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The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
. This mathematical formalism uses mainly a part of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, especially
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early
1900s The 1900s may refer to: * 1900s (decade), the decade from 1900 to 1909 * The century from 1900 to 1999, almost synonymous with the 20th century The 20th (twentieth) century began on January 1, 1901 ( MCMI), and ended on December 31, 2000 ( MM ...
by the use of abstract mathematical structures, such as infinite-dimensional
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s ( ''L''2 space mainly), and
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
on these spaces. In brief, values of physical
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phys ...
s such as
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
were no longer considered as values of functions on
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
, but as
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s; more precisely as spectral values of linear
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
s in Hilbert space. These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of ''
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
'' and ''quantum observables'', which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by Heisenberg through a
thought experiment A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anci ...
, and is represented mathematically in the new formalism by the non-commutativity of operators representing quantum observables. Prior to the development of quantum mechanics as a separate
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...
, the mathematics used in physics consisted mainly of formal
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, beginning with
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, and increasing in complexity up to differential geometry and
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s.
Probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
was used in statistical mechanics. Geometric intuition played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of differential geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (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 dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
.


History of the formalism


The "old quantum theory" and the need for new mathematics

In the 1890s, Planck was able to derive the blackbody spectrum, which was later used to avoid the classical
ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an infinity, unbounded quantity o ...
by making the unorthodox assumption that, in the interaction of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible ...
with
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic par ...
, energy could only be exchanged in discrete units which he called
quanta Quanta is the plural of quantum. Quanta may also refer to: Organisations * Quanta Computer, a Taiwan-based manufacturer of electronic and computer equipment * Quanta Display Inc., a Taiwanese TFT-LCD panel manufacturer acquired by AU Optronic ...
. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, , is now called Planck's constant in his honor. In 1905, Einstein explained certain features of the
photoelectric effect The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid stat ...
by assuming that Planck's energy quanta were actual particles, which were later dubbed photons. All of these developments were
phenomenological Phenomenology may refer to: Art * Phenomenology (architecture), based on the experience of building materials and their sensory properties Philosophy * Phenomenology (philosophy), a branch of philosophy which studies subjective experiences and a ...
and challenged the theoretical physics of the time. Bohr and Sommerfeld went on to modify
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
in an attempt to deduce the
Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Sy ...
from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
, 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. In 1923,
de Broglie Louis Victor Pierre Raymond, 7th Duc de Broglie (, also , or ; 15 August 1892 – 19 March 1987) was a French physicist and aristocrat who made groundbreaking contributions to quantum theory. In his 1924 PhD thesis, he postulated the wave na ...
proposed that
wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical concepts "particle" or "wave" to fully describe th ...
applied not only to photons but to electrons and every other physical system. The situation changed rapidly in the years 1925–1930, when working mathematical foundations were found through the groundbreaking work of
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
,
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a ...
, Pascual Jordan, and the foundational work of
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
,
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
and
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Unive ...
, and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
discovered the uncertainty relations and
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
introduced the idea of complementarity.


The "new quantum theory"

Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
's matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra. Later in the same year, Schrödinger created his wave mechanics. Schrödinger's formalism was considered easier to understand, visualize and calculate as it led to
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
, which physicists were already familiar with solving. Within a year, it was shown that the two theories were equivalent. Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary partic ...
should be interpreted as the
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
of an object smeared out over an extended, possibly infinite, volume of space. It was
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a ...
who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a ''pointlike'' object. Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of the
Copenhagen interpretation The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as feat ...
of quantum mechanics. Schrödinger's
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
can be seen to be closely related to the classical Hamilton–Jacobi equation. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. In his PhD thesis project,
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Unive ...
discovered that 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, when one expresses them through Poisson brackets, a procedure now known as canonical quantization. To be more precise, already before Schrödinger, the young postdoctoral fellow
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
invented his matrix mechanics, which was the first correct quantum mechanics–– the essential breakthrough. Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, a very radical formulation in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even aware that his "index-schemes" were matrices, as Born soon pointed out to him. In fact, in these early years,
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...
was not generally popular with physicists in its present form. Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Unive ...
, who wrote a lucid account in his 1930 classic ''
The Principles of Quantum Mechanics ''The Principles of Quantum Mechanics'' is an influential monograph on quantum mechanics written by Paul Dirac and first published by Oxford University Press in 1930. Dirac gives an account of quantum mechanics by "demonstrating how to cons ...
''. He is the third, and possibly most important, pillar of that field (he soon was the only one to have discovered a relativistic generalization of the theory). In his above-mentioned account, he introduced the
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathem ...
, together with an abstract formulation in terms of the
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
used in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
; 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. His work was particularly fruitful in many types of generalizations of the field. The first complete mathematical formulation of this approach, known as the Dirac–von Neumann axioms, is generally credited to
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
's 1932 book '' Mathematical Foundations of Quantum Mechanics'', although
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
had already referred to Hilbert spaces (which he called ''unitary spaces'') in his 1927 classic paper and book. It was developed in parallel with a new approach to the mathematical
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
based on linear operators rather than the
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
s that were David Hilbert's approach a generation earlier. 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 John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
. 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.


