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90 relations: Algebra over a field, Angular momentum, Angular momentum operator, Baker–Campbell–Hausdorff formula, Biquaternion, Bloch sphere, Cartan decomposition, Classical mechanics, Clifford algebra, Commutator, Complex number, Covering group, Cyclic permutation, Determinant, Dirac equation, Division algebra, Dot product, Eigenvalues and eigenvectors, Einstein notation, Electromagnetic field, Euclidean space, Euler's formula, Euler's four-square identity, Exponential map (Lie theory), Four-tensor, Fundamental representation, Gamma matrices, Gell-Mann matrices, Generalizations of Pauli matrices, Greek alphabet, Group (mathematics), Hermitian matrix, Hilbert space, Hilbert–Schmidt operator, Identity matrix, Imaginary unit, Involutory matrix, Isomorphism, Isospin, Kronecker delta, Kronecker product, Levi-Civita symbol, Lie algebra, Lie group, Linear span, List of things named after Charles Hermite, Mathematical formulation of quantum mechanics, Mathematical physics, Mathematics, Matrix (mathematics), ..., Matrix exponential, Observable, Olinde Rodrigues, Orthogonality, Pauli equation, Pauli group, Poincaré group, Projective representation, Quantum information, Quantum logic gate, Quantum mechanics, Quantum state, Quaternion, Qubit, Relativistic angular momentum, Relativistic quantum mechanics, Representation theory of SU(2), Richard Liboff, Rotation, Rotation group SO(3), Sigma, Skew-Hermitian matrix, Special unitary group, Sphere, Spherical law of cosines, Spin (physics), Spin-½, Spinors in three dimensions, Structure constants, Sylvester's formula, Tau, Tensor, Tensor contraction, Tensor product, Theory of relativity, Trace (linear algebra), Unitary matrix, Vector space, Wave function, Wolfgang Pauli. Expand index (40 more) »
Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
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Angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum.
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Angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum.
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Baker–Campbell–Hausdorff formula
In mathematics, the Baker–Campbell–Hausdorff formula is the solution to the equation for possibly noncommutative and in the Lie algebra of a Lie group.
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Biquaternion
In abstract algebra, the biquaternions are the numbers, where, and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group.
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Bloch sphere
In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.
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Cartan decomposition
The Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory.
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Classical mechanics
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
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Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.
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Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Covering group
In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p: G → H is a continuous group homomorphism.
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Cyclic permutation
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of X. If S has k elements, the cycle is called a k-cycle.
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.
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Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928.
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Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
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Dot product
In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Einstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity.
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Electromagnetic field
An electromagnetic field (also EMF or EM field) is a physical field produced by electrically charged objects.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
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Euler's four-square identity
In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.
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Exponential map (Lie theory)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.
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Four-tensor
In physics, specifically for special relativity and general relativity, a four-tensor is an abbreviation for a tensor in a four-dimensional spacetime.
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Fundamental representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight.
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Gamma matrices
In mathematical physics, the gamma matrices, \, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ1,3(R).
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Gell-Mann matrices
The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3x3 traceless Hermitian matrices used in the study of the strong interaction in particle physics.
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Generalizations of Pauli matrices
In mathematics and physics, in particular quantum information, the term generalized Pauli matrices refers to families of matrices which generalize the (linear algebraic) properties of the Pauli matrices.
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Greek alphabet
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and: Hermitian matrices can be understood as the complex extension of real symmetric matrices.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm where \|\ \| is the norm of H, \ an orthonormal basis of H, and Tr is the trace of a nonnegative self-adjoint operator.
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Identity matrix
In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.
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Imaginary unit
The imaginary unit or unit imaginary number is a solution to the quadratic equation.
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Involutory matrix
In mathematics, an involutory matrix is a matrix that is its own inverse.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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Isospin
In nuclear physics and particle physics, isospin is a quantum number related to the strong interaction.
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Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.
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Kronecker product
In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.
