Encyclopedia > Mathematical formulation of quantum mechanics
**Quantum physics** | | *Quantum mechanics* | Introduction to... **Mathematical formulation of...** Fig. ...
Quantum mechanics (QM, or quantum theory) is a physical science dealing with the behaviour of matter and energy on the scale of atoms and subatomic particles / waves. ...
| *Fundamental concepts* | Decoherence · Interference Uncertainty · Exclusion Transformation theory Ehrenfest theorem · Measurement In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ...
Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ...
In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ...
The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ...
The term transformation theory refers to a procedure used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927. ...
The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. ...
The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ...
| *Experiments* | Double-slit experiment Davisson-Germer experiment Stern–Gerlach experiment EPR paradox · Popper's experiment Schrödinger's cat Double-slit diffraction and interference pattern The double-slit experiment consists of letting light diffract through two slits, which produces fringes or wave-like interference patterns on a screen. ...
In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow moving electrons at a crystalline Nickel target. ...
In quantum mechanics, the Sternâ€“Gerlach experiment, named after Otto Stern and Walther Gerlach, is a celebrated experiment in 1920 on deflection of particles, often used to illustrate basic principles of quantum mechanics. ...
In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. ...
Poppers experiment is an experiment proposed by the 20th century philosopher of science Karl Popper, to test the standard interpretation (the Copenhagen interpretation) of Quantum mechanics. ...
SchrÃ¶dingers Cat: If the nucleus in the bottom left decays, the Geiger counter on its right will sense it and trigger the release of the gas. ...
| *Equations* | Schrödinger equation Pauli equation Klein-Gordon equation Dirac equation For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...
The Pauli equation is a SchrÃ¶dinger equation which handles spin. ...
The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the SchrÃ¶dinger equation. ...
In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ...
| *Advanced theories* | Quantum field theory Wightman axioms Quantum electrodynamics Quantum chromodynamics Quantum gravity Feynman diagram Quantum field theory (QFT) is the quantum theory of fields. ...
In physics the Wightman axioms are an attempt of mathematically stringent, axiomatic formulation of quantum field theory. ...
Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...
Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ...
This article does not cite any references or sources. ...
In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ...
| *Interpretations* | Copenhagen · Ensemble Hidden variables · Transactional Many-worlds · Consistent histories Quantum logic Consciousness causes collapse It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ...
The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ...
The Ensemble Interpretation, or Statistical Interpretation of Quantum Mechanics, is an interpretation that can be viewed as a minimalist interpretation. ...
In physics, a hidden variable theory is urged by a minority of physicists who argue that the statistical nature of quantum mechanics implies that quantum mechanics is incomplete; it is really applicable only to ensembles of particles; new physical phenomena beyond quantum mechanics are needed to explain an individual event. ...
The transactional interpretation of quantum mechanics (TIQM) by Professor John Cramer is an unusual interpretation of quantum mechanics that describes quantum interactions in terms of a standing wave formed by retarded (forward in time) and advanced (backward in time) waves. ...
The many-worlds interpretation of quantum mechanics or MWI (also known as the relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome...
In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology. ...
In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ...
Consciousness causes collapse is the theory that observation by a conscious observer is responsible for the wavefunction collapse in quantum mechanics. ...
| *Scientists* | Planck · Schrödinger Heisenberg · Bohr · Pauli Dirac · Bohm · Born de Broglie · von Neumann Einstein · Feynman Everett · Others Max Karl Ernst Ludwig Planck (April 23, 1858 in Kiel, Germany â€“ October 4, 1947 in GÃ¶ttingen, Germany) was a German physicist. ...
Bust of SchrÃ¶dinger, in the courtyard arcade of the main building, University of Vienna, Austria. ...
Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ...
Niels (Henrik David) Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in 1922. ...
This article is about Austrian-Swiss physicist Wolfgang Pauli. ...
Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...
David Bohm. ...
Max Born (December 11, 1882 in Breslau â€“ January 5, 1970 in GÃ¶ttingen) was a mathematician and physicist. ...
Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892â€“March 19, 1987), was a French physicist and Nobel Prize laureate. ...
John von Neumann (Hungarian Margittai Neumann JÃ¡nos Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics...
â€œEinsteinâ€ redirects here. ...
Richard Phillips Feynman (May 11, 1918 â€“ February 15, 1988; IPA: ) was an American physicist known for expanding the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and particle theory. ...
