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The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
The term spin matrix refers to a number of matrices, which are related to Spin ... Pauli matrices, also called the "Pauli spin matrices". Generalizations of Pauli ...
Multi-qubit Pauli matrices can be written as products of single-qubit Paulis on disjoint qubits. Alternatively, when it is clear from context, the tensor product symbol can be omitted, i.e. unsubscripted Pauli matrices written consecutively represents tensor product rather than matrix product. For example:
Suppose there is a spin 1/2 particle in a state = [].To determine the probability of finding the particle in a spin up state, we simply multiply the state of the particle by the adjoint of the eigenspinor matrix representing spin up, and square the result.
The operator S u has eigenvalues of ± ħ / 2 , just like the usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot product of the direction with a vector of the three operators for the three x-, y-, z-axis directions.
The two-component helicity eigenstates satisfy ^ (^) = (^) where are the Pauli matrices, ^ is the direction of the fermion momentum, = depending on whether spin is pointing in the same direction as ^ or opposite.
Products of these Pauli spin operators can be used to fill the 3×3 table such that each row and each column contains a mutually commuting set of observables with eigenvalues +1 and −1, and with the product of the observables in each row being the identity operator, and the product of observables in each column equating to minus the identity ...