# 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.

### Partition function (statistical mechanics)

**partition functionpartition functionsgrand partition function**

::where is the partition function.

### 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.