The

**mathematical formulations of quantum mechanics** are those

mathematical formalismsIn foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....

that permit a rigorous description of

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional

Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

s and operators on these spaces. Many of these structures are drawn from

functional analysisFunctional 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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, a research area within

pure mathematicsBroadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

that was influenced in part by the needs of quantum mechanics. In brief, values of physical observables such as

energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

and

momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

were no longer considered as values of

functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s on

phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

, but as eigenvalues; more precisely: as

spectral valuesIn functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the...

(point spectrum plus absolute continuous plus singular continuous spectrum) of linear

operatorIn 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, very often along withsome extra relevant information....

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* and

*quantum observable* which are radically different from those used in previous

modelsA mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

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 experimentA thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

, and is represented mathematically in the new formalism by the non-commutativity of quantum observables.

Prior to the emergence of quantum mechanics as a separate

theoryThe English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...

, the mathematics used in physics consisted mainly of formal

Mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, beginning with

calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

; and, increasing in complexity up to differential geometry and

partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s.

Probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

was used in

statistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

. Geometric intuition clearly played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called

classical physicsWhat "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

, 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 spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

.

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

In the 1890s,

PlanckMax Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

was able to derive the blackbody spectrum which was later used to avoid the classical

ultraviolet catastropheThe ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was a prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power....

by making the unorthodox assumption that, in the interaction of

radiationIn physics, radiation is a process in which energetic particles or energetic waves travel through a medium or space. There are two distinct types of radiation; ionizing and non-ionizing...

with

matterMatter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

, energy could only be exchanged in discrete units which he called

quantaIn physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

. 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 honor.

In 1905,

EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

explained certain features of the

photoelectric effectIn the photoelectric effect, electrons are emitted from matter as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as photoelectrons...

by assuming that Planck's energy quanta were actual particles, which were later dubbed photons.

All of these developments were

phenomenologicalThe term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...

and flew in the face of the theoretical physics of the time.

Bohr and SommerfeldThe old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...

went on to modify classical mechanics in an attempt to deduce the

Bohr modelIn atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...

from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its

phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed. The most sophisticated version of this formalism was the so-called Sommerfeld–Wilson–Ishiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body problem) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.

In 1923 de Broglie proposed that wave-particle duality 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 SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

,

Werner HeisenbergWerner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

,

Max BornMax Born was a German-born 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 number of notable physicists in the 1920s and 30s...

,

Pascual Jordan

, and the foundational work of

John von NeumannJohn von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

,

Hermann WeylHermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...

and

Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

, 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 discovered the uncertainty relations and

Niels BohrNiels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

introduced the idea of complementarity.

### The "new quantum theory"

Erwin Schrödinger'sErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

wave mechanicsThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

originally was the first successful attempt at replicating the observed quantization of atomic spectra with the help of a precise mathematical realization of de Broglie's wave-particle duality. Schrödinger's wave mechanics were created independently, uniquely based on de Broglie's concepts, less formal and easier to understand, visualize and exploit. 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

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

should be interpreted as the

charge densityThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

of an object smeared out over an extended, possibly infinite, volume of space, but

Max BornMax Born was a German-born 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 number of notable physicists in the 1920s and 30s...

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 interpretationThe Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

of quantum mechanics. Schrödinger's wave function can be seen to be closely related to the classical

Hamilton–Jacobi equationIn mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. In physics, the Hamilton–Jacobi equation is a reformulation of classical mechanics and, thus, equivalent to other formulations such as...

. 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.

To be more precise: already before Schrödinger the young student

Werner HeisenbergWerner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

invented his

matrix mechanicsMatrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...

, which was the first correct quantum mechanics, i.e. the essential breakthrough. Heisenberg's

matrix mechanicsMatrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...

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. In fact, in these early years

linear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

was not generally known to 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 is generally attributed to

Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

, 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 notationBra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

, together with an abstract formulation in terms of the

Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

used in

functional analysisFunctional 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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

; 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 all kind of generalizations of the field. Concerning quantum mechanics,

DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

's method is now called

canonical quantizationIn physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...

.

The first complete mathematical formulation of this approach is generally credited to

John von NeumannJohn von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

's 1932 book

*Mathematical Foundations of Quantum Mechanics*, although

Hermann WeylHermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...

had already referred to Hilbert spaces (which he called

*unitary spaces*) in his 1927 classic book. It was developed in parallel with a new approach to the mathematical

spectral theoryIn 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 of studies of linear algebra and the solutions of...

based on linear operators rather than the

quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

s that were

David HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

'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 NeumannJohn von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

. In other words, discussions about

*interpretation* of the theoryAn interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations...

