Pauli matrices - Citizendia

The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted A Hermitian matrix (or self-adjoint matrix) is a Square matrix with complex entries which is equal to its own Conjugate transpose &mdash that In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^* In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Usually indicated by the Greek letter 'sigma' (σ), they are occasionally denoted with a 'tau' (τ) when used in connection with isospin symmetries. In Physics, and specifically Particle physics, isospin ( isotopic spin, isobaric spin) is a Quantum number related to the They are:

$\sigma_1 = \sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$

$\sigma_2 = \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}$

$\sigma_3 = \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$

The name refers to Wolfgang Pauli.

1 Algebraic properties
- 1.1 Commutation relations
2 SU(2)
3 Physics
- 3.1 Quantum mechanics
- 3.2 Quantum information
4 See also
5 References

Algebraic properties

$\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I$

where I is the identity matrix. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main

The determinants and traces of the Pauli matrices are:

$\begin{matrix} \det (\sigma_i) &=& -1 & \\[1ex] \operatorname{Tr} (\sigma_i) &=& 0 & \quad \hbox{for}\ i = 1, 2, 3 \end{matrix}$

From above we can deduce that the eigenvalues of each σ_i are ±1. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n 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 In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

Together with the identity matrix I (which is sometimes written as σ₀), the Pauli matrices form an orthogonal basis, in the sense of Hilbert-Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices. In Mathematics, a Hilbert–Schmidt operator is a Bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm This article assumes some familiarity with Analytic geometry and the concept of a limit.

Commutation relations

The Pauli matrices obey the following commutation and anticommutation relations:

$\begin{matrix} [\sigma_a, \sigma_b] &=& 2 i \sum_c \varepsilon_{a b c}\,\sigma_c \\[1ex] \{\sigma_a, \sigma_b\} &=& 2 \delta_{a b} \cdot I \end{matrix}$

where $\varepsilon_{abc}$ is the Levi-Civita symbol, $δ a b$ is the Kronecker delta, and I is the identity matrix. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. The Levi-Civita symbol, also called the Permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in Tensor In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

The above two relations can be summarized as:

$\sigma_a \sigma_b = \delta_{ab} \cdot I + i \sum_c \varepsilon_{abc} \sigma_c \,$ .

For example,

$\begin{matrix} \sigma_1\sigma_2 &=& i\sigma_3,\\ \sigma_2\sigma_3 &=& i\sigma_1,\\ \sigma_2\sigma_1 &=& -i\sigma_3,\\ \sigma_1\sigma_1 &=& I.\\ \end{matrix}$

The Pauli vector is defined by

$\vec{\sigma} = \sigma_1 \hat{x} + \sigma_2 \hat{y} + \sigma_3 \hat{z} \,$

and the summary equation for the commutation relations can be used to prove

$(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = \vec{a} \cdot \vec{b} + i \vec{\sigma} \cdot ( \vec{a} \times \vec{b} ) \quad \quad \quad \quad (1) \,$

(as long as the vectors a and b commute with the pauli matrices)

as well as

$e^{i (\vec{a} \cdot \vec{\sigma})} = \cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a} \quad \quad \quad \quad \quad \quad (2) \,$

for $\vec{a} = a \hat{n}$ .

Proof of (1)

$(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) \,$	$= a_i \sigma_i b_j \sigma_j \,$
	$= a_i b_j \sigma_i \sigma_j \,$
	$= a_i b_j \left( \delta_{ij} \cdot I+ i \varepsilon_{ijk} \sigma_k \right) \,$
	$= a_i b_j \delta_{ij} \cdot I+ i \sigma_k \varepsilon_{ijk} a_i b_j \,$
	$= \vec{a} \cdot \vec{b} + i \vec{\sigma} \cdot ( \vec{a} \times \vec{b} )\,$

Proof of (2)

First notice that for even powers

$(\hat{n} \cdot \vec{\sigma})^{2n} = I \,$

but for odd powers

$(\hat{n} \cdot \vec{\sigma})^{2n+1} = \hat{n} \cdot \vec{\sigma} \,$

Combine these two facts with the knowledge of the relation of the exponential to sine and cosine:

$e^{ix} \,$	$= \sum_{n=0}^\infty{\frac{i^n x^n}{n!}} \,$
	$= \sum_{n=0}^\infty{\frac{(-1)^n x^{2n}}{2n!}} + i\sum_{n=0}^\infty{\frac{(-1)^n x^{2n+1}}{(2n+1)!}} \,$

Which, when we use $x = a (\hat{n} \cdot \vec{\sigma}) \,$ gives us

$= \sum_{n=0}^\infty{\frac{(-1)^n (a\hat{n}\cdot \vec{\sigma})^{2n}}{2n!}} + i\sum_{n=0}^\infty{\frac{(-1)^n (a\hat{n}\cdot \vec{\sigma})^{2n+1}}{(2n+1)!}} \,$

$= \sum_{n=0}^\infty{\frac{(-1)^n a^{2n}}{2n!}} + i (\hat{n}\cdot \vec{\sigma}) \sum_{n=0}^\infty{\frac{(-1)^n a^{2n+1}}{(2n+1)!}} \,$

The sum on the left is cosine, and the sum on the right is sine so finally,

$e^{i a(\hat{n} \cdot \vec{\sigma})} = \cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a} \,$

SU(2)

The matrix group SU(2) is a Lie group, and its Lie algebra is the set of the anti-Hermitian 2×2 matrices with trace 0. Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Direct calculation shows that the Lie algebra SU(2) is the 3 dimensional real algebra spanned by the set {i σ_j}. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector In symbols,

$\; \operatorname{SU}(2) = \operatorname{span} \{ i \sigma_1, i \sigma_2 , i \sigma_3 \}.$

As a result, i σ_js can be seen as infinitesimal generators of SU(2). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

A Cartan decomposition of SU(2)

This relationship between the Pauli matrices and SU(2) can be explored further, as can be seen from the following simple example. We can write

$\; \operatorname{SU}(2) = \operatorname{span} \{i \sigma_2\} \oplus \operatorname{span} \{ i \sigma_1, i \sigma_3\}.$

We put

$\; \mathfrak{k} = \operatorname{span} \{i \sigma_3\},$

and

$\; \mathfrak{p} = \operatorname{span} \{ i \sigma_1, i \sigma_2\}$

Using the algebraic identities listed in the previous section, it can be verified that $\mathfrak{k}$ and $\mathfrak{p}$ form a Cartan pair of the Lie algebra SU(2). 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 Furthermore,

$\; \mathfrak{a} = \operatorname{span} \{ i \sigma_2\}$

is a maximal abelian subalgebra of $\mathfrak{p}$ . Now, a version of Cartan decomposition states that any element U in the Lie group SU(2) can be expressed in the form

$U = e^{k_1} e^a e^{k_2}\,\!$ where $k_1, k_2 \in \mathfrak{k}$ and $a \in \mathfrak{a}.$

In other words, any unitary U of determinant 1 is of the form

$U = e^{i \alpha \sigma_1} e^{i \beta \sigma_2} e^{i \gamma \sigma_3}\,\!$

for some real numbers α, β, and γ. 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

Extending to unitary matrices gives that any unitary 2 × 2 U is of the form

$U = e^{i \delta} e^{i \alpha \sigma_1} e^{i \beta \sigma_2} e^{i \gamma \sigma_3}\,\!$

where the additional parameter δ is also real.

SO(3)

The Lie algebra SU(2) is isomorphic to the Lie algebra SO(3), which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective This article is about rotations in three-dimensional Euclidean space In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In other words, one can say that $i σ j$ 's are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. It might be of interest here to note that even though their infinitesimal generators SU(2) and SO(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a two-to-one homomorphism from SU(2) to SO(3). In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism

Quaternions

Consider the real linear span S of {I, σ₁ σ₂, σ₂ σ₃, σ₃ σ₁}. S is isomorphic to the real algebra of quaternions H. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician The isomorphism from H to S is given by

$1 \simeq 1, i \simeq \sigma_1 \sigma_2, j \simeq \sigma_3 \sigma_1, k \simeq \sigma_2 \sigma_3.$

As the quaternions of unit norm is group-isomorphic to SU(2), this gives yet another way of describing SU(2) via the Pauli matrices. The two-to-one homomorphism from SU(2) to SO(3) can also be explicitly given in terms of the Pauli matrices in this formulation.