Hugh Everett III (November 11, 1930 â€“ July 19, 1982) was an American physicist who first proposed the many-worlds interpretation(MWI) of quantum physics, which he called his relative state formulation. ...
Below is a list of famous physicists. ...
| This box: view • talk • edit | The **mathematical formulation of quantum mechanics** is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. It is 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 were drawn from functional analysis, a research area within pure mathematics that developed in parallel with, and was influenced 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 of linear operators. Fig. ...
Äž: For the film, see: 1900 (film). ...
The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ...
In classical mechanics, momentum (pl. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
Phase space of a dynamical system with focal stability. ...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ...
This formulation of quantum mechanics continues to be used today. At the heart of the description are ideas of *quantum state* and *quantum observable* which, for systems of atomic scale, 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. In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ...
In philosophy, physics, and other fields, a thought experiment (from the German Gedankenexperiment) is an attempt to solve a problem using the power of human imagination. ...
In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of 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. The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ...
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...
Probability theory is the branch of mathematics concerned with analysis of random phenomena. ...
Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...
Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ...
Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ...
The Bohr model of the atom In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by waves of electrons in orbit â€” similar in structure to the solar system, but with electrostatic forces providing attraction, rather than gravity, and with waves spread over entire...
Phase space of a dynamical system with focal stability. ...
## History of the formalism
### The "old quantum theory" and the need for new mathematics *Main article: Old quantum theory* In the decade of 1890, Planck was able to derive the blackbody spectrum which was later used to solve the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of radiation with matter, energy could only be exchanged in discrete units which he called quanta. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, *h* is now called Planck's constant in his honour. The Bohr model of the atom The Bohr Model is a physical model that depicts the atom as a small positively charged nucleus with electrons in orbit at different levels, similar in structure to the solar system. ...
Max Karl Ernst Ludwig Planck (April 23, 1858 in Kiel, Germany â€“ October 4, 1947 in GÃ¶ttingen, Germany) was a German physicist. ...
As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ...
The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. ...
Radiation as used in physics, is energy in the form of waves or moving subatomic particles. ...
In physics, matter is commonly defined as the substance of which physical objects are composed, not counting the contribution of various energy or force-fields, which are not usually considered to be matter per se (though they may contribute to the mass of objects). ...
In physics, a quantum (plural: quanta) is an indivisible entity of energy. ...
A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...
In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's light quanta were actual particles, which are called photons. â€œEinsteinâ€ redirects here. ...
A diagram illustrating the emission of electrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material. ...
In physics, the photon (from Greek Ï†Ï‰Ï‚, phÅs, meaning light) is the quantum of the electromagnetic field; for instance, light. ...
A sketch to justify spectroscopy observations for hydrogen atoms In 1913, Bohr calculated the spectrum of the hydrogen atom with the help of a new model of the atom in which the electron could orbit the proton only on a discrete set of classical orbits, determined by the condition that angular momentum was an integer multiple of Planck's constant. Electrons could make quantum leaps from one orbit to another, emitting or absorbing single quanta of light at the right frequency. Image File history File links No higher resolution available. ...
Image File history File links No higher resolution available. ...
Extremely high resolution spectrogram of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of the interaction between radiation (electromagnetic radiation, or light, as well as particle radiation) and matter. ...
Niels (Henrik David) Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in 1922. ...
Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. ...
The Bohr model of the hydrogen atom, where negatively charged electrons confined to atomic shells encircle a small positively charged atomic nucleus, and that an electron jump between orbits must be accompanied by an emitted or absorbed amount of electromagnetic energy hÎ½. The orbits that the electrons travel in are...
e- redirects here. ...
In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit (1. ...
Quantum Leap is a science fiction television series that ran for 97 episodes from March 1989 to May 1993 on NBC. It follows the adventures of Dr. Samuel Beckett (played by Scott Bakula), a brilliant scientist who after researching time-travel, and doing experiments in something he calls The Imaging...
All of these developments were phenomenological and flew in the face of the theoretical physics of the time. Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed. The most sophisticated version of this formalism was the so-called Sommerfeld-Wilson-Ishiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body problem) could not be predicted. The mathematical status of quantum theory remained uncertain for some time. The term phenomenology in modern science, especially in physics, is used to describe a body of knowledge which relates several different empirical observations of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. ...
