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Quantum mechanics
\Delta x \, \Delta p \ge \frac{\hbar}{2}
Uncertainty principle
Introduction to...

Mathematical formulation of. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Quantum mechanics (QM or quantum theory) is a physical science dealing with the behavior of Matter and Energy on the scale of Atoms . .

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The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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. Year 1900 ( MCM) was an exceptional Common year starting on Monday (link will display the full calendar of the Gregorian calendar This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that 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. For functional analysis as used in psychology see the Functional analysis (psychology article Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application 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. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of Eigenvalues for matrices In physics an operator is a function acting on the space of Physical states As a resultof its application on a physical state another physical state is obtained

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 states that locating a particle in a small region of space makes the Momentum of the particle uncertain A thought experiment (from the German Gedankenexperiment) is a proposal for an Experiment that would test a Hypothesis or Theory In Mathematics, commutativity is the ability to change the order of something without changing the end result

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. The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i 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 Geometric intuition clearly played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of geometric concepts. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. 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. In Atomic physics, the Bohr model created by Niels Bohr depicts the Atom as a small positively charged nucleus surrounded by Electrons In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System

Contents

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. The old quantum theory was a collection of results from the years 1900-1925 which predate modern Quantum mechanics. In Physics, a black body is an object that absorbs all light that falls on it The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe was a prediction of early 20th century Classical physics that an ideal Black body at Radiation, as in Physics, is Energy in the form of waves or moving Subatomic particles emitted by an atom or other body as it changes from a higher energy Matter is commonly defined as being anything that has mass and that takes up space. 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 Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta.

In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's light quanta were actual particles, which are called photons. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Introduction When a Metallic surface is exposed to Electromagnetic radiation above a certain threshold Frequency, the light is absorbed and Electrons In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena

A sketch to justify spectroscopy observations for hydrogen atoms
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. Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ Niels Henrik David Bohr (nels ˈb̥oɐ̯ˀ in Danish 7 October 1885 – 18 November 1962 was a Danish Physicist who made fundamental contributions to understanding A hydrogen atom is an atom of the chemical element Hydrogen. The electrically neutral In Atomic physics, the Bohr model created by Niels Bohr depicts the Atom as a small positively charged nucleus surrounded by Electrons The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive Electrons could make quantum leaps from one orbit to another, emitting or absorbing single quanta of light at the right frequency. In Physics, a quantum leap or quantum jump is a change of an Electron from one energy state to another within an Atom.

All of these developments were phenomenological and flew in the face of the theoretical physics of the time. The term phenomenology in Science is used to describe a body of knowledge which relates several different empirical observations of phenomena to each other Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. In Atomic physics, the Bohr model created by Niels Bohr depicts the Atom as a small positively charged nucleus surrounded by Electrons 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. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System The most sophisticated version of this formalism was the so-called Sommerfeld-Wilson-Ishiwara quantization. In Atomic physics, the Bohr model created by Niels Bohr depicts the Atom as a small positively charged nucleus surrounded by Electrons 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 n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by The mathematical status of quantum theory remained uncertain for some time.

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, FRS (də bʁœj ( August 15 1892 &ndash March 19 1987) was a French In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and

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. Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician.

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. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system

To be more precise: already before Schrödinger the young student Werner Heisenberg invented his matrix mechanics, which was the first correct quantum mechanics, i. Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the Matrix mechanics is a formulation of Quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925 e. the essential breakthrough. But Schrödinger's wave mechanics was created independently, was uniquely based on de Broglie's concepts, less formal and easier to understand, visualize and exploit. Originally the equivalence of Schrödinger's theory with that of Heisenberg was not seen; to show it, was also an important accomplishment of Schrödinger himself, performed in 1926, some months after the first publication of his theory:

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. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In Functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of Eigenvalues for matrices However, Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the square a of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The linear surface or volume charge density is the amount of Electric charge in a line, Surface, or Volume.

It was Max Born, who introduced the interpretation of the square of the wave function as the probability distribution of the position of a pointlike object. Max Born (11 December 1882 &ndash 5 January 1970 was a German Physicist and Mathematician who was instrumental in the development of Quantum Born's idea was soon taken over by Niels Bohr in Copenhagen, who then became the "father" of the Copenhagen interpretation of quantum mechanics which held until the Many Worlds Interpretation which resolved its many paradoxes. The Copenhagen interpretation is an interpretation of Quantum mechanics. The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, parallel universes,

With hindsight, Schrödinger's wave function can be seen to be closely related to the classical Hamilton-Jacobi equation. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system In Physics, the Hamilton–Jacobi equation (HJE is a reformulation of Classical mechanics and thus equivalent to other formulations such as Newton's laws of The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. I. e. , the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, where one uses Poisson brackets. In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition

