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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.
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]
In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which ...
The number "2s + 1" is the multiplicity of the spin system. For example, there are only two possible values for a spin- 1 / 2 particle: s z = + 1 / 2 and s z = − 1 / 2 . These correspond to quantum states in which the spin component is pointing in the +z or −z directions respectively, and are often referred to as "spin ...
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 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 , and , which are mutually noncommuting, i.e., [,] = and its cyclic permutations.
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average.
The vacuum state or | is the state of lowest energy and the expectation values of and † vanish in this state: a | 0 = 0 = 0 | a † {\displaystyle a|0\rangle =0=\langle 0|a^{\dagger }} The electrical and magnetic fields and the vector potential have the mode expansion of the same general form: