(1. Hilbert Space The words "Hilbert space" here will alwa...)
1. Hilbert Space The words "Hilbert space" here will always denote what math ematicians call a separable Hilbert space. It is composed of vectors each with a denumerable infinity of coordinates ql' q2' Q3, .... Usually the coordinates are considered to be complex numbers and each vector has a squared length ~rIQrI2. This squared length must converge in order that the q's may specify a Hilbert vector. Let us express qr in terms of real and imaginary parts, qr = Xr + iYr' Then the squared length is l:.r(x; + y;). The x's and y's may be looked upon as the coordinates of a vector. It is again a Hilbert vector, but it is a real Hilbert vector, with only real coordinates. Thus a complex Hilbert vector uniquely determines a real Hilbert vector. The second vector has, at first sight, twice as many coordinates as the first one. But twice a denumerable in finity is again a denumerable infinity, so the second vector has the same number of coordinates as the first. Thus a complex Hilbert vector is not a more general kind of quantity than a real one.
Dirac received his early education in Bristol and in 1921 graduated from Bristol University with first-class honors in electrical engineering.
For the next two years he studied mathematics at Bristol and completed the equivalent of a second bachelor's degree.
In 1923 he entered the University of Cambridge as a research student under the direction of Ralph Fowler, who opened for him new horizons in atomic physics.
In the late spring of 1926 he was awarded the Ph. D. for his work.
Dirac's first remarkable contribution along kinetic energy came before he earned his doctorate in 1926.
In his paper "The Fundamental Equations of Quantum Mechanics" (1925), Dirac decided to extricate the fundamental point in Heisenberg's now famous paper. Dirac also obtained the same formula, but through a more fundamental approach to the problem.
Dirac's crucial insight consisted in finding that a very simple operation formed the basis of the formula in question.
He taught physics at the University of Wisconsin in Madison and gave lectures on quantum mechanics at other American universities.
Dirac showed the connection between these statistical properties and the antisymmetry of the wave function in quantum mechanics. The Dirac Delta.
In 1927 Dirac showed that the quantum mechanics of Heisenberg and the wave mechanics of SchrödingerSchrodinger were not at odds with each other or with his own formulation; rather they were equivalent ways of describing nature in the small.
This insight was a result of Dirac's transformation theory, his own favorite piece of work; in old age he called it his "darling. "
This object is now called the Dirac delta, and its use has pervaded mathematical physics.
Later, the Dirac delta was subjected to rigorous analysis and Dirac's applications were found to be correct.
On his return to England he was appointed praelector in mathematical physics (1929 - 1932).
Dirac realized that they behaved like holes in an infinite sea of negative electrons and that these holes moved like particles with positive electric charge.
At first he thought they might be protons, but by 1931 he realized that they had to be positrons, antiparticles of electrons. In his 1931 paper predicting the positron Dirac showed that an isolated magnetic pole (monopole) could exist without leading to any inconsistency with quantum mechanics.
He also showed that if one monopole exists anywhere in the universe it can account for the quantization of electrical charge.
His prediction was verified in 1932 with the discovery of the positron by Carl David Anderson. The Dirac equation and the prediction of antimatter are generally considered to be Dirac's most important contributions to physics.
Since the charge on the electron (or possibly on the quark) is the fundamental unit, Dirac could calculate the magnetic strength of the monopole: it is impressively large. So far, no convincing evidence for monopoles has turned up.
The experimental clarification came when C. D. Anderson, doing cosmic-ray research in R. A. Millikan's laboratory in Pasadena, Calif. , obtained on August 2, 1932, the photograph of an electron path, the curvature of which could be accounted for only if the electron had a positive charge. The positively charged electron, or positron, was, however, still unconnected with the negative energy states implied in Dirac's theory of the electron. The work needed in this respect was largely done by Dirac, though not without some promptings from others.
A most lucid summary of the results was given by Dirac in the lecture which he delivered on December 12, 1933, in Stockholm, when he received the Nobel Prize in physics jointly with Schrödinger.
The year 1934-1935 was spent at the Institute for Advanced Study, Princeton, N. J.
Working by himself, he developed the subject in a highly original way.
He returned to Europe by traveling around the world, accompanied part of the way by Heisenberg.
In the late 1930's Dirac suggested that the presence of large dimensionless numbers in physics was no accident and that their values were related to the age of the universe.
In 1939 Dirac developed a notation for quantum mechanics in which the state of a system.
Dirac suggested that the presence of large dimensionless numbers in physics was no accident and that their values were related to the age of the universe.
The most important consequence of the large-numbers hypothesis is that G, the Newton-Cavendish constant of gravitation, is not really constant but decreases slowly with time.
In 1969 Dirac retired from Cambridge.
Royal Society (1932)
Dirac married Margit Wigner on January 2, 1937. They had two daughters, Mary and Florence.