Schrödinger, Erwin

Schrödinger, Erwin (1887-1961), Austrian physicist and Nobel laureate. Schrödinger formulated the theory of wave mechanics, which describes the behavior of the tiny particles that make up matter in terms of waves. Schrödinger formulated the Schrödinger wave equation to describe the behavior of electrons (tiny, negatively charged particles) in atoms. For this achievement, he was awarded the 1933 Nobel Prize in physics with British physicist Paul Dirac and German physicist Werner Heisenberg, who also made important advances in the theory of atomic structure. See also Quantum Theory; Atom.

Schrödinger was born in Vienna, Austria. His father was an oilcloth manufacturer who had studied chemistry, and his mother was the daughter of a chemistry professor. He attended an elementary school in Innsbruck for a few weeks, but Schrödinger received most of his early education from a private tutor. In 1898 he entered the Gymnasium in Vienna, where he studied mathematics, physics, and ancient languages. He then attended the University of Vienna from 1906 to 1910, specializing in physics. Schrödinger obtained his doctoral degree in physics in 1910. After a year in military training, he returned to the university to teach a first-year physics laboratory class. His early research ranged over many topics in experimental and theoretical physics.

During World War I (1914-1918) Schrödinger served as an artillery officer and then returned to his previous post at Vienna. Conditions were difficult in Austria after the war, and in 1920 Schrödinger decided to go to Germany. After a series of short-lived posts at the University of Jena, Stuttgart University, and the University of Breslau (now Wrocław, Poland) in 1920 and 1921, he became a professor of physics at the University of Zürich in Switzerland in 1921.

Schrödinger's most important work was done at Zürich, and his work received much attention. He succeeded German physicist Max Planck as professor of theoretical physics at the University of Berlin in 1927. Schrödinger remained there until the rise of the National Socialist (Nazi) movement in 1933, when he went to the University of Oxford in England. There he became a fellow of Magdalen College. Homesick, he returned to Austria in 1936 to take up a post at Graz University, but the Nazi takeover of Austria in 1938 placed Schrödinger in danger. Schrödinger was not Jewish, but his opposition to Nazi policies made him a potential target. The prime minister of Ireland, Eamon de Valera, helped Schrödinger get out of Austria. De Valera’s help also led to an appointment to a post at the Institute for Advanced Studies in Dublin in 1939. Schrödinger continued work in theoretical physics in Dublin until 1956, when he returned to Austria to a chair at the University of Vienna. He stayed at the University of Vienna until his death.

Schrödinger's great discovery of wave mechanics originated with the work of French physicist Louis de Broglie. In 1923 de Broglie used ideas from German American physicist Albert Eintstein’s special theory of relativity to show that an electron, or any other particle, has a wave associated with it (see Albert Einstein). De Broglie’s work resulted in the equation λ = h/p, where λ is the wavelength of the associated wave, h is a number called Planck's constant, and p is the momentum of the particle. Physicists immediately deduced that if particles (particularly electrons) have waves, then a particular type of partial differential equation known as a wave equation should be able to describe their behavior. These ideas were taken up by both de Broglie and Schrödinger, and in 1926 each published the same wave equation. Unfortunately, while the equation is true, it was of very little help in explaining the behavior of particles.

Later the same year Schrödinger used a new approach. He studied the mathematics of partial differential equations and the Hamiltonian function, a powerful idea in mechanics developed by British mathematician Sir William Rowan Hamilton in the mid-1800s. Schrödinger formulated an equation in terms of the energy of the electron and the energy of the electric field in which it was situated. Partial differential equations have many solutions, but solutions to Schrödinger’s equation had to meet strict conditions to be useful in describing the electron. Among other things, they had to be finite and possess only one value. These solutions were associated with special values of the electron’s energy level, known as proper values or eigenvalues.

Schrödinger solved the equation for the hydrogen atom, V = –e²/r, in which V is the energy of the electric field surrounding the electron, e is the electron's charge, and r is its distance from the atom’s nucleus. He found that the eigenvalues of the electron’s energy corresponded with those of the energy levels given in the older theory of Danish physicist Niels Bohr. Bohr’s theory of the atom described electrons orbiting atoms in strict circular orbits at particular distances that corresponded to specific levels of energy. In the hydrogen atom (which consists of one electron and one proton), the wave function Schrödinger derived instead describes where physicists are most likely to find the electron. The electron is most likely to be where Bohr predicted it to be, but it does not follow a strictly circular orbit. The electron is described by the more complicated notion of an orbital—a region in space where the electron has varying degrees of probability of being found.