Werner Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, being certainly very radical in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even knowing that his "index-schemes" were matrices. Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the Matrix mechanics is a formulation of Quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925 In fact, in these early years linear algebra was not generally known to physicists in its present form. Linear algebra is the branch of Mathematics concerned with

Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, 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, being the third, and perhaps most important, person working independently in that field (he soon was the only one, who found a relativistic generalization of the theory). In his above-mentioned account, he introduced the bra-ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory and found a third, most general one, which represented the dynamics of the system. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical This article assumes some familiarity with Analytic geometry and the concept of a limit. For functional analysis as used in psychology see the Functional analysis (psychology article His work was particularly fruitful in all kind of generalizations of the field. Concerning quantum mechanics, Dirac's method is now called canonical quantization. In Physics, canonical quantization is one of many procedures for quantizing a Classical theory.

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. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Year 1927 ( MCMXXVII) was a Common year starting on Saturday (link will display full calendar of the Gregorian calendar. 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. 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 map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most

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. An interpretation of quantum mechanics is a statement which attempts to explain how Quantum mechanics informs our Understanding of Nature.

Later developments

The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. In quantum field theory (QFT the forces between particles are mediated by other particles 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.

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. This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. In Physics the Wightman axioms are an attempt at a mathematically rigorous formulation of Quantum field theory. The Haag-Kastler Axiomatic framework for Quantum field theory, named after Rudolf Haag and Daniel Kastler, is an application to local In Mathematical physics, constructive quantum field theory is the field devoted to showing that quantum theory is mathematically compatible with Special In Mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given Classical theory. Quantum field theory in curved spacetime is an extension of standard Quantum field theory to curved spacetime. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. 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, wave function collapse is one of two processes by which Quantum systems apparently evolve according to the laws of Over the intervening 70 years, the problem of measurement became an active research area and itself spawned some new formulations of quantum mechanics.

A related topic is the relationship to classical mechanics. The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, parallel universes, In Quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior In Quantum mechanics, the consistent histories approach is intended to give a modern Interpretation of quantum mechanics, generalising the conventional Copenhagen In Mathematical physics and Quantum mechanics, quantum logic is a set of rules for Reasoning about propositions which takes the principles of 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. The classical limit is the ability of a physical theory to approximate or "recover" Classical mechanics when considered over special values of its parameters 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.

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 Bohm interpretation of Quantum mechanics, sometimes called Bohmian mechanics, the ontological interpretation, or the causal interpretation The issue of hidden variables has become in part an experimental issue with the help of quantum optics. Quantum optics is a field of research in Physics, dealing with the application of Quantum mechanics to phenomena involving Light and its interactions

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. Louis-Victor-Pierre-Raymond 7th duc de Broglie, FRS (də bʁœj ( August 15 1892 &ndash March 19 1987) was a French David Joseph Bohm ( December 20 1917, Wilkes-Barre Pennsylvania – October 27 1992, London) was an American John Stewart Bell ( June 28 1928 &ndash October 1 1990) was a Physicist, and the originator of Bell's Theorem, one of the In Theoretical physics, pilot wave theory was the first known example of a Hidden variable theory, presented by Louis de Broglie in 1927 Bell's theorem is a theorem that shows that the predictions of Quantum mechanics (QM are not intuitive and touches upon fundamental philosophical issues that relate to modern In Quantum mechanics, the Kochen-Specker (KS theorem is a "no go" theorem provedby Simon Kochen and Ernst Specker in 1967. 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 In physics the term dynamics customarily refers to the time evolution of physical processes Time evolution is the change of state brought about by the passage of Time, applicable to systems with internal state (also called stateful systems) Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations 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. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System Scientific modelling is the process of generating abstract, conceptual, Graphical and or mathematical models. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix In Mathematics, Stone's theorem on one-parameter Unitary groups is a basic theorem of Functional analysis which establishes a One-to-one

Postulates of quantum mechanics

The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann's postulates.

The expected value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector \left|\psi\right\rangle\in H is
\langle\psi\mid A\mid\psi\rangle
By spectral theory, we can associate a probability measure to the values of A in any state ψ. In Mathematics, spectral theory is an inclusive term for theories extending the Eigenvector and Eigenvalue theory of a single Square matrix. A probability space, in Probability theory, is the conventional Mathematical model of Randomness. We can also show that the possible values of the observable A in any state must belong to the spectrum of A. In Functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of Eigenvalues for matrices In the special case A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues. In Physics, discrete spectrum is a Finite set or a Countable set of Eigenvalues of an Operator. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes
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. In Mathematics, a trace class operator is a Compact operator for which a trace may be defined such that the trace is finite and independent of the choice The expected value of A in the state ρ is
\operatorname{tr}(A\rho)
If ρψ is the orthogonal projector onto the one-dimensional subspace of H spanned by \left|\psi\right\rangle, then
\operatorname{tr}(A\rho_\psi)=\left\langle\psi\mid A\mid\psi\right\rangle
Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. In Mathematics, the convex hull or convex envelope for a set of points X in a Real Vector space V is the minimal Convex 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 Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain

Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin and Pauli's exclusion principle, see below.