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Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers, for some positive integer.
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Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Linear span
In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set.
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List of things named after Charles Hermite
Numerous things are named after the French mathematician Charles Hermite (1822–1901).
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Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics.
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Mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.
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Observable
In physics, an observable is a dynamic variable that can be measured.
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Olinde Rodrigues
Olinde Rodrigues Benjamin Olinde Rodrigues (6 October 1795 – 17 December 1851), more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer.
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Orthogonality
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.
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Pauli equation
In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field.
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Pauli group
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices \begin 0&1\\ 1&0 \end,\quad Y.
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Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Minkowski (1908) as the group of Minkowski spacetime isometries.
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Projective representation
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity; scalar transformations). In more concrete terms, a projective representation is a collection of operators \rho(g),\, g\in G, where it is understood that each \rho(g) is only defined up to multiplication by a constant. These should satisfy the homomorphism property up to a constant: for some constants c(g,h). Since each \rho(g) is only defined up to a constant anyway, it does not strictly speaking make sense to ask whether the constants c(g,h) are equal to 1. Nevertheless, one can ask whether it is possible to choose a particular representative of each family \rho(g) of operators in such a way that the \rho(g)'s satisfy the homomorphism property on the nose, not just up to a constant. If such a choice is possible, we say that \rho can be "de-projectivized," or that \rho can be "lifted to an ordinary representation." This possibility is discussed further below.
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Quantum information
In physics and computer science, quantum information is information that is held in the state of a quantum system.
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Quantum logic gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits.
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Quantum mechanics
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
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Quantum state
In quantum physics, quantum state refers to the state of an isolated quantum system.
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Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers.
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Qubit
In quantum computing, a qubit or quantum bit (sometimes qbit) is a unit of quantum information—the quantum analogue of the classical binary bit.
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Relativistic angular momentum
In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR).
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Relativistic quantum mechanics
In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM).
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Representation theory of SU(2)
In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups.
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Richard Liboff
Richard Lawrence Liboff (December 30, 1931 – March 9, 2014) was an American physicist who authored five books and over 100 other publications in variety of fields, including plasma physics, planetary physics, cosmology, quantum chaos, and quantum billiards.
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Rotation
A rotation is a circular movement of an object around a center (or point) of rotation.
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Rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.
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Sigma
Sigma (upper-case Σ, lower-case σ, lower-case in word-final position ς; σίγμα) is the eighteenth letter of the Greek alphabet.
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Skew-Hermitian matrix
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or antihermitian if its conjugate transpose is equal to the original matrix, with all the entries being of opposite sign.
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Special unitary group
In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.
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Sphere
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
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Spherical law of cosines
In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.
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Spin (physics)
In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.
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Spin-½
In quantum mechanics, spin is an intrinsic property of all elementary particles.
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Spinors in three dimensions
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product.
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Structure constants
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination.
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Sylvester's formula
In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function of a matrix as a polynomial in, in terms of the eigenvalues and eigenvectors of.
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Tau
Tau (uppercase Τ, lowercase τ; ταυ) is the 19th letter of the Greek alphabet.
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Tensor
In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.
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Tensor contraction
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual.
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Tensor product
In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.
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Theory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity.
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Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.
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Unitary matrix
In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if where is the identity matrix.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.
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Wolfgang Pauli
Wolfgang Ernst Pauli (25 April 1900 – 15 December 1958) was an Austrian-born Swiss and American theoretical physicist and one of the pioneers of quantum physics.
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Redirects here:
Pauli Gate, Pauli Gates, Pauli Group, Pauli Matrices, Pauli Spin Matrices, Pauli algebra, Pauli gate, Pauli gates, Pauli matricies, Pauli matrix, Pauli operator, Pauli operators, Pauli spin matrices, Pauli spin matrix, Pauli vector, Sigma matrices.
References
[1] https://en.wikipedia.org/wiki/Pauli_matrices