Later developments

The application of the new quantum theory to electromagnetism resulted in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the ones presented here are simple special cases. *
Path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional ...
*
Phase-space formulation The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mom ...
of quantum mechanics & geometric quantization *
quantum field theory in curved spacetime In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory treats spacetime as a fixed, classical background, while givin ...
*
axiomatic An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
, algebraic and constructive quantum field theory *
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...
formalism * Generalized statistical model 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. 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 Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...
.


Postulates of quantum mechanics

A physical system is generally described by three basic ingredients: states;
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phys ...
s; 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 In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
model of mechanics: states are points in a phase space formulated by
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called s ...
, 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 normally consists of a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
of states, observables are self-adjoint operators on the space of states, time evolution is given by a
one-parameter group In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :\varphi : \mathbb \rightarrow G from the real line \mathbb (as an additive group) to some other topological group G. If \varphi is in ...
of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations. (It is possible, to map this Hilbert-space picture to a phase space formulation, invertibly. See below.) The following summary of the mathematical framework of quantum mechanics can be partly traced back to the Dirac–von Neumann axioms.


Description of the state of a system

Each isolated physical system is associated with a (topologically) separable complex
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
with
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
.
Ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (g ...
s (that is, subspaces of ''complex'' dimension 1) in are associated with quantum states of the system. In other words, quantum states can be identified with equivalence classes (rays) of vectors of length 1 in , 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. A quantum mechanical state is a ''ray'' in projective Hilbert space, not a ''vector''. Many textbooks fail to make this distinction, which could be partly a result of the fact that the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
itself involves Hilbert-space "vectors", with the result that the imprecise use of "state vector" rather than ''ray'' is very difficult to avoid. Accompanying Postulate I is the composite system postulate: In the presence of
quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state o ...
, the quantum state of the composite system cannot be factored as a tensor product of states of its local constituents; Instead, it is expressed as a sum, or superposition, of tensor products of states of component subsystems. A subsystem in an entangled composite system generally can't be described by a state vector (or a ray), but instead is described by a density operator; Such quantum state is known as a mixed state. The density operator of a mixed state is a
trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace- ...
, nonnegative ( positive semi-definite) self-adjoint operator normalized to be of trace 1. In turn, any density operator of a mixed state can be represented as a subsystem of a larger composite system in a pure state (see
purification theorem In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the action ...
). In the absence of quantum entanglement, the quantum state of the composite system is called a
separable state In quantum mechanics, separable states are quantum states belonging to a composite space that can be factored into individual states belonging to separate subspaces. A state is said to be entangled if it is not separable. In general, determinin ...
. The density matrix of a bipartite system in a separable state can be expressed as \rho=\sum_k p_k \rho_1^k \otimes \rho_2^k , where \; \sum_k p_k = 1 . If there is only a single non-zero p_k, then the state can be expressed just as \rho = \rho_1 \otimes \rho_2 , and is called simply separable or product state.


Measurement on a system


Description of physical quantities

Physical
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phys ...
s are represented by Hermitian matrices on . Since these operators are Hermitian, their eigenvalues are always real, and represent the possible outcomes/results from measuring the corresponding observable. If the spectrum of the observable is discrete, then the possible results are ''quantized''.


Results of measurement

By
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
, we can associate a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
to the values of in any state . We can also show that the possible values of the observable in any state must belong to the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
of . The expectation value (in the sense of probability theory) of the observable for the system in state represented by the unit vector ∈ ''H'' is \langle\psi, A, \psi\rangle. If we represent the state in the basis formed by the eigenvectors of , then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue. For a mixed state , the expected value of in the state is \operatorname(A\rho), and the probability of obtaining an eigenvalue a_n in a discrete, nondegenerate spectrum of the corresponding observable A is given by \mathbb P(a_n)=\operatorname(, a_n\rangle\langle a_n, \rho)=\langle a_n, \rho, a_n\rangle . If the eigenvalue a_n has degenerate, orthonormal eigenvectors \ , then the projection operator onto the eigensubspace can be defined as the identity operator in the eigensubspace: P_n=, a_\rangle\langle a_, +, a_\rangle\langle a_, + \dots + , a_\rangle\langle a_, , and then \mathbb P(a_n)=\operatorname(P_n\rho) . Postulates II.a and II.b are collectively known as the Born rule of quantum mechanics.