The Bohr model of the atom In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by waves of electrons in orbit â€” similar in structure to the solar system, but with electrostatic forces providing attraction, rather than gravity, and with waves spread over entire...
Phase space of a dynamical system with focal stability. ...
The Bohr model of the atom In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by waves of electrons in orbit â€” similar in structure to the solar system, but with electrostatic forces providing attraction, rather than gravity, and with waves spread over entire...
The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i. ...
In 1923 de Broglie proposed that wave-particle duality applied not only to photons but to electrons and every other physical system. Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892â€“March 19, 1987), was a French physicist and Nobel Prize laureate. ...
In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ...
The situation changed rapidly in the years 1925-1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger and Werner Heisenberg and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas. Bust of SchrÃ¶dinger, in the courtyard arcade of the main building, University of Vienna, Austria. ...
Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ...
John von Neumann (Hungarian Margittai Neumann JÃ¡nos Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics...
Hermann Klaus Hugo Weyl (November 9, 1885 â€“ December 9, 1955) was a German mathematician. ...
Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...
### The "new quantum theory" Erwin Schrödinger's wave mechanics originally was the first successful attempt at replicating the observed quantization of atomic spectra with the help of a precise mathematical realization of de Broglie's wave-particle duality. Schrödinger proposed an equation (now bearing his name) for the wave associated to an electron in an atom according to de Broglie, and explained energy quantization by the well-known fact that differential operators of the kind appearing in his equation had a discrete spectrum. However, Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the (squared amplitude of the) wavefunction of an electron must be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. It was Max Born who introduced the probabilistic interpretation of the (squared amplitude of the) wave function as the probability distribution of the position of a *pointlike* object. With hindsight, Schrödinger's wave function can be seen to be closely related to the classical Hamilton-Jacobi equation. Bust of SchrÃ¶dinger, in the courtyard arcade of the main building, University of Vienna, Austria. ...
The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. ...
For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...
In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ...
e- redirects here. ...
Charge density is the amount of electric charge per unit volume. ...
Max Born (December 11, 1882 in Breslau â€“ January 5, 1970 in GÃ¶ttingen) was a mathematician and physicist. ...
A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ...
The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ...
Werner Heisenberg's matrix mechanics formulation, introduced contemporaneously to Schrödinger's wave mechanics and based on algebras of infinite matrices, was certainly very radical in light of the mathematics of classical physics. In fact, at the time linear algebra was not generally known to physicists in its present form. Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ...
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
The reconciliation of the two approaches is generally associated to Paul Dirac, who wrote a lucid account in his 1930 classic *Principles of Quantum Mechanics*. In it, he introduced the bra-ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis, and showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory. Dirac's method is now called canonical quantization. The first complete mathematical formulation of this approach is generally credited to John von Neumann's 1932 book *Mathematical Foundations of Quantum Mechanics*, although Hermann Weyl had already referred to Hilbert spaces (which he called *unitary spaces*) in his 1927 classic book. It was developed in parallel with a new approach to the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert's approach a generation earlier. Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...
Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...
The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...
In physics, canonical quantization is one of many procedures for quantizing a classical theory. ...
John von Neumann (Hungarian Margittai Neumann JÃ¡nos Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics...
Hermann Klaus Hugo Weyl (November 9, 1885 â€“ December 9, 1955) was a German mathematician. ...
Year 1927 (MCMXXVII) was a common year starting on Saturday (link will display full calendar) of the Gregorian calendar. ...
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...
David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ...
Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions about *interpretation* of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations. John von Neumann (Hungarian Margittai Neumann JÃ¡nos Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics...
It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ...
### Later developments The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the one presented here is a simple special case. In fact, the difficulties involved in implementing any of the following formulations cannot be said yet to have been solved in a satisfactory fashion except for ordinary quantum mechanics. Quantum field theory (QFT) is the quantum theory of fields. ...
On a different front, von Neumann originally dispatched quantum measurement with his infamous postulate on the collapse of the wavefunction, raising a host of philosophical problems. Over the intervening 70 years, the *problem of measurement* became an active research area and itself spawned some new formulations of quantum mechanics. This article is about a formulation of quantum mechanics. ...
In physics the Wightman axioms are an attempt of mathematically stringent, axiomatic formulation of quantum field theory. ...
The Haag-Kastler axiomatic framework for quantum field theory is an application to local quantum physics of C-star algebra theory. ...