Quaternions form a division algebra - there always is an inverse - whereas Pauli matrices do not.

Physics

Quantum mechanics

In quantum mechanics, each Pauli matrix represents an observable describing the spin of a spin ½ particle in the three spatial directions. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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 In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Quantum mechanics, spin is an intrinsic property of all elementary particles. Also, as an immediate consequence of the Cartan decomposition mentioned above, $i σ j$ are the generators of rotation acting on non-relativistic particles with spin ½. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. The state of the particles are represented as two-component spinors. The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the An interesting property of spin ½ particles is that they must be rotated by an angle of 4 $π$ in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north/south pole on the 2-sphere S², they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics, two Vectors are orthogonal if they are Perpendicular, i This article assumes some familiarity with Analytic geometry and the concept of a limit.

For a spin ¹⁄₂ particle, the spin operator is given by $\mathbf{J} =\frac\hbar2\boldsymbol{\sigma}$ . The Pauli matrices can be generalized to describe higher spin systems in three spatial dimensions. The spin matrices for spin 1 and spin ³⁄₂ are given below:

j=1:

$J_x = \frac\hbar\sqrt{2} \begin{pmatrix} 0&1&0\\ 1&0&1\\ 0&1&0 \end{pmatrix}$

$J_y = \frac\hbar\sqrt{2} \begin{pmatrix} 0&-i&0\\ i&0&-i\\ 0&i&0 \end{pmatrix}$

$J_z = \hbar \begin{pmatrix} 1&0&0\\ 0&0&0\\ 0&0&-1 \end{pmatrix}$

j=³⁄₂:

$J_x = \frac\hbar2 \begin{pmatrix} 0&\sqrt{3}&0&0\\ \sqrt{3}&0&2&0\\ 0&2&0&\sqrt{3}\\ 0&0&\sqrt{3}&0 \end{pmatrix}$

$J_y = \frac\hbar2 \begin{pmatrix} 0&-i\sqrt{3}&0&0\\ i\sqrt{3}&0&-2i&0\\ 0&2i&0&-i\sqrt{3}\\ 0&0&i\sqrt{3}&0 \end{pmatrix}$

$J_z = \frac\hbar2 \begin{pmatrix} 3&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-3 \end{pmatrix}$

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_n is defined to consist of all n-fold tensor products of Pauli Matrices. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually

The fact that any 2 × 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states (2 × 2 positive semidefinite matrices with trace 1). In Quantum mechanics, the Bloch sphere is a geometrical representation of the Pure state space of a two-level quantum mechanical system named after This can be seen by simply first writing a Hermitian matrix as a real linear combination of {σ₀, σ₁, σ₂, σ₃} then impose the positive semidefinite and trace 1 assumptions. 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

Quantum information

In quantum information, single-qubit quantum gates are 2 × 2 unitary matrices. In Quantum mechanics, quantum information is Physical information that is held in the "state" of a Quantum system. A qubit is not to be confused with a Cubit, which is an ancient measure of length A quantum gate or quantum logic gate is a basic Quantum circuit operating on a small number of Qubits They are the analogues for Quantum computers The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the Z-Y decomposition of a single-qubit gate. Choosing a different Cartan pair gives a similar X-Y decomposition of a single-qubit gate.

References

Liboff, Richard L. (2002). In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the infinitesimal generators of the Special unitary group called In Mathematics and Physics, in particular Quantum information, the term generalized Pauli matrices refers to families of matrices which generalize In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime The Pauli Equation, also known as the Schrödinger-Pauli equation is the formulation of the Schrödinger equation for spin one-half particles which takes into account Richard L Liboff is a US physicist who has authored five books and nearly 150 other publications in variety of fields including Plasma physics, Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.

Schiff, Leonard I. (1968). Quantum Mechanics. McGraw-Hill. ISBN 007-Y85643-5.

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