodinger equation for the hydrogen atom predicts a series of discrete states, each with a characteristic distribution of the electron's probability around the nucleus, what is known as an "orbital." In contrast to the precisely defined orbits in the Bohr model, inconsistent with the uncertainty principle, orbitals can be pictured as clouds of electron charge.

Atomic orbitals can be labeled by a set of four so-called quantum numbers. One of these quantum numbers refers to the electron spin, which has one of two possible values. Wolfgang Pauli proposed the exclusion principle in 1925, whereby no two electrons in an atom could share the same set of four quantum numbers. In other words, every orbital occupied by an electron must be different from every other one in at least one of the quantum numbers. The exclusion principle very elegantly accounts for the periodic structure of the elements. In concept, if one electron at a time could be added to an atom, while at the same time increasing the atomic number by one, the electron would go into the lowest-energy orbital available to it. Those orbitals which are already occupied are no longer available.

The existence of chemical bonding can likewise be accounted for by quantum mechanics. Walter Heitler and Fritz London showed in 1927 that two hydrogen atoms can link together to form a hydrogen molecule, with the chemical formula H2. A key aspect of this bonding mechanism is an exchange interaction, whereby an electron originally located on one of the atoms can, with equal probability, be associated with the other atom. This is again a purely quantum mechanical effect, for classical particles can, in principle, be given permanent labels. It also becomes clear why a chemical bond almost always involves a pair of electrons, as was proposed on empirical grounds by Gilbert Newton Lewis in 1916. Other atoms can form more than one chemical bond. Carbon, for example, can participate in four bonds and is said to have a valence of 4. The possible valences of an atom (more than one can exist) are determined by the occupancy of its orbitals and how its electrons can be rearranged to form bonding pairs. The geometric arrangement of bonds can also be deduced from the shapes of atomic orbitals. For example, if a carbon atom forms single bonds to four other atoms, the latter will lie at the corners of a tetrahedron.

Quantum mechanics has also been applied to the structure of solids. This has led to an understanding of the ways in which atoms are bonded together in various types of crystals. It can be explained why a diamond is so hard, why salt dissolves in water, why metals can be drawn into thin wires and why they conduct electricity, why other substances are insulators or semiconductors, and so on. It has become possible to apply theoretical principles to engineer new materials with desired characteristics. The most spectacular breakthrough of this nature was the invention of the transistor by John Bardeen, Walter H. Brattain, and William Shockley in 1947.

This has spawned the revolutions in microelectronics and computers, which will shape the world well into the twenty-first century.

Context

Classical mechanics, the laws of motion first enunciated by 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 Maxwell and by an understanding of statistical mechanics, the work of Maxwell and Ludwig Boltzmann. It appeared to many physicists that a comprehensive theory of their subject was nearly at hand. This turned out to be highly premature, however, for developments in the twentieth century were to undermine the foundations of classical physics completely. The quantum theory had its origin in 1900 with the work of Planck on black body radiation. Further advances came with Einstein's explanation of the photoelectric effect in 1905 and 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. Since that time, the quantum theory has become the cornerstone of the understanding of nature, revolutionizing the fields of physics, chemistry, material science, and molecular biology.

Principal terms

OBSERVABLE: a measurable property of a physical system, such as position, momentum, and energy

PARTICLE: a fundamental unit of matter, usually idealized as having negligible size and being located at a single point in space

PROBABILITY: a fraction expressing the likelihood of a certain occurrence

QUANTIZED: having only certain discrete values, such as the energy of an atom

SCHRODINGER EQUATION: the fundamental equation of nonrelativistic quantum mechanics; its solution gives the wave function for a quantum system

UNCERTAINTY: the intrinsic error in a physical measurement; cannot be reducible beyond a certain limit, according to quantum mechanics

WAVE: an extended excitation ranging over a region of empty space or in a material medium; for example, light, sound, and ocean waves

WAVE FUNCTION: the solution of Schrodinger's equation describing the state of a quantum mechanical system

Bibliography

French, A. P. AN INTRODUCTION TO QUANTUM MECHANICS. New York: W. W. Norton, 1978. A clearly written, introductory college-level text.

Gamow, George. MR. TOMPKINS IN PAPERBACK. Cambridge, England: Cambridge University Press, 1965. An entertaining series of stories showing what life might be like in an imaginary world in which quantum and relativistic effects were part of everyday experience.

Gribbin, John. IN SEARCH OF SCHRODINGER'S CAT. New York: Bantam Books, 1984. A popular account of the history of quantum mechanics. Discusses some alternative, unconventional interpretations of quantum phenomena.

Hey, Tony, and Patrick Walters. THE QUANTUM UNIVERSE. Cambridge, England: Cambridge University Press, 1987. A popular account of quantum physics making extensive use of pictures rather than mathematical formulas. Contains personal glimpses of the creators of the theory.

Pagels, Heinz R. THE COSMIC CODE. New York: Simon & Schuster, 1982. A lucidly written exposition for the general reader on quantum theory and other developments in modern physics, from the subatomic to the cosmic level.

Electrons and Atoms