Superselection sectors. The correspondence between states and rays needs to be refined somewhat to take into account so-called superselection sectors. A superselection sector is a concept used in Quantum mechanics when a representation of a *-algebra is decomposed into irreducible components States in different superselection sectors cannot influence each other, and the relative phases between them are unobservable.

Pictures of dynamics

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. In Quantum mechanics, a state function is a linear combination (a superposition of eigenstates. Time evolution is the change of state brought about by the passage of Time, applicable to systems with internal state (also called stateful systems) This map is characterized by a differential equation as follows: If \left|\psi\left(t\right)\right\rangle denotes the state of the system at any one time t, the following Schrödinger equation holds:

i\hbar\frac{d}{d t}\left|\psi(t)\right\rangle=H\left|\psi(t)\right\rangle

where H is a densely-defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and \hbar is the reduced Planck constant. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. As an observable, H corresponds to the total energy of the system. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός

Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary group U(t): H → H such that

\left|\psi(t+s)\right\rangle=U(t)\left|\psi(s)\right\rangle

for all times s, t. In Mathematics, Stone's theorem on one-parameter Unitary groups is a basic theorem of Functional analysis which establishes a One-to-one The existence of a self-adjoint Hamiltonian H such that

U(t)=e^{-(i/\hbar)t H}

is a consequence of Stone's theorem on one-parameter unitary groups. In Mathematics, Stone's theorem on one-parameter Unitary groups is a basic theorem of Functional analysis which establishes a One-to-one (It is assumed that H does not depend on time and that the perturbation starts at t0 = 0; otherwise one must use the Dyson series, formally written as U(t)={\mathcal{T}}\, \,\exp{-(i/\hbar )\int\limits_{t_0}^t \,\rm dt'\, H(t')}, where {\mathcal{T}} is Dyson's time-ordering symbol. In Scattering theory, the Dyson series is a Perturbative series and each term is represented by Feynman diagrams This series diverges asymptotically )

\left|\psi\right\rangle = \left|\psi(0)\right\rangle
A(t) = U(-t)AU(t). \quad

It is then easily checked that the expected values of all observables are the same in both pictures

\langle\psi\mid A(t)\mid\psi\rangle=\langle\psi(t)\mid A\mid\psi(t)\rangle

and that the time-dependent Heisenberg operators satisfy

i\hbar{d\over dt}A(t) = [A(t),H].

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.

i\hbar\frac{d }{dt}\left|\psi(t)\right\rangle ={H}_{\rm int}(t) \left|\psi(t)\right\rangle
i\hbar{d \over d t}A(t) = [A(t),H_{0}].

The interaction picture does not always exist, though. In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist. Rudolf Haag postulatedthat the Interaction picture does not exist in an interacting relativistic Quantum field theory, something now commonly known as Haag's This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e. g. H = H0 + V, in the interaction picture it does, at least, if V does not commute with H0, since H_{\rm int}(t)\equiv e^{{(i/\hbar})tH_0}\,V\,e^{{(-i/\hbar})tH_0}. So the above-mentioned Dyson-series has to be used anyhow.

The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets); but this is already rather "high-browed", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory and many-body physics. In quantum field theory (QFT the forces between particles are mediated by other particles Many-Body Theory (or Many-body physics) is an area of Physics which provides the framework for understanding the collective behavior of vast assemblies of Interacting

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. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the In Physics, the canonical commutation relation is the relation between Canonical conjugate quantities (quantities which are related by definition such that one is The Stone-von Neumann theorem states all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. 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 This is related to quantization and the correspondence between classical and quantum mechanics, and is therefore not strictly part of the general mathematical framework.

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. The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. 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.

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 ( Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the

This is related to quantization of constrained systems and quantization of gauge theories. See Gauge theory for the classical preliminaries In order to quantize a gauge theory like for example Yang-Mills theory Chern-Simons or BF model It is also possible to formulate a quantum theory of "events" where time becomes an observable( see D. Edwards ).