Effect of measurement on the state

When a measurement is performed, only one result is obtained (according to some interpretations of quantum mechanics). This is modeled mathematically as the processing of additional information from the measurement, confining the probabilities of an immediate second measurement of the same observable. In the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first. Therefore the state vector must change as a result of measurement, and ''collapse'' onto the eigensubspace associated with the eigenvalue measured. For a mixed state , after obtaining an eigenvalue a_n in a discrete, nondegenerate spectrum of the corresponding observable A , the updated state is given by \rho'=\frac . If the eigenvalue a_n has degenerate, orthonormal eigenvectors \ , then the projection operator onto the eigensubspace is P_n=, a_\rangle\langle a_, +, a_\rangle\langle a_, + \dots + , a_\rangle\langle a_, . Postulates II.c is sometimes called the "state update rule" or "collapse rule"; Together with the Born rule (Postulates II.a and II.b), they form a complete representation of measurements, and are sometimes collectively called the measurement postulate(s). Note that the projection-valued measures (PVM) described in the measurement postulate(s) can be generalized to positive operator-valued measures (POVM), which is the most general kind of measurement in quantum mechanics. A POVM can be understood as the effect on a component subsystem when a PVM is performed on a larger, composite system (see Naimark's dilation theorem).


Time evolution of a system

Though it is possible to derive the Schrödinger equation, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the DeBroglie hypothesis or path integrals. Equivalently, the time evolution postulate can be stated as: For a closed system in a mixed state , the time evolution is \rho(t)=U(t;t_0)\rho(t_0) U^\dagger(t;t_0). The evolution of an open quantum system can be described by quantum operations (in an operator sum formalism) and quantum instruments, and generally does not have to be unitary.


Other implications of the postulates

* Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem (
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
is another matter entirely). * 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 In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
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. * Recent research has shown that the composite system postulate (tensor product postulate) can be derived from the state postulate (Postulate I) and the measurement postulates (Postulates II); Moreover, it has also been shown that the measurement postulates (Postulates II) can be derived from "unitary quantum mechanics", which includes only the state postulate (Postulate I), the composite system postulate (tensor product postulate) and the unitary evolution postulate (Postulate III). 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.


Spin

In addition to their other properties, all particles possess a quantity called spin, an ''intrinsic angular momentum''. Despite the name, particles do not literally spin around an axis, and quantum mechanical spin has no correspondence in classical physics. In the position representation, a spinless wavefunction has position and time as continuous variables, . For spin wavefunctions the spin is an additional discrete variable: , where takes the values; \sigma = -S \hbar , -(S-1) \hbar , \dots, 0, \dots ,+(S-1) \hbar ,+S \hbar \,. That is, the state of a single particle with spin is represented by a -component spinor of complex-valued wave functions. Two classes of particles with ''very different'' behaviour are
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s which have integer spin (), and
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s possessing half-integer spin ().


Pauli's principle

The property of spin relates to another basic property concerning systems of identical particles: Pauli's exclusion principle, which is a consequence of the following permutation behaviour of an -particle wave function; again in the position representation one must postulate that for the transposition of any two of the particles one always should have i.e., on transposition of the arguments of any two particles the wavefunction should ''reproduce'', apart from a prefactor which is for bosons, but () for fermions. Electrons are fermions with ; quanta of light are bosons with . 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. Only in dimension can one construct entities where is replaced by an arbitrary complex number with magnitude 1, called
anyon In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchan ...
s. 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.


Mathematical structure of quantum mechanics


Pictures of dynamics

Summary:


Representations

The original form of the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
depends on choosing a particular representation of Heisenberg's
canonical commutation relations In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p ...
. The
Stone–von Neumann theorem In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after ...
dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to the phase space formulation of quantum mechanics, which works in full
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
instead of
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
, so then with a more intuitive link to the classical limit thereof. This picture also simplifies considerations of quantization, the deformation extension from classical to quantum mechanics. The
quantum harmonic oscillator 量子調和振動子 は、調和振動子, 古典調和振動子 の 量子力学, 量子力学 類似物です。任意の滑らかな ポテンシャル エネルギー, ポテンシャル は通常、安定した 平衡点 の近くで ...
is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrödinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the Segal–Bargmann (Fock-space or coherent state) representation (named after
Irving Segal Irving Ezra Segal (1918–1998) was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for h ...
and Valentine Bargmann). All four 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 with a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter , and in that case the time ''t'' becomes an additional generalized coordinate of the physical system. At the quantum level, translations in would be generated by a "Hamiltonian" , where ''E'' is the energy operator and 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 (this requires the use of a rigged Hilbert space and a renormalization of the norm). This is related to the 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).


