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Pauli Matrix
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary.[1] Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries
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Wolfgang Pauli
Wolfgang Ernst Pauli (/ˈpɔːli/;[5] German: [ˈpaʊli]; 25 April 1900 – 15 December 1958) was an Austrian-born Swiss and American theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein,[6] Pauli received the Nobel Prize in Physics
Nobel Prize in Physics
for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or Pauli principle". The discovery involved spin theory, which is the basis of a theory of the structure of matter.Contents1 Biography1.1 Early years 1.2 Scientific research 1.3 Personality and reputation 1.4 Personal life2 Bibliography 3 References 4 Further reading 5 External linksBiography[edit] Early years[edit] Pauli was born in Vienna
Vienna
to a chemist Wolfgang Joseph Pauli (né Wolf Pascheles, 1869–1955) and his wife Bertha Camilla Schütz; his sister was Hertha Pauli, the writer and actress
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Eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that only changes by a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v
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Imaginary Unit
The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication
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Kronecker Delta
In mathematics, the Kronecker delta
Kronecker delta
(named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: δ i j = 0 if  i ≠ j , 1 if  i = j . displaystyle delta _ ij = begin cases 0& text if ineq j,\1& text if i=j.end cases where the Kronecker delta
Kronecker delta
δij is a piecewise function of variables i and j
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Involutory Matrix
In mathematics, an involutory matrix is a matrix that is its own inverse. That is, multiplication by matrix A is an involution if and only if A2 = I. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any nonsingular matrix multiplied by its inverse is the identity.[1]Contents1 Examples 2 Symmetry 3 Properties 4 See also 5 ReferencesExamples[edit] The 2 × 2 real matrix ( a b c − a ) displaystyle begin pmatrix a&b\c&-aend pmatrix is involutory provided that a 2 + b c = 1. displaystyle a^ 2 +bc=1
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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. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or A
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Trace Of A Matrix
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., tr ⁡ ( A ) = ∑ i = 1 n a i i = a 11 + a 22 + ⋯ + a n n displaystyle operatorname tr (A)=sum _ i=1 ^ n a_ ii =a_ 11 +a_ 22 +dots +a_ nn 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. This characterization can be used to define the trace of a linear operator in general
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Eigenvalues
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that only changes by a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v
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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
Hilbert space
H with finite Hilbert–Schmidt norm ‖ A ‖ H S 2 = T r ( A ∗ A ) := ∑ i ∈ I ‖ A e i ‖
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Hermitian
Numerous things are named after the French mathematician Charles Hermite (1822–1901):Contents1 Hermite 2 Hermite's 3 Hermitian 4 Astronomical objectsHermite[edit]Cubic Hermite spline, a type of third-degree spline Gauss–Hermite quadrature, an extension of Gaussian quadrature
Gaussian quadrature
method Hermite distribution, a parametrized family of discrete probability distributions Hermite–Lindemann theorem, theorem about transcendental numbers Hermite constant, a constant related to the geometry of certain lattices
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Normalisable Wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi, respectively). The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform
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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. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in applications in physics that do not distinguish between tangent and cotangent spaces
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Isomorphic
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 admits an inverse.[note 1] Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide
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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. There are different definitions used in group theory and ring theory.Contents1 Group theory1.1 Identities (group theory)2 Ring theory2.1 Identities (ring theory)3 Graded rings and algebras 4 Derivations4.1 General Leibniz rule5 See also 6 Notes 7 References 8 Further reading 9 External linksGroup theory[edit] The commutator of two elements, g and h, of a group G, is the element[g, h] = g−1h−1gh.It is equal to the group's identity if and only if g and h commute (i.e., if and only if gh = hg). The subgroup of G generated by all commutators is called the derived group or the commutator subgroup of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation
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