Spin

In addition to their other properties all particles possess a quantity, which has no correspondence at all in conventional physics, namely the spin, which is some kind of intrinsic angular momentum (therefore the name). In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In the position representation, instead of a wavefunction without spin, \psi = \psi(\mathbf r), one has with spin:  \psi =\psi(\mathbf r,\sigma), where σ belongs to the following discrete set of values: \sigma \in\{ -S\cdot\hbar , -(S-1)\cdot\hbar , ... ,+(S-1)\cdot\hbar ,+S\cdot\hbar\}. One distinguishes bosons (S=0 or 1 or 2 or . In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein . . ) and fermions (S=1/2 or 3/2 or 5/2 or . In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. . . )

Pauli's principle

The property of spin relates to another basic property concerning systems of N identical particles: Pauli's exclusion principle, which is a consequence of the following permutation behaviour of an N-particle wave function; again in the position representation one must postulate that for the transposition of any two of the N particles one always should have

\psi ( \,...\, ;\,\mathbf r_i,\sigma_i\,;\, ...\,;\mathbf r_j,\sigma_j\,;\,...) \stackrel{!}{=}(-1)^{2S}\cdot \psi ( \,...\, ;\,\mathbf r_j,\sigma_j\,;\, ...\,;\mathbf r_i,\sigma_i\,;\,...)

i. e. , on transposion of the arguments of any two particles the wavefunction should reproduce, apart from a prefactor ( − 1)2S which is +1 for bosons, but (-1) for fermions. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. Electrons are fermions with S=1/2; quanta of light are bosons with S=1. In nonrelativistic quantum mechanics all particles are either bosons or fermions; in relativistic quantum theories also "supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic part. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that Only in dimension d=2 one can construct entities where ( − 1)2S is replaced by an arbitrary complex number with magnitude 1 ( -> anyons). In Mathematics and Physics, an anyon is a type of particle that only occurs in two-dimensional systems

Although spin and the Pauli principle can only be derived from relativistic generalizations of quantum mechanics the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e. g. the periodic system of chemistry, are consequences of the two properties. The periodic table of the chemical elements is a tabular method of displaying the Chemical elements Although precursors to this table exist its invention is

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. Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following:

A = \int \lambda \, d \operatorname{E}_A(\lambda),

where EA is the resolution of the identity (also called projection-valued measure) associated to A. In Mathematics, particularly Functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint Then the probability of the measurement outcome lying in an interval B of R is |EA(B) ψ|2. In other words, the probability is obtained by integrating the characteristic function of B against the countably additive measure

\langle \psi \mid \operatorname{E}_A \psi \rangle.

For example, suppose the state space is the n-dimensional complex Hilbert space Cn and A is a Hermitian matrix with eigenvalues λi, with corresponding eigenvectors ψi. The projection-valued measure associated with A, EA, is then

\operatorname{E}_A (B) = | \psi_i\rangle \langle \psi_i|,

where B is a Borel set containing only the single eigenvalue λi. If the system is prepared in state

| \psi \rangle \,

Then the probability of a measurement returning the value λi can be calculated by integrating the spectral measure

\langle \psi \mid \operatorname{E}_A \psi \rangle

over Bi. This gives trivially

\langle \psi| \psi_i\rangle \langle \psi_i \mid \psi \rangle = | \langle \psi \mid \psi_i\rangle | ^2.

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). In Functional analysis and quantum measurement theory, a POVM (Positive Operator Valued Measure is a measure whose values are non-negative Self-adjoint To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections

| \psi_i\rangle \langle \psi_i| \,

by a finite set of positive operators

F_i F_i^* \,

whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {λ1 . . .  λn} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is λi. Instead of collapsing to the (unnormalized) state

| \psi_i\rangle \langle \psi_i |\psi\rangle \,

after the measurement, the system now will be in the state

F_i |\psi\rangle. \,

Since the Fi Fi* 's need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.

The same formulation applies to general mixed states.

In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. Time evolution is the change of state brought about by the passage of Time, applicable to systems with internal state (also called stateful systems) For example, time evolution is deterministic 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. 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, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite dimensional (matrix C*-algebras An

In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).

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 (also known as relative state formulation, theory of the universal wavefunction, parallel universes, The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, parallel universes,

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. History The concept of folklore developed as part of the 19th century ideology of Romantic nationalism, leading to the reshaping of oral traditions to serve modern ideological Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. 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 &ndash January 27, 1972) was a German American Mathematician. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most The University of Göttingen ( German: Georg-August-Universität Göttingen) is a University in the city of Göttingen, Germany. 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. Matrix mechanics is a formulation of Quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925 Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, where the physics was radically new.

The main tools include:

See also: list of mathematical topics in quantum theory. Linear algebra is the branch of Mathematics concerned with Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes For functional analysis as used in psychology see the Functional analysis (psychology article This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, spectral theory is an inclusive term for theories extending the Eigenvector and Eigenvalue theory of a single Square matrix. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i 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 In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its In Mathematics and its applications a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855 and Joseph Liouville In Mathematics, an eigenfunction of a Linear operator, A, defined on some Function space is any non-zero function f in Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and This is a list of mathematical topics in quantum theory, by Wikipedia page

References

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