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.G. Greenstein and A. Zajonc
/ref> The von Neumann description of quantum measurement of an observable , when the system is prepared in a pure state is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton–Simon experiment; it is not applicable to most present-day measurements within the quantum domain): * Let have spectral resolution A = \int \lambda \, d \operatorname_A(\lambda), where is the
resolution of the identity In functional analysis, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. ...
(also called projection-valued measure) associated with . Then the probability of the measurement outcome lying in an interval of is . In other words, the probability is obtained by integrating the characteristic function of against the countably additive measure \langle \psi \mid \operatorname_A \psi \rangle. * If the measured value is contained in , then immediately after the measurement, the system will be in the (generally non-normalized) state . If the measured value does not lie in , replace by its complement for the above state. For example, suppose the state space is the -dimensional complex Hilbert space and is a Hermitian matrix with eigenvalues , with corresponding eigenvectors . The projection-valued measure associated with , , is then \operatorname_A (B) = , \psi_i\rangle \langle \psi_i, , where is a Borel set containing only the single eigenvalue . If the system is prepared in state , \psi \rangle Then the probability of a measurement returning the value can be calculated by integrating the spectral measure \langle \psi \mid \operatorname_A \psi \rangle over . 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). 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 is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is . 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 operators 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. 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 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 physics.


List of mathematical tools

Part of the folklore of the subject concerns the
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
textbook Methods of Mathematical Physics put together by Richard Courant from David Hilbert's
Göttingen University Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. General information The orig ...
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, whereas the physics was radically new. The main tools include: *
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...
:
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s,
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s,
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s *
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
:
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s, linear operators,
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
*
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
:
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
,
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
,
ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
,
Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form: for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...
,
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, ...
s *
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an e ...
:
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s


Notes


References

*
J. von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cover ...
, ''Mathematical Foundations of Quantum Mechanics'' (1932), Princeton University Press, 1955. Reprinted in paperback form. * H. Weyl, ''The Theory of Groups and Quantum Mechanics'', Dover Publications, 1950. * 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). *
R. F. Streater Raymond Frederick "Ray" Streater (born 1936) is a British physicist, and professor emeritus of Applied Mathematics at King's College London. He is best known for co-authoring a text on quantum field theory, the 1964 ''PCT, Spin and Statistics ...
and A. S. Wightman, ''PCT, Spin and Statistics and All That'', Benjamin 1964 (Reprinted by Princeton University Press) * R. Jost, ''The General Theory of Quantized Fields'', American Mathematical Society, 1965. * J. M. Jauch, ''Foundations of quantum mechanics'', Addison-Wesley Publ. Cy., Reading, Massachusetts, 1968. * G. Emch, ''Algebraic Methods in Statistical Mechanics and Quantum Field Theory'', Wiley-Interscience, 1972. * M. Reed and B. Simon, ''Methods of Mathematical Physics'', vols I–IV, Academic Press 1972. * T. S. Kuhn, '' Black-Body Theory and the Quantum Discontinuity, 1894–1912'', Clarendon Press, Oxford and Oxford University Press, New York, 1978. * D. Edwards, ''The Mathematical Foundations of Quantum Mechanics'', Synthese, 42 (1979),pp. 1–70. * R. Shankar, "Principles of Quantum Mechanics", Springer, 1980. * E. Prugovecki, ''Quantum Mechanics in Hilbert Space'', Dover, 1981. * S. Auyang, ''How is Quantum Field Theory Possible?'', Oxford University Press, 1995. * N. Weaver, ''Mathematical Quantization'', Chapman & Hall/CRC 2001. * G. Giachetta, L. Mangiarotti, G. Sardanashvily, ''Geometric and Algebraic Topological Methods in Quantum Mechanics'', World Scientific, 2005. * D. McMahon, ''Quantum Mechanics Demystified'', 2nd Ed., McGraw-Hill Professional, 2005. * G. Teschl, ''Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators'', https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009. * V. Moretti, ''Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation'', 2nd Edition, Springer, 2018. * B. C. Hall, ''Quantum Theory for Mathematicians'', Springer, 2013. * V. Moretti, ''Fundamental Mathematical Structures of Quantum Theory'', Springer, 2019. * K. Landsman, ''Foundations of Quantum Theory'', Springer 2017 {{Functional analysis Mathematical physics History of physics