In mathematical physics, constructive quantum field theory is the field devoted to attempts to put quantum field theory on a basis of completely defined concepts from functional analysis. ...
In mathematical physics, geometric quantization is a mathematical approach to define a quantum theory corresponding to a given classical theory in such a way that certain analogies between the classical theory and the quantum theory remain manifest, for example the similarity between the Heisenberg equation in the Heisenberg picture of...
Quantum field theory in curved spacetimes is an extension of the standard quantum field theory to curved spacetimes. ...
C*-algebras are an important area of research in functional analysis, a branch of mathematics. ...
The term formalism describes an emphasis on form over content or meaning in the arts, literature, or philosophy. ...
The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ...
In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ...
A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called classical limit of quantum mechanics. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, quantization, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself. The many-worlds interpretation of quantum mechanics or MWI (also known as the relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome...
Quantum decoherence is the general term for the consequences of irreversible quantum entanglement. ...
In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology. ...
In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ...
The classical limit is the ability of a physical theory to approximate or recover classical mechanics when considered over special values of its parameters. ...
Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ...
Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called hidden-variable theories. The issue of hidden variables has become in part an experimental issue with the help of quantum optics. In physics, a hidden variable theory is urged by a minority of physicists who argue that the statistical nature of quantum mechanics implies that quantum mechanics is incomplete; it is really applicable only to ensembles of particles; new physical phenomena beyond quantum mechanics are needed to explain an individual event. ...
Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ...
Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892 â€“ March 19, 1987), was a French physicist and Nobel Prize laureate. ...
David Bohm. ...
This article or section is not written in the formal tone expected of an encyclopedia article. ...
In theoretical physics, pilot wave theory was the first known example of a hidden variable theory, presented by Louis de Broglie in 1927. ...
This article may be too technical for most readers to understand. ...
The Kochen-Specker theorem was invented by Simon Kochen and Ernst Specker in 1967. ...
## Mathematical structure of quantum mechanics A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, more generally, a group of physical symmetries. A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description consists of a Hilbert space of states, observables are self adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations. In physics, the term state is used in several related senses, each of which expresses something about the way a physical system is. ...
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ...
In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. ...
For a system with internal state (also called stateful system), time evolution means the change of state brought about by the passage of time. ...
Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...
Phase space of a dynamical system with focal stability. ...
An abstract model (or conceptual model) is a theoretical construct that represents something, with a set of variables and a set of logical and quantitative relationships between them. ...
A symplectic space is either a symplectic manifold or a symplectic vector space. ...
This picture illustrates how the hours on a clock form a group under modular addition. ...
The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...
On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...
Stones theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators which are strongly continuous, that is and are homomorphisms: Such one-parameter...
### Postulates of quantum mechanics The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann's postulates. - Each physical system is associated with a (topologically) separable complex Hilbert space
*H* with inner product . Rays (one-dimensional subspaces) in *H* are associated with states of the system. In other words, physical states can be identified with equivalence classes of vectors of length 1 in *H*, where two vectors represent the same state if they differ only by a phase factor. *Separability* is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state. - The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
- Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily (supersymmetry is another matter entirely).
- Physical observables are represented by densely-defined self-adjoint operators on
*H*. - The expected value (in the sense of probability theory) of the observable
*A* for the system in state represented by the unit vector is - By spectral theory, we can associate a probability measure to the values of
*A* in any state ψ. We can also show that the possible values of the observable *A* in any state must belong to the spectrum of *A*. In the special case *A* has only discrete spectrum, the possible outcomes of measuring *A* are its eigenvalues. - More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator ρ normalized to be of trace 1. The expected value of
*A* in the state ρ is - If ρ
_{ψ} is the orthogonal projector onto the one-dimensional subspace of *H* spanned by , then - Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors
*pure states* and other density operators *mixed states*. One can in this formalism state Heisenberg's uncertainty principle and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article. In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...
The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...
In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...
In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ...
A unitary transformation is an isomorphism between two Hilbert spaces. ...
This article or section is in need of attention from an expert on the subject. ...
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ...
On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...
In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ...
In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ...
In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ...
In mathematics and physics, discrete spectrum of an operator on Hilbert space is the part of the spectrum which corresponds to discrete spectral measures. ...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. ...
In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ...
Convex hull: elastic band analogy In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. // For planar objects, i. ...
