Mathematical Formulation of Quantum Mechanics

Mathematical Formulation Of Quantum Mechanics

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces. Many of these structures are drawn from functional analysis, a research area within pure mathematics that was influenced in part by the needs of quantum mechanics. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely: as spectral values (point spectrum plus absolute continuous plus singular continuous spectrum) of linear operators in Hilbert space.

These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of quantum state and quantum observable which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by Heisenberg through a thought experiment, and is represented mathematically in the new formalism by the non-commutativity of quantum observables.

Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of formal mathematical analysis, beginning with calculus; and, increasing in complexity up to differential geometry and partial differential equations. Probability theory was used in statistical mechanics. Geometric intuition clearly played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures. The most sophisticated example of this is the Sommerfeld–Wilson–Ishiwara quantization rule, which was formulated entirely on the classical phase space.

Read more about Mathematical Formulation Of Quantum Mechanics:  Mathematical Structure of Quantum Mechanics, The Problem of Measurement, List of Mathematical Tools

Other articles related to "mathematical formulation of quantum mechanics, mathematical, quantum mechanics, mechanics, quantum":

Mathematical Formulation Of Quantum Mechanics - List of Mathematical Tools
... Part of the folklore of the subject concerns the mathematical physics textbook Methods of Mathematical Physics put together by Richard Courant from David Hilbert's Göttingen ... At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it ... It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential ...
List Of Mathematical Topics In Quantum Theory - Mathematical Formulation of Quantum Mechanics
... set of commuting observables Heisenberg picture Hilbert space Interaction picture Measurement in quantum mechanics quantum field theory quantum logic quantum operation Schr ...
John Von Neumann - Career and Abilities - Mathematical Formulation of Quantum Mechanics
... See also von Neumann entropy, Density matrix, Quantum mutual information, von Neumann measurement scheme, and Wave function collapse Quantum mechanics ...

Famous quotes containing the words mechanics, quantum, formulation and/or mathematical:

    It is only the impossible that is possible for God. He has given over the possible to the mechanics of matter and the autonomy of his creatures.
    Simone Weil (1909–1943)

    The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.
    Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    You do not mean by mystery what a Catholic does. You mean an interesting uncertainty: the uncertainty ceasing interest ceases also.... But a Catholic by mystery means an incomprehensible certainty: without certainty, without formulation there is no interest;... the clearer the formulation the greater the interest.
    Gerard Manley Hopkins (1844–1889)

    What he loved so much in the plant morphological structure of the tree was that given a fixed mathematical basis, the final evolution was so incalculable.
    —D.H. (David Herbert)