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In what follows we will show how to map a 1D spin chain of spin-1/2 particles to fermions. Take spin-1/2 Pauli operators acting on a site of a 1D chain, +,,.Taking the anticommutator of + and , we find {+,} =, as would be expected from fermionic creation and annihilation operators.
That is, the resulting spin operators for higher-spin systems in three spatial dimensions can be calculated for arbitrarily large s using this spin operator and ladder operators. For example, taking the Kronecker product of two spin- 1 / 2 yields a four-dimensional representation, which is separable into a 3-dimensional spin-1 ( triplet ...
The Ehrenfest theorem, named after Austrian theoretical physicist Paul Ehrenfest, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force = ′ on a massive particle moving in a scalar potential (), [1]
The metric defines an ABC score as a triplet of values that represent the size of a set of source code statements. An ABC score is calculated by counting the number of assignments (A), number of branches (B), and number of conditionals (C) in a program. ABC score can be applied to individual methods, functions, classes, modules or files within ...
The Kubo formula, named for Ryogo Kubo who first presented the formula in 1957, [1] [2] is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation.
The dimension of the on-site basis is 2, because the state of each spin can be described as a superposition of spin-up and spin-down, denoted | and | . The full system has dimension 2 N {\displaystyle 2^{N}} , and the Hamiltonian represented as a matrix has size 2 N × 2 N {\displaystyle 2^{N}\times 2^{N}} .
The basic idea can be illustrated for the basic example of spin operators of quantum mechanics. For any set of right-handed orthogonal axes, define the components of this vector operator as S x {\displaystyle S_{x}} , S y {\displaystyle S_{y}} and S z {\displaystyle S_{z}} , which are mutually noncommuting , i.e., [ S x , S y ] = i ℏ S z ...
In 1949, José Enrique Moyal elucidated how the Wigner function provides the integration measure (analogous to a probability density function) in phase space, to yield expectation values from phase-space c-number functions g(x, p) uniquely associated to suitably ordered operators Ĝ through Weyl's transform (see Wigner–Weyl transform and ...