En física matemática y matemáticas , las matrices de Pauli son un conjunto de tres matrices complejas de 2 × 2 que son hermitianas y unitarias . [1] Usualmente indicados por la letra griega sigma ( σ ), ocasionalmente son denotados por tau ( τ ) cuando se usan en conexión con simetrías isospin .
Estas matrices llevan el nombre del físico Wolfgang Pauli . En mecánica cuántica , ocurren en la ecuación de Pauli que tiene en cuenta la interacción del espín de una partícula con un campo electromagnético externo .
Cada matriz de Pauli es hermitiana y, junto con la matriz de identidad I (a veces considerada como la matriz cero de Pauli σ 0 ), las matrices de Pauli forman una base para el espacio vectorial real de las matrices hermitianas de 2 × 2 . Esto significa que cualquier matriz hermitiana de 2 × 2 puede escribirse de forma única como una combinación lineal de matrices de Pauli, siendo todos los coeficientes números reales.
Hermitianos representan observables en la mecánica cuántica, por lo que las matrices de Pauli abarcan el espacio de observables de la 2 complejo -dimensional espacio de Hilbert . En el contexto del trabajo de Pauli, σ k representa el observable correspondiente al giro a lo largo del k- ésimo eje de coordenadas en el espacio euclidiano tridimensional ℝ 3 .
Las matrices de Pauli (después de multiplicarlas por i para hacerlas antihermitianas ) también generan transformaciones en el sentido de las álgebras de Lie : las matrices iσ 1 , iσ 2 , iσ 3 forman una base para el álgebra de Lie real, que se expone al grupo unitario especial SU (2) . [nb 1] El álgebra generada por las tres matrices σ 1 , σ 2 , σ 3 es isomórfica al álgebra de Clifford de ℝ 3 , y el álgebra (asociativa unital) generada por iσ 1 , iσ 2 , iσ 3 es isomorfo de cuaterniones .
Propiedades algebraicas
Las tres matrices de Pauli se pueden compactar en una sola expresión:
donde i = √ −1 es la unidad imaginaria , y δ ab es el delta de Kronecker , que es igual a +1 si a = by 0 en caso contrario. Esta expresión es útil para "seleccionar" cualquiera de las matrices numéricamente sustituyendo valores de a = 1, 2, 3 , a su vez útil cuando cualquiera de las matrices (pero ninguna en particular) se va a utilizar en manipulaciones algebraicas.
Las matrices son involutivas :
donde yo es la matriz de identidad .
Los determinantes y trazas de las matrices de Pauli son:
De lo cual, podemos deducir que los valores propios de cada σ i son ± 1 .
Con la inclusión de la matriz identidad, I (a veces denotada como σ 0 ), las matrices de Pauli forman una base ortogonal (en el sentido de Hilbert-Schmidt ) del espacio real de Hilbert de matrices hermitianas complejas de 2 × 2 ,, y el complejo espacio de Hilbert de todas las matrices 2 × 2 ,.
Autovectores y autovalores
Cada una de las matrices de Pauli ( hermitianas ) tiene dos valores propios , +1 y -1 . Usando una convención en la que antes de la normalización, el 1 se coloca en las posiciones superior e inferior de las funciones de onda + y - respectivamente, los vectores propios normalizados correspondientes son:
Una ventaja de usar esta convención es que las funciones de onda + y - pueden estar relacionadas entre sí, usando las propias matrices de Pauli, por ψ x + = iσ y ψ x - , ψ y + = σ z ψ y - , y ψ z + = σ x ψ z - .
Vector de Pauli
El vector de Pauli está definido por [nb 2]
y proporciona un mecanismo de mapeo desde una base de vector a una base de matriz de Pauli [2] de la siguiente manera,
usando la convención de suma . Más,
sus valores propios son , y además (ver completitud, a continuación)
Sus vectores propios normalizados son
Relaciones de conmutación
Las matrices de Pauli obedecen a las siguientes relaciones de conmutación :
y relaciones anticonmutación :
donde la constante de estructura ε abc es el símbolo de Levi-Civita , se usa la notación sumatoria de Einstein, δ ab es el delta de Kronecker e I es la matriz identidad 2 × 2 .