Schrödinger's wave equation can describe atoms other than hydrogen as well as molecules and ions (atoms or molecules with electric charge), but such cases are very difficult to solve. In a few such cases physicists have found approximate solutions, usually with a computer carrying out the numerical work.

Schrödinger's mathematical description of electron waves found immediate acceptance. The mathematical description matched what scientists had learned about electrons by observing them and their effects. In 1925, a year before Schrödinger published his results, German-British physicist Max Born and German physicist Werner Heisenberg developed a mathematical system called matrix mechanics. Matrix mechanics also succeeded in describing the structure of the atom, but it was totally theoretical. It gave no picture of the atom that physicists could verify observationally. Schrödinger's vindication of de Broglie's idea of electron waves immediately overturned matrix mechanics, though later physicists showed that wave mechanics is equivalent to matrix mechanics.

During his later years Schrödinger became increasingly worried by the uncertain nature of quantum mechanics, of which wave mechanics is a part. Schrödinger believed he had produced a defining description of the atom in the same way that the three laws of English physicist Isaac Newton defined classical mechanics and the way that the equations of British physicist James Clerk Maxwell described electrodynamics. Instead, each new discovery about the structure of the atom only made atomic structure more complicated. Much of Schrödinger’s later work was concerned with philosophy, particularly as applied to physics and the atom.

Schrödinger, Erwin

b. Aug. 12, 1887, Vienna, Austria

d. Jan. 4, 1961, Vienna

Austrian theoretical physicist who contributed to the wave theory of matter and to other fundamentals of quantum mechanics. He shared the 1933 Nobel Prize for Physics with the British physicist P.A.M. Dirac.

Schrödinger entered the University of Vienna in 1906 and obtained his doctorate in 1910, upon which he accepted a research post at the university's Second Physics Institute. He saw military service in World War I and then went to the University of Zürich in 1921, where he remained for the next six years. There, in a six-month period in 1926, at the age of 39, a remarkably late age for original work by theoretical physicists, he produced the papers that gave the foundations of quantum wave mechanics. In those papers he described his partial differential equation that is the basic equation of quantum mechanics and bears the same relation to the mechanics of the atom as Newton's equations of motion bear to planetary astronomy. Adopting a proposal made by Louis de Broglie in 1924 that particles of matter have a dual nature and in some situations act like waves, Schrödinger introduced a theory describing the behaviour of such a system by a wave equation that is now known as the Schrödinger equation. The solutions to Schrödinger's equation, unlike the solutions to Newton's equations, are wave functions that can only be related to the probable occurrence of physical events. The definite and readily visualized sequence of events of the planetary orbits of Newton is, in quantum mechanics, replaced by the more abstract notion of probability. (This aspect of the quantum theory made Schrödinger and several other physicists profoundly unhappy, and he devoted much of his later life to formulating philosophical objections to the generally accepted interpretation of the theory that he had done so much to create.)

In 1927 Schrödinger accepted an invitation to succeed Max Planck, the inventor of the quantum hypothesis, at the University of Berlin, and he joined an extremely distinguished faculty that included Albert Einstein. He remained at the university until 1933, at which time he reached the decision that he could no longer live in a country in which the persecution of Jews had become a national policy. He then began a seven-year odyssey that took him to Austria, Great Britain, Belgium, the Pontifical Academy of Science in Rome, and--finally in 1940--the Dublin Institute for Advanced Studies, founded under the influence of Premier Eamon de Valera, who had been a mathematician before turning to politics. Schrödinger remained in Ireland for the next 15 years, doing research both in physics and in the philosophy and history of science. During this period he wrote What Is Life? (1944), an attempt to show how quantum physics can be used to explain the stability of genetic structure. Although much of what Schrödinger had to say in this book has been modified and amplified by later developments in molecular biology, his book remains one of the most useful and profound introductions to the subject. In 1956 Schrödinger retired and returned to Vienna as professor emeritus at the university.

