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I just observed that Ehrenfest theorem:

$$\frac{d}{dt}\langle A\rangle = \frac{1}{i\hbar}\langle[A,H]\rangle + \left<\frac{\partial A}{\partial t}\right>$$

And Kubo's formula:

$$\langle A(t)\rangle = \langle A \rangle_0 - \frac{i}{\hbar}\int_{t_0}^{t} dt'\left<\left[A(t'),H(t')\right]\right>$$

Are formally very similar, i.e., Kubo's formula looks like the integral version of Ehrenfest's theorem. However, I am aware that Kubo's formula is derived from the interaction picture from the linear response to a small time-dependent perturbation, which is not how the Ehrenfest is derived. So I was wondering why, if so, they look related. Is the Ehrenfest theorem somehow illustrating some kind of "excitations"?

Aesku
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1 Answers1

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TL;DR Wikipedia article on Kubo formula is wrong. See article on Green-Kubo relations.

The Heisenberg equation of motion for an arbitrary operator is $$ \frac{d}{dt}A = \frac{1}{i\hbar}[A,H] + \frac{\partial A}{\partial t}.$$ Averaging this over the wave function or density matrix produces a generalization of Ehrenfest theorem (which in strict sense refers to the equations for position and momentum.) It is essentially a statement that quantum dynamics reduces to classical one, provided that the width of the wave packet (in space and momentum space) is smaller than the required precision or characteristic length scales (e.g., of the potential).

Note also that in the Schrödinger picture the above equation defines the operator of time derivative of $A$.

Kubo formula is a relation between the linear response coefficients to a perturbation and the correlation functions of the conjugate variables (to the perturbation variable.) In a sense, Kubo formulate is generalization of the Einstein relation, closely related the fluctuation-dissipation theorem.

Derivations of Kubo formula indeed often start with the averaged Heisenberg equation of motion - but claiming that this is the statement of the Kubo theorem, as Wikipedia article does, is simply wrong. As a minimum, it is better to consult the article on Green-Kubo relations.

A typical derivation of Kubo formula would proceed as follows:

  • Writing up perturbation series for operator of interest in terms of the perturbation (e.g., series for the electric current $\mathbf{j}$ in terms of electric field $E$). These series are typically derived from the above mentioned Heisenberg equation of motion.
  • Truncating these series at the linear term (in perturbation) and averaging them (here we have the "Wikipedia formula".)
  • Extracting the expression for the linear response coefficient from the formula (e.g., in case of an electric current, we are interested in conductance: $\mathbf{j}=\sigma \mathbf{E}$.) This often involves Fourier transnform or related manipulations.
  • It is also common to express the response coefficient in terms of a symmetric correlation - e.g., current-current correlation (rather than current-charge correlation or something of the kind.) This is typically achieved by using appropriate gauge and done in the very beginning of the derivation (without warning students - proving that it works in other gauges often takes some independent work with pen and paper.)

Importantly, Kubo formula implies presence of interactions, which would produce the finite and meaningful results - calculating average for non-interacting electrons is not of much interest (but instructive.)

Standard references for Kubo would be Mahan's Many-body physics (in standard second quantization language) and the appendix of Imry's Introduction to mesoscopic physics (without second quantization.)

Related: Fluctuation-dissipation theorem in the Keldysh formalism

Roger V.
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