Pauli matrices

The Pauli matrices (after Wolfgang Pauli ), together with the 2 × 2 unit matrix, which is referred to in this context, a base of 4- dimensional complex vector space of complex Hermitian 2 × 2 matrices.

In quantum physics, the Pauli matrices represent the effect of the spin operators on spin- ½ - states ( such as electrons ), where the spin operators with respect to the basis defined below have the representation:

Are the Pauli matrices

( At this point, to discriminate the index of the imaginary unit. )

Representation

The Pauli matrices can be represented as matrices using the Dirac notation next to the display: Either the standard basis vectors or eigenvectors of the Pauli matrices can be used for the linear combination.

The vectors used are defined as follows, where the used of Bras are presented, which is indicated by "" by vectors:

Properties

The Pauli matrices are Hermitian and unitary. It follows with the fourth basic element defined by

The determinants and traces of the Pauli matrices are

From the above it follows that each of the Pauli matrix eigenvalues ​​having 1 and -1.

In addition:

The Pauli matrices satisfy the algebraic relation

( Is the Levi- Civita symbol ), ie, in particular up to a factor 2, the same relations as the angular momentum algebra

And the Clifford - Dirac algebra or

The Pauli matrices belong to the special case of angular momentum operators, based on the vectors of angular momentum - work multiplets with quantum numbers in measurement systems with the following:

It is a natural number, and for contact to the various quantum numbers. For the angular momentum operators act on the components of linear combinations of the two basis vectors and, therefore, by multiplying by the following matrices

With and then results that the angular momentum operators act on the components of Spin-1/2-Zuständen by multiplying by half the Pauli matrices.

Assigned rotation group, associated with Spin-1/2-Systemen

The linear hull of the multiplied with Pauli matrices with the usual matrix multiplication a Lie algebra, and because of with for each unit vector and all applicable real identity

These three matrices, the generators of the complex rotation group SU (2).

The factor 1/2 in the above equation is mathematically dispensable. The equation is frequently required in the physical application in precisely this form. Because ( as mentioned in the Introduction) provide in quantum physics is the matrix operators that describe the change of state of a Spin-1/2-Systems (eg an electron) when measuring the different spin components. On the other hand, describes the given by the exponential matrix, the change of the spin state at a spatial rotation. this corresponds to the directional angle of rotation about the axis of rotation oriented given by the unit vector. For arises; i.e. Spin-1/2-System is not rotating by the angle, but only by rotation at the angle to the initial state again transferred ( "spin ordre Hungen ").

Eigenvectors

The matrix, the eigenvectors of

As can easily be seen:

Corresponding to the eigenvalues. The eigenvectors of are

And the eigenvectors of

Isomorphic to the quaternions

The imaginary unit multiplied with the Pauli matrices together with the unit matrix, ie the quantity span a 4 -dimensional R- algebra which is isomorphic to the quaternions H. An isomorphic mapping, for example:

Known as the Einheitsquaternionen. Prior to this assignment allows each of the 24 automorphisms of the quaternion group Q8 turn. Thus, also an isomorphism be built " in reverse order":

Kronecker product of Pauli matrices

Pauli matrices can be used for the preparation of Hamiltonians and for approximating the exponential function of such operators. Are the four Pauli matrices, one can generate higher-dimensional matrices using the Kronecker product.

Properties of the Pauli matrices will also apply to these matrices. Are and two Kronecker products of Pauli matrices, then:

• Are matrices
• ( The identity matrix )
• Or ( commutativity )
• The Kronecker product of Pauli matrices are linearly independent and form a basis in the vector space of matrices. Hamiltonians of many physical models can be due to the base property as the sum of such matrices express ( linear combination ). In particular, let creation and annihilation operators of fermions that can finally take many states, simply expressed by them.

Examples of such models are the Hubbard model, Heisenberg model ( quantum mechanics) and Anderson model.

The Kronecker product of Pauli matrices occurs in the description of Spin-1/2-Systemen, which are composed of several subsystems. The relationship is given by the fact that the tensor product of two operators is given in the associated matrix representation just by the Kronecker product of matrices ( see Kronecker product # associated with tensor products ).

Approximation of the exponential of the Hamiltonian

Often one is interested in the exponential of the Hamiltonian.

Due to the commutativity of the matrices can be arranged arbitrarily in one product. Is a permutation, then:

Therefore, there are rational numbers with:

These rational numbers are, with some exceptions, difficult to calculate.

A first approximation is obtained by considering only summands which consist of commuting matrices.

The approximation can be further improved by, ... considered pairs, triples of non- commuting matrices.