Schrödinger was born in Vienna on 12 August 1887. His father was an oilcloth manufacturer who had studied chemistry and his mother was the daughter of a chemistry professor. Apart from a few weeks when he attended an elementary school in Innsbruck, Schrödinger received his early education from a private tutor. In 1898 he entered the Gymnasium in Vienna where he enjoyed mathematics, physics, and ancient languages. He then attended the University of Vienna, specializing in physics. Schrödinger obtained his doctorate in 1910 and a year later he became an assistant in the university's Second Physics Institute. His early research ranged over many topics in experimental and theoretical physics.

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

Schrödinger's most productive work was done at Zürich and it resulted in his succeeding Max Planck as professor of theoretical physics at Berlin in 1927. He remained there until the rise of the Nazis in 1933, when Schrödinger went to Oxford, England, where he became a fellow of Magdalen College. Homesick, he returned to Austria in 1936 to take up a post at Graz, but the Nazi takeover of Austria in 1938 placed Schrödinger in danger. The intervention of the prime minister of Ireland, Éamon de Valera (1882–1975), led to his appointment in 1939 to a post at the Institute for Advanced Studies in Dublin. Schrödinger continued work in theoretical physics there until 1956, when he returned to Austria to a chair at the University of Vienna. In the following year Schrödinger suffered a severe illness from which he never fully recovered. He died in Vienna on 4 January 1961.

The origin of Schrödinger's great discovery of wave mechanics began with the work of Louis de Broglie, who, in 1924, using ideas from Albert Einstein's special theory of relativity, showed that an electron or any other particle has a wave associated with it. The fundamental result was that

λ = h/p

where λ is the wavelength of the associated wave, h is Planck's constant, and p is the momentum of the particle. An immediate deduction from this discovery was that if particles, and particularly electrons, have waves then – like sound and other kinds of waves – their behaviour should be capable of description by a particular type of partial differential equation known as a wave equation. These ideas were taken up by both de Broglie and Schrödinger and in 1926 each published the same wave equation which, when written in relativistic terms, is:

where ψ is the wave function, t is the time, m is the mass of the electron, c is the velocity of light, h is Planck's constant, and x, y, and z represent the position of the electron in Cartesian coordinates. Unfortunately, while the equation is true, it was of very little help in developing further facts and explanations.

Later the same year, however, Schrödinger used a new approach. After spending some time studying the mathematics of partial differential equations and using the Hamiltonian function, a powerful idea in mechanics due to William Rowan Hamilton, he formulated an equation in terms of the energies of the electron and the field in which it was situated. His new equation was:

where E is the total energy of the electron and V is the potential of the field in which the electron is moving. This equation neglects the small effects of special relativity. Partial differential equations have many solutions and very stringent conditions had to be fulfilled by the individual solutions of this equation in order for it 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 E, known as proper values or eigenvalues. Schrödinger solved the equation for the hydrogen atom, where:

V = −e²/r

e being the electron's charge and r is its distance from the nucleus, and found that the values of E corresponded with those of the energy levels given in the older theory of Niels Bohr. Also, to each value of E there corresponded a finite number of particular solutions for the wave function ψ, and these could be associated with lines in the spectrum of atomic hydrogen. In the hydrogen atom the wave function describes where we can expect to find the electron, and it turns out that while it is most likely to be where Bohr predicted it to be, it does not follow a circular orbit but is described by the more complicated notion of an orbital, a region in space where the electron can be found with varying degrees of probability.

Atoms other than hydrogen and also molecules and ions can be described by Schrödinger's wave equation but such cases are very difficult to solve. In certain cases approximations have been used, usually with the numerical work being carried out on a computer.

Schrödinger's mathematical description of electron waves found immediate acceptance because these waves could be visualized as standing waves around the nucleus. In 1925, a year before Schrödinger published his results, a mathematical system called matrix mechanics, developed by Max Born and Werner Heisenberg had also succeeded in describing the structure of the atom but it was totally theoretical and gave no picture of the atom. Schrödinger's vindication of de Broglie's picture of electron waves immediately overturned matrix mechanics, though it was later shown that wave mechanics is equivalent to matrix mechanics.

During his later years, Schrödinger became increasingly worried by the way quantum mechanics, of which wave mechanics is a part, was interpreted, in particular with the probabilistic nature of the wave function. Schrödinger believed he had given a great description to the atom in the same way that Isaac Newton's laws described mechanics and James Clerk Maxwell's equations described electrodynamics, only to find that the structure of the atom became increasingly more difficult to describe explicitly with each new discovery. Much of his later work was concerned with philosophy, particularly as applied to physics and the atom.

Schrödinger made a fundamental contribution to physics in finally producing a solid mathematical explanation of the quantum theory first advanced by Planck in 1900, and the subsequent structures of the atom formulated by Bohr and de Broglie.

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