|
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. They are : These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. Each Pauli matrix is Hermitian, and together with the identity matrix (sometimes considered as the zeroth Pauli matrix ), the Pauli matrices (multiplied by ''real'' coefficients) span the full vector space of Hermitian matrices. In the language of quantum mechanics, Hermitian matrices are observables, so the Pauli matrices span the space of observables of the -dimensional complex Hilbert space. In the context of Pauli's work, is the observable corresponding to spin along the th coordinate axis in three-dimensional Euclidean space . The Pauli matrices (after multiplication by to make them anti-Hermitian), also generate transformations in the sense of Lie algebras: the matrices form a basis for , which exponentiates to the special unitary group SU(2). The algebra generated by the three matrices is isomorphic to the Clifford algebra of , called the algebra of physical space. == Algebraic properties == All three of the Pauli matrices can be compacted into a single expression: : where is the imaginary unit, and is the Kronecker delta, which equals +1 if and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of , in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations. The matrices are involutory: : where is the identity matrix. *The determinants and traces of the Pauli matrices are: : From above we can deduce that the eigenvalues of each are . *Together with the identity matrix (sometimes written as ), the Pauli matrices form an orthogonal basis, in the sense of Hilbert–Schmidt, for the real Hilbert space of complex Hermitian matrices, or the complex Hilbert space of all matrices. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pauli matrices」の詳細全文を読む スポンサード リンク
|