An extreme point or an extremal point in general sense is a point that belongs to the extremity of something. ...
In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ...
**Superselection sectors**. The correspondence between states and rays needs to be refined somewhat to take into account so-called superselection sectors. States in different superselection sectors cannot influence each other, and the relative phases between them are unobservable. A superselection sector is a concept used in quantum mechanics. ...
### Pictures of dynamics In the so-called Schrödinger picture of quantum mechanics, the dynamics is given as follows: Heisenbergs form for the equations of motion We have seen that in SchrÃ¶dingers scheme the dynamical variables of the system remain fixed during a period of undisturbed motion. ...
The time evolution of the state is given by a differentiable function from the real numbers **R**, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If denotes the state of the system at any one time *t*, the following Schrödinger equation holds: For a system with internal state (also called stateful system), time evolution means the change of state brought about by the passage of time. ...
For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...
where H is a densely-defined self-adjoint operator, called the system Hamiltonian, *i* is the imaginary unit and is the reduced Planck constant. As an observable, H corresponds to the total energy of the system. The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...
Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ...
Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary group *U*(*t*): *H* → *H* such that In mathematics, Stones theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators which are strongly continuous, that is and are homomorphisms: Such...
for all times *s*, *t*. The existence of a self-adjoint Hamiltonian H such that is a consequence of Stone's theorem on one-parameter unitary groups. Stones theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators which are strongly continuous, that is and are homomorphisms: Such one-parameter...
The Heisenberg picture of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus: The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ...
It is then easily checked that the expected values of all observables are the same in both pictures and that the time-dependent Heisenberg operators satisfy This assumes A is not time dependent in the Schrödinger picture. Notice the commutator expression is purely formal when one of the operators is unbounded. One would specify a representation for the expression to make sense of it. The so-called Dirac picture or interaction picture has time-dependent *states* and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the states can be solved exactly, confining any complications to the evolution of the operators. For this reason, the Hamiltonian for states is called "free Hamiltonian" and the Hamiltonian for observables is called "interaction Hamiltonian". In symbols: In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. ...
In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the SchrÃ¶dinger picture and the Heisenberg picture. ...
The interaction picture does not always exist, though. In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Rudolf Haag showed in 1955 that the interaction picture cannot be rigorously defined in quantum field theory, a result now commonly cited as Haags Theorem. ...
The Heisenberg picture is the closest to classical mechanics, but the Schrödinger picture is considered easiest to understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory. Quantum field theory (QFT) is the quantum theory of fields. ...
Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated to the symmetry (for instance, angular or linear momentum).
### Representations The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's canonical commutation relations. The Stone-von Neumann theorem states all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent .This is related to quantization and the correspondence between classical and quantum mechanics, and is therefore not strictly part of the general mathematical framework. For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...
Werner Heisenberg Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics. ...
In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...
In mathematics and in theoretical physics, the Stone-von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. ...
Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ...
The quantum harmonic oscillator is an exactly-solvable system where the possibility of choosing among more than one representation can be seen in all its glory. There, apart from the Schrödinger (position or momentum) representation one encounters the Fock (number) representation and the Bargmann-Segal (phase space or coherent state) representation. All three are unitarily equivalent. The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...
### Time as an operator The framework presented so far singles out time as *the* parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated to a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter *s*, and in that case the time *t* becomes an additional generalized coordinate of the physical system. At the quantum level, translations in *s* would be generated by a "Hamiltonian" *H*-*E*, where *E* is the energy operator and *H* is the "ordinary" Hamiltonian. However, since *s* is an unphysical parameter, *physical* states must be left invariant by "*s*-evolution", and so the physical state space is the kernel of *H*-*E* (this requires the use of a rigged Hilbert space and a renormalization of the norm). In mathematics, a rigged Hilbert space is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis. ...
This is related to quantization of constrained systems and quantization of gauge theories. It is also possible to formulate a quantum theory of "events" where time becomes an observable( see D. Edwards ). See gauge theory for the classical prelimanaries. ...
## The problem of measurement The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is the effects of measurement. The von Neumann description of quantum measurement of an observable *A*, when the system is prepared in a pure state *ψ* is the following: Various meters Measurement is an observation that reduces an uncertainty expressed as a quantity. ...