, 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 theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the one presented here is a simple special case. In fact, the difficulties involved in implementing any of the following formulations cannot be said yet to have been solved in a satisfactory fashion except for ordinary quantum mechanics.

- Feynman path integrals
- axiomatic
In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.The axioms exist in...

, algebraic and constructive quantum field theoryIn mathematical physics, constructive quantum field theory is the field devoted to showing that quantum theory is mathematically compatible with special relativity. This demonstration requires new mathematics, in a sense analogous to Newton developing calculus in order to understand planetary...

- Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...

& geometric quantizationIn mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory...

- quantum field theory in curved spacetime
Quantum field theory in curved spacetime is an extension of standard, Minkowski-space quantum field theory to curved spacetime. A general prediction of this theory is that particles can be created by time dependent gravitational fields , or by time independent gravitational fields that contain...

- C* algebra formalism
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....

- Generalized Statistical Model of Quantum Mechanics
In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...

On a different front, von Neumann originally dispatched quantum measurement with his infamous postulate on the collapse of the wavefunction, raising a host of philosophical problems. Over the intervening 70 years, the

*problem of measurement* became an active research area and itself spawned some new formulations of quantum mechanics.

- Relative state/Many-worlds interpretation
The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction, but denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternative histories and futures are real, each representing an...

of quantum mechanics
- Decoherence
- Consistent histories
In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology...

formulation of quantum mechanics
- Quantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...

formulation 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,

quantizationIn physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...

, 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 opticsQuantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter.- History of quantum optics :...

.

- de Broglie
Louis-Victor-Pierre-Raymond, 7th duc de Broglie, FRS was a French physicist and a Nobel laureate in the year 1929. He was the sixteenth member elected to occupy seat 1 of the Académie française in 1944, and served as Perpetual Secretary of the Académie des sciences, France.-Biography :Louis de...

–BohmDavid Joseph Bohm FRS was an American-born British quantum physicist who contributed to theoretical physics, philosophy, neuropsychology, and the Manhattan Project.-Youth and college:...

–BellJohn Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...

pilot waveIn theoretical physics, the Pilot Wave theory was the first known example of a hidden variable theory, presented by Louis de Broglie in 1927. Its more modern version, the Bohm interpretation,...

formulation of quantum mechanics
- Bell's inequalities
- Kochen–Specker theorem

## Mathematical structure of quantum mechanics

A physical system is generally described by three basic ingredients:

states;

observableIn physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...

s; and

dynamicsIn the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...

(or law of

time evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...

) or, more generally, a group of physical symmetries. A classical description can be given in a fairly direct way by a

phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

model of mechanics: states are points in a

symplecticA symplectic space is either a symplectic manifold or a symplectic vector space. The latter is a special case of the former....

phase space, observables are real-valued functions on it, time evolution is given by a one-parameter

groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description consists of a

Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

of states, observables are self adjoint operators on the space of states, time evolution is given by a

one-parameter groupIn mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators...

of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.

### Postulates of quantum mechanics

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

- Each physical system is associated with a (topologically) separable complex
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

*H* with inner product . Rays (one-dimensional subspaces) in *H* are associated with states of the system. In other words, physical states can be identified with equivalence classes of vectors of length 1 in *H*, where two vectors represent the same state if they differ only by a phase factorFor any complex number written in polar form , the phase factor is the exponential part, i.e. eiθ. As such, the term "phase factor" is similar to the term phasor, although the former term is more common in quantum mechanics. This phase factor is itself a complex number of absolute value 1...

. *Separability* is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state.
- The Hilbert space of a composite system is the Hilbert space tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

of the state spaces associated with the component systems (for instance, J.M. Jauch, *Foundations of quantum mechanics*, section 11-7). For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
- Physical symmetries act on the Hilbert space of quantum states unitarily
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...

or antiunitarily due to Wigner's theoremWigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT act on the Hilbert space of states....

(supersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

is another matter entirely).
- Physical observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...

s are represented by densely-defined self-adjoint operatorIn mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

s on *H*.

- The expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

(in the sense of probability theory) of the observable *A* for the system in state represented by the unit vector *H* is

- 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 of studies of linear algebra and the solutions of...