Of all of the physicists of his generation, Schrödinger stands out because of his extraordinary intellectual versatility. He was at home in the philosophy and literature of all of the Western languages, and his popular scientific writing in English, which he had learned as a child, is among the best of its kind. His study of ancient Greek science and philosophy, summarized in his Nature and the Greeks (1954), gave him both an admiration for the Greek invention of the scientific view of the world and a skepticism toward the relevance of science as a unique tool with which to unravel the ultimate mysteries of human existence. Schrödinger's own metaphysical outlook, as expressed in his last book, Meine Weltansicht (1961; My View of the World), closely paralleled the mysticism of the Vedanta.

Because of his exceptional gifts, Schrödinger was able in the course of his life to make significant contributions to nearly all branches of science and philosophy, an almost unique accomplishment at a time when the trend was toward increasing technical specialization in these disciplines.

Schrödinger equation

the fundamental equation of the science of submicroscopic phenomena known as quantum mechanics. The equation, developed (1926) by the Austrian physicist Erwin Schrödinger, has the same central importance to quantum mechanics as Newton's laws of motion have for the large-scale phenomena of classical mechanics.

Essentially a wave equation, the Schrödinger equation describes the form of the probability waves (or wave functions [see de Broglie wave]) that govern the motion of small particles, and it specifies how these waves are altered by external influences. Schrödinger established the correctness of the equation by applying it to the hydrogen atom, predicting many of its properties with remarkable accuracy. The equation is used extensively in atomic, nuclear, and solid-state physics.

Schrödinger's wave mechanics

from quantum mechanics

Schrödinger's wave mechanics

Schrödinger expressed Broglie's hypothesis concerning the wave behaviour of matter in a mathematical form that is adaptable to a variety of physical problems without additional arbitrary assumptions. He was guided by a mathematical formulation of optics, in which the straight-line propagation of light rays can be derived from wave motion when the wavelength is small compared to the dimensions of the apparatus employed. In the same way, Schrödinger set out to find a wave equation for matter that would give particle-like propagation when the wavelength becomes comparatively small. According to classical mechanics, if a particle of mass m_e is subjected to a force such that its potential energy is V(x,y,z) at position x,y,z, then the sum of V(x,y,z) and the kinetic energy p²/2m_e is equal to a constant, the total energy E of the particle. Thus,

It is assumed that the particle is bound--i.e., confined by the potential to a certain region in space because its energy E is insufficient for it to escape. Since the potential varies with position, two other quantities do also: the momentum and, hence, by extension from the Broglie relation, the wavelength of the wave. Postulating a wave function (x,y,z) that varies with position, Schrödinger replaced p in the above energy equation with a differential operator that embodied the Broglie relation. He then showed that satisfies the partial differential equation

This is the (time-independent) Schrödinger wave equation of 1926, which established quantum mechanics in a widely applicable form. An important advantage of Schrödinger's theory is that no further arbitrary quantum conditions need be postulated. The required quantum results follow from certain reasonable restrictions placed on the wave function--for example, that it should not become infinitely large at large distances from the centre of the potential.

Schrödinger applied his equation to the hydrogen atom, for which the potential function, given by classical electrostatics, is proportional to -e²/r, where -e is the charge on the electron. The nucleus (a proton of charge e) is situated at the origin, and r is the distance from the origin to the position of the electron. Schrödinger solved the equation for this particular potential with straightforward, though not elementary, mathematics. Only certain discrete values of E lead to acceptable functions . These functions are characterized by a trio of integers n, l, m, termed quantum numbers. The values of E depend only on the integers n (1, 2, 3, etc.) and are identical with those given by the Bohr theory. The quantum numbers l and m are related to the angular momentum of the electron; l(l + 1) is the magnitude of the angular momentum, and m is its component along some physical direction.

The square of the wave function, ², has a physical interpretation. Schrödinger originally supposed that the electron was spread out in space and that its density at point x,y,z was given by the value of ² at that point. Almost immediately Born proposed what is now the accepted interpretation--namely, that ² gives the probability of finding the electron at x,y,z. The distinction between the two interpretations is important. If ² is small at a particular position, the original interpretation implies that a small fraction of an electron will always be detected there. In Born's interpretation, nothing will be detected there most of the time, but, when something is observed, it will be a whole electron. Thus, the concept of the electron as a point particle moving in a well-defined path around the nucleus is replaced in wave mechanics by clouds that describe the probable locations of electrons in different states.