- Let
*A* have spectral resolution where E_{A} is the resolution of the identity (also called projection-valued measure) associated to *A*. Then the probability of the measurement outcome lying in an interval *B* of **R** is |E_{A}(*B*) *ψ*|^{2}. In other words, the probability is obtained by integrating the characteristic function of *B* against the countably additive measure In mathematics, projection-valued measures are used to express results in spectral theory. ...
- If the measured valued is contained in
*B*, then immediately after the measurement, the system will be in the (generally non-normalized) state E_{A}(*B*) *ψ*. If the measured value does not lie in *B*, replace *B* by its complement for the above state. For example, suppose the state space is the *n*-dimensinal complex Hilbert space **C**^{n} and *A* is a Hermitian matrix with eigenvalues *λ*_{i}, with corresponding eigenvectors *ψ*_{i}. The projection-valued measure associated with *A*, E_{A}, is then where *B* is a Borel set containing only the single eigenvalue *λ*_{i}. If the system is prepared in state Then the probability of a measurement returning the value *λ*_{i} can be calculated by integrating the spectral measure over *B*_{i}. This gives trivially The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the *projection postulate*. A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections In functional analysis and quantum measurement theory, a POVM (Positive Operator Value Measure) is a measure whose values are non-negative self-adjoint operators on a Hilbert space. ...
by a finite set of positive operators whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {*λ*_{1} ... *λ*_{n}} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is *λ*_{i}. Instead of collapsing to the (unnormalized) state after the measurement, the system now will be in the state Since the *F*_{i} F_{i}* 's need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds. The same formulation applies to general mixed states. The term mixed state refers to a concept in physics, particularly quantum mechanics. ...
In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is determinisic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other quantum operations, which are described by completely positive maps which do not increase the trace. For a system with internal state (also called stateful system), time evolution means the change of state brought about by the passage of time. ...
In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. ...
In mathematics, Chois theorem on completely positive maps is a result that classifies completely positive maps between finite dimensional (matrix) C*-algebras. ...
### The *relative state* interpretation An alternative interpretation of measurement is Everett's relative state interpretation, which was later dubbed the "many-worlds interpretation" of quantum mechanics. The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics, based on Hugh Everetts relative-state formulation. ...
The many-worlds interpretation of quantum mechanics or MWI (also known as the relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome...
## List of mathematical tools Part of the folklore of the subject concerns the mathematical physics textbook *Courant-Hilbert*, put together by Richard Courant from David Hilbert's Göttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation. 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 equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, where the physics was radically new. This article does not cite any references or sources. ...
Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ...
Methoden der mathematischen Physik was a 1924 book, in two volumes totalling around 1000 pages, published under the names of David Hilbert and Richard Courant. ...
Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ...
David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ...
The Georg-August University of GÃ¶ttingen (Georg-August-UniversitÃ¤t GÃ¶ttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ...
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ...
The main tools include: See also: list of mathematical topics in quantum theory. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...
In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ...
In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...
In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ...
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation. ...
In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. ...
In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles FranÃ§ois Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset...
In mathematics, an eigenfunction of a linear operator A defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. ...
Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...
In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...
This is a list of mathematical topics in quantum theory, by Wikipedia page. ...
## References - S. Auyang,
*How is Quantum Field Theory Possible?*, Oxford University Press, 1995. - D. Edwards,
*The Mathematical Foundations of Quantum Mechanics*, Synthese, 42 (1979),pp.1-70. - G. Emch,
*Algebraic Methods in Statistical Mechanics and Quantum Vield Theory*, Wiley-Interscience, 1972. - R. Jost,
*The General Theory of Quantized Fields*, American Mathematical Society, 1965. - A. Gleason,
*Measures on the Closed Subspaces of a Hilbert Space*, Journal of Mathematics and Mechanics, 1957. - G. Mackey,
*Mathematical Foundations of Quantum Mechanics*, W. A. Benjamin, 1963 (paperback reprint by Dover 2004). - J. von Neumann,
*Mathematical Foundations of Quantum Mechanics*, Princeton University Press, 1955. Reprinted in paperback form. - R. F. Streater and A. S. Wightman,
*PCT, Spin and Statistics and All That*, Benjamin 1964 (Reprinted by Princeton University Press) - M. Reed and B. Simon,
*Methods of Mathematical Physics*, vols 1-IV, Academic Press 1972. - H. Weyl,
*The Theory of Groups and Quantum Mechanics*, Dover Publications, 1950. |