, we can associate a probability measureIn 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...

to the values of *A* in any state ψ. We can also show that the possible values of the observable *A* in any state must belong to the spectrum of *A*. In the special case *A* has only discrete spectrumIn physics, an elementary explanation of a discrete spectrum is that it is an emission spectrum or absorption spectrum for which there is only an integer number of intensities. Atomic electronic absorption and emission spectrum are discrete, as contrasted with, for example, the emission spectrum...

, the possible outcomes of measuring *A* are its eigenvalues.

- More generally, a state can be represented by a so-called density operator, which is a trace class
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 of basis....

, nonnegative self-adjoint operator normalized to be of trace 1. The expected value of *A* in the state is

- If is the orthogonal projector onto the one-dimensional subspace of
*H* spanned by , then

- Density operators are those that are in the closure of the convex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....

of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme pointIn mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S...

s 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 principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

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.

Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

and Pauli's

exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

, see below.

**Superselection sectors**. The correspondence between states and rays needs to be refined somewhat to take into account so-called

superselection sectorIn Quantum mechanics, superselection extends the concept of selection rules.Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables....

s. States in different superselection sectors cannot influence each other, and the relative phases between them are unobservable.

### Pictures of dynamics

- In the so-called Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...

of quantum mechanics, the dynamics is given as follows:

The

time evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...

of the state is given by a differentiable function from the real numbers

**R**, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows:

If

denotes the state of the system at any one time

*t*, the following

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

holds:

where H is a densely-defined self-adjoint operator, called the system

HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

,

*i* is the

imaginary unitA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

and

is the reduced Planck constant. As an observable, H corresponds to the total

energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

of the system.

Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary group

*U*(

*t*):

*H* →

*H* such that

for all times

*s*,

*t*. The existence of a self-adjoint Hamiltonian H such that

is a consequence of

Stone's theorem on one-parameter unitary groupsIn mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators...

. (It is assumed that

*H* does not depend on time and that the perturbation starts at

; otherwise one must use the

Dyson seriesIn scattering theory, the Dyson series, formulated by British-born American physicist Freeman Dyson, is a perturbative series, and each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics at the second order the difference from...

, formally written as

where

is Dyson's time-ordering symbol.

(This symbol permutes a product of noncommuting operators of the form

into the uniquely determined re-ordered expression

with

The result is a causal chain, the primary

*cause* in the past on the utmost r.h.s., and finally the present

*effect* on the utmost l.h.s. .)

- The Heisenberg picture
In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...

of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus:

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

and that the time-dependent Heisenberg operators satisfy

This assumes A is not time dependent in the Schrödinger picture. Notice the commutator expression is purely formal when one of the operators is unbounded. One would specify a representation for the expression to make sense of it.

- The so-called Dirac picture or interaction picture
In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...

has time-dependent *states* and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the observables can be solved exactly, confining any complications to the evolution of the states. For this reason, the Hamiltonian for the observables is called "free Hamiltonian" and the Hamiltonian for the states is called "interaction Hamiltonian". In symbols:

The interaction picture does not always exist, though. In interacting quantum field theories,

Haag's theoremRudolf Haag postulatedthat the interaction picture does not exist in an interacting, relativistic quantum field theory , something now commonly known as Haag's Theorem...

states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e.g.

, in the interaction picture it does, at least, if

*V* does not commute with

, since

.

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. The Dirac picture is the one used in perturbation theory, and is specially associated to

quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

and

many-body physicsThe many-body theory is an area of physics which provides the framework for understanding the collective behavior of vast assemblies of interacting particles. In general terms, the many-body theory deals with effects that manifest themselves only in systems containing large numbers of constituents...

.

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 equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

depends on choosing a particular representation of Heisenberg's canonical commutation relations. The

Stone–von Neumann theoremIn mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators...

states all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. This is related to

quantizationIn physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...

and the correspondence between classical and quantum mechanics, and is therefore not strictly part of the general mathematical framework.

The

quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

is an exactly-solvable system where the possibility of choosing among more than one representation can be seen in all its glory. There, apart from the Schrödinger (position or momentum) representation one encounters the Fock (number) representation and the Segal–Bargmann (phase space or coherent state) representation (named after

Irving SegalIrving Ezra Segal was a mathematician known for work on theoretical quantum mechanics.He was at the Massachusetts Institute of Technology...

and

Valentine BargmannValentine Bargmann was a German born American mathematician and physicist.Born in Berlin, Germany, Bargmann studied there from 1925 to 1933. After the Machtergreifung he moved to Switzerland to the University of Zürich where he received his Ph.D...

