Kubo formula

The Kubo formula, named for Ryogo Kubo, is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation.wikipedia
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Linear response function

linear response theorylinear responsesusceptibility
The Kubo formula, named for Ryogo Kubo, is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation.
This defines particularly the Kubo formula, which considers the general case that the "force" h(t) is a perturbation of the basic operator of the system, the Hamiltonian, where \hat B corresponds to a measurable quantity as input, while the output x(t) is the perturbation of the thermal expectation of another measurable quantity \hat A(t).

Ryogo Kubo

Kubo
The Kubo formula, named for Ryogo Kubo, is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation.

Perturbation theory (quantum mechanics)

perturbation theoryperturbativeperturbation
The Kubo formula, named for Ryogo Kubo, is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation.

Heaviside step function

Heaviside functionunit step functionHeaviside unit step function
The perturbation is described by an additional time dependence in the Hamiltonian: where \theta (t) is the Heaviside function ( = 1 for positive times, =0 otherwise) and \hat V(t) is hermitian and defined for all t, so that \hat H(t) has for positive t-t_0 again a complete set of real eigenvalues E_n(t).

Density matrix

density operatordensity matricesvon Neumann equation
However, one can again find the time evolution of the density matrix rsp.

Schrödinger equation

Schrödinger's equationwave mechanicsSchrödinger wave equation
The time dependence of the states is governed by the Schrödinger equation which thus determines everything, corresponding of course to the Schrödinger picture.

Schrödinger picture

Schrödinger representationSchrödinger operatorswave-mechanical
The time dependence of the states is governed by the Schrödinger equation which thus determines everything, corresponding of course to the Schrödinger picture.

Interaction picture

Dirac pictureinteraction HamiltonianSchwinger–Tomonaga equation
But since \hat{V}(t) is to be regarded as a small perturbation, it is convenient to now use instead the interaction picture representation, in lowest nontrivial order.

Second quantization

second-quantization formalismcanonically quantizedCovariant formulation
(see also Second quantization)

Green–Kubo relations

Green-Kubo relationsforce-flux lawGreen-Kubo
* Green–Kubo relations

Free electron model

free electronselectron gasDrude–Sommerfeld model
More exact values for the electrical conductivity and Wiedemann–Franz law can be obtained by softening the relaxation-time approximation by appealing to the Boltzmann transport equations or the Kubo formula.