NationMaster - Encyclopedia: Pauli matrices

The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. Usually indicated by the Greek letter 'sigma' (σ), they are occasionally denoted with a 'tau' (τ) when used in connection with isospin symmetries. They are: In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose â€” that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... Isospin (isotopic spin, isobaric spin) is a physical quantity which is mathematically analogous to spin. ...

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

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

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

The name refers to Wolfgang Pauli. This article is about Austrian-Swiss physicist 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

$sigma_1^2 = sigma_2^2 = sigma_3^2 = begin{pmatrix} 1&00&1end{pmatrix} = I$

where I is the identity matrix. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

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 nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... 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. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...

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 H1->H2 such that there exists an orthonormal basis of H1 such that is finite. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

Commutation relations

$sigma_1sigma_2 = isigma_3,!$

$sigma_3sigma_1 = isigma_2,!$

$sigma_2sigma_3 = isigma_1,!$

$sigma_isigma_j = -sigma_jsigma_imbox{ for }ine j,!$

The Pauli matrices obey the following commutation and anticommutation relations:

$begin{matrix} [sigma_i, sigma_j] &=& 2 i,varepsilon_{i j k},sigma_k [1ex] {sigma_i, sigma_j} &=& 2 delta_{i j} cdot I end{matrix}$

where $varepsilon_{ijk}$ is the Levi-Civita symbol,

δ i j

is the Kronecker delta, and I is the identity matrix.

The above two relations can be summarized as: In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if... The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

$sigma_i sigma_j = delta_{ij} cdot I + i varepsilon_{ijk} sigma_k ,$ .

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 (for $vec{a} = a hat{n}$ )

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

Proof of (1)

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

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:

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

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

$= sum_{n=0}^infty{frac{(-1)^n (ahat{n}cdot sigma)^{2n}}{2n!}} + isum_{n=0}^infty{frac{(-1)^n (ahat{n}cdot sigma)^{2n+1}}{(2n+1)!}} ,$

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

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. Direct calculation shows that the Lie algebra su(2) is the 3 dimensional real algebra spanned by the set {i σ_j}. In symbols, In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

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

As a result, i σ_js can be seen as infinitesimal generators of SU(2). In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

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). Furthermore, In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups. ...

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

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 In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups. ...

$U = e^{k_1} e^a e^{k_2},!$ 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

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

for some real numbers α, β, and γ.

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

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

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 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, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... This picture illustrates how the hours in a clock form a group. ... A sphere rotating around its axis. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint...

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. The isomorphism from H to S is given by In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

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

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. 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. The state of the particles are represented as two-component spinors. 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.

For a spin ¹⁄₂ particle, the spin operator is given by $mathbf{J} =frachbar2boldsymbol{sigma}$ . 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: Fig. ... 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. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... Two-dimensional analogy of space-time curvature described in General Relativity. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... For other uses, see sphere (disambiguation). ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

$J_x = frachbarsqrt{2} begin{pmatrix} 0&1&0 1&0&1 0&1&0 end{pmatrix}$

$J_y = frachbarsqrt{2} begin{pmatrix} 0&-i&0 i&0&-i 0&i&0 end{pmatrix}$

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

j=³⁄₂:

$J_x = frachbar2 begin{pmatrix} 0&sqrt{3}&0&0 sqrt{3}&0&2&0 0&2&0&sqrt{3} 0&0&sqrt{3}&0 end{pmatrix}$

$J_y = frachbar2 begin{pmatrix} 0&-isqrt{3}&0&0 isqrt{3}&0&-2i&0 0&2i&0&-isqrt{3} 0&0&isqrt{3}&0 end{pmatrix}$

$J_z = frachbar2 begin{pmatrix} 3&0&0&0 0&1&0&0 0&0&-1&0 0&0&0&-3 end{pmatrix}$

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.

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). 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.

Fig. ... This article is about Austrian-Swiss physicist Wolfgang Pauli. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Bloch sphere In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a 2-level quantum mechanical system. ... The term mixed state refers to a concept in physics, particularly quantum mechanics. ... 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. ...

Quantum information

In quantum information, single-qubit quantum gates are 2 × 2 unitary matrices. 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.

In quantum mechanics, quantum information is physical information that is held in the state of a quantum system. ... To meet Wikipedias quality standards and make it more accessible, this article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... A quantum gate or quantum logic gate is a rudimentary quantum circuit operating on a small number of qubits. ...

References

Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.

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

Categories: Lie groups | Matrices | Rotational symmetry Richard L. Liboff is a U.S. physicist who has authored five books and nearly 150 other publications in variety of fields, including plasma physics, planetary physics, cosmology, quantum chaos, and quantum billiards. ...

Results from FactBites:

Wolfgang Pauli (647 words)
	Pauli was born in Vienna, Austria on August 25, 1900.
	Pauli moved to the United States in 1940, where he was Professor of Theoretical Physics at Princeton.
	In 1945, Pauli received the Nobel Prize in Physics for his "decisive contribution through his discovery in 1925 of a new law of Nature, the exclusion principle or Pauli principle." He had been nominated for the prize by Einstein.

Pauli matrices - Wikipedia, the free encyclopedia (811 words)
	The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices.
	The determinants and traces of the Pauli matrices are:
	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).

More results at FactBites »

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