). 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 spaceIn mathematics, a rigged Hilbert space is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense...

and a renormalization of the norm).

This is related to quantization of constrained systems and quantization of gauge theories. It

is also possible to formulate a quantum theory of "events" where time becomes an observable (see D. Edwards).

### Spin

In addition to their other properties all particles possess a quantity, which has no correspondence at all in conventional physics, namely the

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

, which is some kind of

*intrinsic angular momentum* (therefore the name). In the position representation, instead of a wavefunction without spin,

, one has with spin:

, where

belongs to the following discrete set of values

.

One distinguishes

bosonIn particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s (

*S* = 0 or 1 or 2 or ...) and

fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s (

*S* = 1/2 or 3/2 or 5/2 or ...)

### Pauli's principle

The property of spin relates to another basic property concerning systems of N identical particles: Pauli's

exclusion principleThe Exclusion principle is a philosophical principle that states:-In physicalism:The exclusion principle is most commonly applied when one poses this scenario; One usually considers that the desire to lift one’s arm as a mental event, and the lifting on one's arm, a physical event...

, 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

i.e., on transposition 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.

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"In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension

*d=2* can one construct entities where

is replaced by an arbitrary complex number with magnitude 1 ( -> anyons).

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 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

measurementMeasurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...

. The von Neumann description of quantum measurement of an observable

*A*, when the system is prepared in a pure state

*ψ* 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 experimentIn physics, Compton scattering is a type of scattering that X-rays and gamma rays undergo in matter. The inelastic scattering of photons in matter results in a decrease in energy of an X-ray or gamma ray photon, called the Compton effect...

; it is not applicable to most present-day measurements within the quantum domain):

- Let
*A* have spectral resolution

where E

_{A} is the resolution of the identity (also called

projection-valued measureIn 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 projections on a Hilbert space...

) associated to

*A*. Then the probability of the measurement outcome lying in an interval

*B* of

**R** is |E

_{A}(

*B*)

*ψ*|

^{2}. In other words, the probability is obtained by integrating the characteristic function of

*B* against the countably additive measure

- If the measured value is contained in
*B*, then immediately after the measurement, the system will be in the (generally non-normalized) state E_{A}(*B*) *ψ*. If the measured value does not lie in *B*, replace *B* by its complement for the above state.

For example, suppose the state space is the

*n*-dimensional complex Hilbert space

**C**^{n} and

*A* is a Hermitian matrix with eigenvalues

*λ*_{i}, with corresponding eigenvectors

*ψ*_{i}. The projection-valued measure associated with

*A*, E

_{A}, is then

where

*B* is a Borel set containing only the single eigenvalue

*λ*_{i}. If the system is prepared in state

Then the probability of a measurement returning the value

*λ*_{i} can be calculated by integrating the spectral measure

over

*B*_{i}. This gives trivially

The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the

*projection postulate*.

A more general formulation replaces the projection-valued measure with a

positive-operator valued measure (POVM)In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...

. To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections

by a finite set of positive operators

whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {

*λ*_{1} ...

*λ*_{n}} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is

*λ*_{i}. Instead of collapsing to the (unnormalized) state

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

Since the

*F*_{i} F_{i}* 's need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.

The same formulation applies to general mixed states.

In von Neumann's approach, the state transformation due to measurement is distinct from that due to

time evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...

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 operationIn quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan...

s, 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 interpretationThe many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction, but denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternative histories and futures are real, each representing an...

" of quantum mechanics.

## List of mathematical tools

Part of the

folkloreFolklore consists of legends, music, oral history, proverbs, jokes, popular beliefs, fairy tales and customs that are the traditions of a culture, subculture, or group. It is also the set of practices through which those expressive genres are shared. The study of folklore is sometimes called...

of the subject concerns the

mathematical physicsMathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

textbook Methods of Mathematical Physics put together by

Richard CourantRichard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...

from

David HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

's Göttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his

matrix mechanicsMatrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...

, 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 a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

: complex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, eigenvectors, eigenvalues
- 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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

: Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

s, linear operators, spectral theoryIn 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 of studies of linear algebra and the solutions of...

- differential equations: partial differential equations, separation of variables
In mathematics, separation of variables 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 on a different side of the equation....

, ordinary differential equations, Sturm–Liouville theory, eigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...

s
- harmonic analysis
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 the notions of Fourier series and Fourier transforms...

: Fourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

s

See also:

list of mathematical topics in quantum theory.