Matrices of Pauli

The matrices of Pauli , developed by Wolfgang Pauli, form a bases KNOWN (2).

They are defined like the whole of matrix S complex S of size 2 × 2:

\ sigma_x = \ begin {bmatrix}
0 & 1 \ \ 1 & 0 \ end {bmatrix}
\ sigma_y = \ begin {bmatrix}
0 & - I \ \ I & 0 \ end {bmatrix}
\ sigma_z = \ begin {bmatrix}
1 & 0 \ \ 0 & -1 \ end {bmatrix} (where I is the “imaginary” unit)

These matrices (known as “Matrices of Pauli”) are often used in quantum Mécanique to represent the Spin particle S.

Properties

Identities

\ sigma_1^2 = \ sigma_2^2 = \ sigma_3^2 = \ begin {pmatrix} 1&0 \ \ 0&1 \ end {pmatrix} = I where I is the Matrice identity.
\ sigma_1 \ sigma_2 = I \ sigma_3 \, \!
\ sigma_3 \ sigma_1 = I \ sigma_2 \, \!
\ sigma_2 \ sigma_3 = I \ sigma_1 \, \!
\ sigma_i \ sigma_j = - \ sigma_j \ sigma_i \ mbox {for} I \ J \, \!

Other properties

The Determinant and the trace of the matrices of Pauli are:

\begin{matrix}

\ det (\ sigma_i) &=& -1 & \ \ \ operatorname {Tr} (\ sigma_i) &=& 0 & \ quad \ hbox {for} \ I = 1,2,3 \end{matrix}

Consequently, the eigenvalues of each matrix are \ pm 1.

The matrices of Pauli obey the relation of Commutativité and following Anticommutativité:

\begin{matrix}

\ sigma_j &=& 2 I \, \ epsilon_ {I J K} \, \ sigma_k \ \ \ {\ sigma_i, \ sigma_j \} &=& 2 \ delta_ {I J} \ cdot I \end{matrix}

where εijk is the Symbole of Levi-Civita, δij is the Delta of Kronecker and I is the Matrice identity. Relations Ci-high can be checked while using:

\ sigma_i \ sigma_j = I \ epsilon_ {ijk} \ sigma_k + \ delta_ {ij} \ cdot I.

These relations of commutation are similar to those on the Algèbre of Dregs known (2) and, indeed, known (2) can be interpreted as the argèbre of Dregs of all the linear combinations of imaginary I time the matrices of Pauli iσj, in other words, like the matrices anti Hermitien born 2×2 with trace of 0. In this direction, the matrices of Pauli generate known (2). Consequently, iσj can be seen as the infinitesimal generating of the Groupe of Dregs corresponding KNOWN (2).

The algebra of known (2) is isomorphous with the algebra of Dregs so (3), which correspond to the group of Dregs SO (3), the group of rotations in three dimensions. In other words, the iσj are achievements of “infinitesimal” rotations in a space with three dimensions (in fact, they are the achievements of lower dimension).

Physics

In quantum Mécanique the iσj represent the generators of rotations on the particles not relativists of Spin ½. The state of these particles is represented by Spineur S with two components, which is the fundamental representation of KNOWN (2). An interesting property of the particles of spin ½ is that they must undergo a rotation of 4π radians in order to return in their configuration of origin. This is due to the fact that KNOWN (2) and SO (3) are not globablement isomorphous, in spite of the fact that their infinitesimal generator, known (2) and so (3), is isomorphous. KNOWN (2) is in fact a “coating of degree two” of SO (3): to each element of SO (3) correspond two elements of KNOWN (2).

In quantum mechanics with several particles, the Groupe of Pauli Gn is also useful. It is defined like all the tensor produced with N dimensions of matrices of Pauli.

With the matrix identity I, sometimes indicated σ0, the matrices of Pauli form a bases vector Space real of the square matrices complex 2 × 2. This base is equivalent to the Quaternion S. When that used as bases for the operator of rotation of spin ½, east is identical to that for the representation of corresponding rotation of quaternion.

See too

Reference

  • Liboff, Richard L. (2002). Introductory Mechanics Quantum, Addison-Wesley. ISBN 0805387145.

Random links:Mario Golf: Advance Turn | Air navigator | Síndrome falso de la memoria | Small estate | Ultra vivid scene (album) | Grand Prix of Fall | Graver