Matrices important in quantum mechanics and the study of spin
Wolfgang Pauli (1900–1958), c. 1924. Pauli received the Nobel Prize in physics in 1945, nominated by Albert Einstein, for the Pauli exclusion principle.
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.
These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal / vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left).
Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ0), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices.
This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.
Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex two-dimensional Hilbert space. In the context of Pauli's work, σk represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space
The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 form a basis for the real Lie algebra , which exponentiates to the special unitary group SU(2).[a] The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of [1] and the (unital) associative algebra generated by iσ1, iσ2, iσ3 functions identically (is isomorphic) to that of quaternions ().
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Gull, S.F.; Lasenby, A.N.; Doran, C.J.L. (January 1993). "Imaginary numbers are not Real – the geometric algebra of spacetime" (PDF). Found. Phys. 23 (9): 1175–1201. Bibcode:1993FoPh...23.1175G. doi:10.1007/BF01883676. S2CID 14670523. Retrieved 5 May 2023 – via geometry.mrao.cam.ac.uk.
In mathematical physics and mathematics, the Paulimatrices are a set of three 2 × 2 complex matrices that are Hermitian, involutory and unitary. Usually...
term generalized Paulimatrices refers to families of matrices which generalize the (linear algebraic) properties of the Paulimatrices. Here, a few classes...
{I} \end{bmatrix}}} These two 4×4 matrices are related to the Dirac gamma matrices. Note that 0 and I are 2×2 matrices here. The next step is to look for...
\gamma ^{2},\gamma ^{3}\right\}\ ,} also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they...
introduced the 2×2 Paulimatrices as a basis of spin operators, thus solving the nonrelativistic theory of spin. This work, including the Pauli equation, is...
} 2 identity matrix and the matrices σ k ( k = 1 , 2 , 3 ) {\displaystyle \sigma _{k}\;(k=1,2,3)} are the Paulimatrices. This decomposition simplifies...
consisting of the 2 × 2 identity matrix I {\displaystyle I} and all of the Paulimatrices X = σ 1 = ( 0 1 1 0 ) , Y = σ 2 = ( 0 − i i 0 ) , Z = σ 3 = ( 1 0 0...
through Paulimatrices; see the 2 × 2 derivation for SU(2). For the general n × n case, one might use Ref. The Lie group of n × n rotation matrices, SO(n)...
Hermitian matrices include the Paulimatrices, the Gell-Mann matrices and their generalizations. In theoretical physics such Hermitian matrices are often...
degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants...
description of spins, we replace the spin variables with their respective Paulimatrices. However, depending on the direction of the magnetic field, we can create...
Elementary matrix Exchange matrix Matrix of ones Paulimatrices (the identity matrix is the zeroth Pauli matrix) Householder transformation (the Householder...
In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Paulimatrices and the 2 × 2 identity matrix. Explicitly...
Transparent dry-erase sphere used to teach spherical geometry Paulimatrices – Matrices important in quantum mechanics and the study of spin Quaternionic...
}}\equiv (\sigma _{1},\sigma _{2},\sigma _{3})} is the vector form of Paulimatrices. Matrices of this form have the following properties, which relate them intrinsically...
discussing multi-qubit circuits. The Pauli gates ( X , Y , Z ) {\displaystyle (X,Y,Z)} are the three Paulimatrices ( σ x , σ y , σ z ) {\displaystyle (\sigma...
U(s)=e^{iKs}} Since the Paulimatrices ( σ x , σ y , σ z ) {\displaystyle (\sigma _{x},\sigma _{y},\sigma _{z})} are unitary Hermitian matrices and have eigenvectors...
combination of the Paulimatrices, which together with the identity matrix provide a basis for 2 × 2 {\displaystyle 2\times 2} self-adjoint matrices:: 126 ρ =...
which normalize the n-qubit Pauli group, i.e., map tensor products of Paulimatrices to tensor products of Paulimatrices through conjugation. The notion...
of Pauli matrices Gamma matrices, which can be represented in terms of the Paulimatrices. Higher-dimensional gamma matrices In pure mathematics and physics:...
called the error basis (which is given by the Paulimatrices and the identity). To correct the error, the Pauli operator corresponding to the type of error...
the generators in the Lie algebra involved, express the Paulimatrices in terms of t-matrices, σ → 2i t, so that a ′ ↦ − θ 2 , b ′ ↦ − ϕ 2 . {\displaystyle...
are also used instead). In quantum mechanics, σ is used to indicate Paulimatrices. In astronomy, σ represents velocity dispersion. In astronomy, the prefix...
article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular...
gamma matrices (known as Dirac matrices) and i is the imaginary unit. A second application of the Dirac operator will now reproduce the Pauli term exactly...
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