Por ejemplo,
Relación con el producto puntual y cruzado
Los vectores de Pauli mapean elegantemente estas relaciones de conmutación y anticonmutación con los correspondientes productos vectoriales. Agregar el conmutador al anticonmutador da
así que eso,
Contratando cada lado de la ecuación con componentes de dos 3 -vectores a p y b q (que conmutan con las matrices de Pauli, es decir, a p σ q = σ q a p ) para cada matriz σ q y componente vectorial a p (y igualmente con b q ), y reetiquetar los índices a , b , c → p , q , r , para evitar conflictos de notación, produce
Finalmente, la traducción de la notación de índice para el producto escalar y el producto cruzado da como resultado
|
| ( 1 ) |
Si i se identifica con el pseudoescalar σ x σ y σ z, entonces el lado derecho se convierte en que es también la definición del producto de dos vectores en álgebra geométrica.
Algunas trazas de relaciones
Las siguientes trazas se pueden derivar utilizando las relaciones de conmutación y anticonmutación.
Si también se considera la matriz σ 0 = I , estas relaciones se vuelven
where Greek indices α, β, γ and μ assume values from {0, x, y, z} and the notation is used to denote the sum over the cyclic permutation of the included indices.
Exponential of a Pauli vector
For
one has, for even powers, 2p, p = 0, 1, 2, 3, …
which can be shown first for the p = 1 case using the anticommutation relations. For convenience, the case p = 0 is taken to be I by convention.
For odd powers, 2q + 1, q = 0, 1, 2, 3, …
Matrix exponentiating, and using the Taylor series for sine and cosine,
- .
In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,
|
| (2) |
which is analogous to Euler's formula, extended to quaternions.
Note that
- ,
while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2).
A more abstract version of formula (2) for a general 2 × 2 matrix can be found in the article on matrix exponentials. A general version of (2) for an analytic (at a and −a) function is provided by application of Sylvester's formula,[3]
The group composition law of SU(2)
A straightforward application of formula (2) provides a parameterization of the composition law of the group SU(2).[nb 3] One may directly solve for c in
which specifies the generic group multiplication, where, manifestly,
the spherical law of cosines. Given c, then,
Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to[4]
(Of course, when ̂n is parallel to ̂m, so is ̂k, and c = a + b.)
Adjoint action
It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation effectively by double the angle a,
Completeness relation
An alternative notation that is commonly used for the Pauli matrices is to write the vector index i in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the i-th Pauli matrix is σ iαβ.
In this notation, the completeness relation for the Pauli matrices can be written
- Proof: The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the complex Hilbert space of all 2 × 2 matrices means that we can express any matrix M as
- where c is a complex number, and a is a 3-component complex vector. It is straightforward to show, using the properties listed above, that
- where "tr" denotes the trace, and hence that
- which can be rewritten in terms of matrix indices as
- where summation is implied over the repeated indices γ and δ. Since this is true for any choice of the matrix M, the completeness relation follows as stated above.
As noted above, it is common to denote the 2 × 2 unit matrix by σ0, so σ0αβ = δαβ. The completeness relation can alternatively be expressed as
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' density matrix, (2 × 2 positive semidefinite matrices with unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of {σ0, σ1, σ2, σ3} as above, and then imposing the positive-semidefinite and trace 1 conditions.
For a pure state, in polar coordinates, , the idempotent density matrix
acts on the state eigenvector with eigenvalue 1, hence like a projection operator for it.
Relation with the permutation operator
Let Pij be the transposition (also known as a permutation) between two spins σi and σj living in the tensor product space ,
This operator can also be written more explicitly as Dirac's spin exchange operator,
Its eigenvalues are therefore[5] 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.
SU (2)
The group SU(2) is the Lie group of unitary 2 × 2 matrices with unit determinant; its Lie algebra is the set of all 2 × 2 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra is the 3-dimensional real algebra spanned by the set {iσj}. In compact notation,
As a result, each iσj can be seen as an infinitesimal generator of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper representation of su(2), as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is λ = 1/2, so that
As SU(2) is a compact group, its Cartan decomposition is trivial.
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 the iσj are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though 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 group homomorphism from SU(2) to SO(3), see relationship between SO(3) and SU(2).
Quaternions
The real linear span of {I, iσ1, iσ2, iσ3} is isomorphic to the real algebra of quaternions ℍ. The isomorphism from ℍ to this set is given by the following map (notice the reversed signs for the Pauli matrices):
Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,[6]
As the set of versors U ⊂ ℍ forms a group isomorphic to SU(2), U gives yet another way of describing SU(2). The two-to-one homomorphism from SU(2) to SO(3) may be given in terms of the Pauli matrices in this formulation.
