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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]
For example, consider the operators x(t 1), x(t 2), p(t 1) and p(t 2). The time evolution of those operators depends on the Hamiltonian of the system. Considering the one-dimensional harmonic oscillator, = +, the evolution of the position and momentum operators is given by:
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 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 conventional definition of the spin quantum number is s = n / 2 , where n can be any non-negative integer. Hence the allowed values of s are 0, 1 / 2 , 1, 3 / 2 , 2, etc. The value of s for an elementary particle depends only on the type of particle and cannot be altered in any known way (in contrast to the spin ...
Thus, the spin raising and lowering operators + = + and =, so that [+,] =, correspond (in the sense detailed below) to the bosonic annihilation and creation operators, respectively. The precise relations between the operators must be chosen to ensure the correct commutation relations for the spin operators, such that they act on a finite ...
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.