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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 ...
Pauli matrices, also called the "Pauli spin matrices". Generalizations of Pauli matrices Gamma matrices , which can be represented in terms of the Pauli matrices.
Each Rademacher operator acts on one particular fermion coordinate only, and there it is a Pauli matrix. It may be identified with the observable measuring spin component of that fermion along one of the axes {,,} in spin space. Thus, a Walsh operator measures the spin of a subset of fermions, each along its own axis.
The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli. The Pauli group on n {\displaystyle n} qubits, G n {\displaystyle G_{n}} , is the group generated by the operators described above applied to each of n {\displaystyle n} qubits in the tensor product Hilbert space ( C 2 ) ⊗ n {\displaystyle ...
The Clifford group is defined as the group of unitaries that normalize the Pauli group: = {† =}. Under this definition, C n {\displaystyle \mathbf {C} _{n}} is infinite, since it contains all unitaries of the form e i θ I {\displaystyle e^{i\theta }I} for a real number θ {\displaystyle \theta } and the identity matrix I {\displaystyle I ...
Pauli's equation is derived by requiring minimal coupling, which provides a g-factor g=2. Most elementary particles have anomalous g-factors, different from 2. In the domain of relativistic quantum field theory, one defines a non-minimal coupling, sometimes called Pauli coupling, in order to add an anomalous factor
The simplest case of level splitting is a quantum system with two states whose unperturbed Hamiltonian is a diagonal operator: Ĥ 0 = E 0 I, where I is the 2 × 2 identity matrix. Eigenstates and eigenvalues (energy levels) of a perturbed Hamiltonian