Física
Classical mechanics
In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters.[7] The matrix P corresponding to the position of a point in space is defined in terms of the above Pauli vector matrix,
Consequently, the transformation matrix Qθ for rotations about the x-axis through an angle θ may be written in terms of Pauli matrices and the unit matrix as[7]
Similar expressions follow for general Pauli vector rotations as detailed above.
Quantum mechanics
In quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observable describing the spin of a spin ½ particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, iσj are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin ½. The states of the particles are represented as two-component spinors. In the same way, the Pauli matrices are related to the isospin operator.
An interesting property of spin ½ particles is that they must be rotated by an angle of 4π 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 ½ particle, the spin operator is given by J = ħ/2σ, the fundamental representation of SU(2). By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3)#A note on Lie algebra. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple.[8]
Also useful in the quantum mechanics of multiparticle systems, the general Pauli group Gn is defined to consist of all n-fold tensor products of Pauli matrices.
Relativistic quantum mechanics
In relativistic quantum mechanics, the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as
It follows from this definition that matrices have the same algebraic properties as σi matrices.
However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence needs to be replaced by Σμν, the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the Σμν are also antisymmetric. Hence there are only six independent matrices.
The first three are the The remaining three, , where the Dirac αi matrices are defined as
The relativistic spin matrices Σμν are written in compact form in terms of commutator of gamma matrices as
- .
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.
Ver también
- Spinors in three dimensions
- Gamma matrices
- § Dirac basis
- Angular momentum
- Gell-Mann matrices
- Poincaré group
- Generalizations of Pauli matrices
- Bloch sphere
- Euler's four-square identity
- For higher spin generalizations of the Pauli matrices, see spin (physics) § Higher spins
- Exchange matrix (the second Pauli matrix is an exchange matrix of order two)
Observaciones
- ^ This conforms to the mathematics convention for the matrix exponential, iσ ↦ exp(iσ). In the physics convention, σ ↦ exp(−iσ), hence in it no pre-multiplication by i is necessary to land in SU(2).
- ^ The Pauli vector is a formal device. It may be thought of as an element of M2(ℂ) ⊗ ℝ3, where the tensor product space is endowed with a mapping ⋅ : ℝ3 × (M2(ℂ) ⊗ ℝ3) → M2(ℂ) induced by the dot product on ℝ3.
- ^ N.B. The relation among a, b, c, n, m, k derived here in the 2 × 2 representation holds for all representations of SU(2), being a group identity. Note that, by virtue of the standard normalization of that group's generators as half the Pauli matrices, the parameters a,b,c correspond to half the rotation angles of the rotation group.
Notas
- ^ "Pauli matrices". Planetmath website. 28 March 2008. Retrieved 28 May 2013.
- ^ See the spinor map.
- ^ Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-63235-5. OCLC 43641333.
- ^ cf. J W Gibbs (1884). Elements of Vector Analysis, New Haven, 1884, p. 67. In fact, however, the formula goes back to Olinde Rodrigues, 1840, replete with half-angle: "Des lois géometriques qui regissent les déplacements d' un systéme solide dans l' espace, et de la variation des coordonnées provenant de ces déplacement considérées indépendant des causes qui peuvent les produire", J. Math. Pures Appl. 5 (1840), 380–440; online
- ^ Explicitly, in the convention of "right-space matrices into elements of left-space matrices", it is
- ^ Nakahara, Mikio (2003). Geometry, topology, and physics (2nd ed.). CRC Press. ISBN 978-0-7503-0606-5., pp. xxii.
- ^ a b Goldstein, Herbert (1959). Classical Mechanics. Addison-Wesley. pp. 109–118.
- ^ Curtright, T L; Fairlie, D B; Zachos, C K (2014). "A compact formula for rotations as spin matrix polynomials". SIGMA. 10: 084. arXiv:1402.3541. Bibcode:2014SIGMA..10..084C. doi:10.3842/SIGMA.2014.084. S2CID 18776942.
Referencias
- Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.
- Schiff, Leonard I. (1968). Quantum Mechanics. McGraw-Hill. ISBN 978-0070552876.
- Leonhardt, Ulf (2010). Essential Quantum Optics. Cambridge University Press. ISBN 978-0-521-14505-3.