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The dynamic electric polarizability relates the induced electric dipole moment at a driving frequency to the applied electric field at that frequency in linear response theory. A complex coefficient can encode both a dispersive phase and absorption. For a charged one-dimensional quantum harmonic oscillator with charge magnitude $e$, mass $m$, and natural frequency ${\omega }_0$, the undamped response isOne derivation uses the oscillator electric-dipole selection rule and first-order time-dependent perturbation theory. The same result follows from the classical Lorentz oscillator model: $m\ddot{z}+m{\omega }_0^{2}z=-eE$ and $d=-ez$ imply $\ddot{d}+{\omega }_0^{2}d=e^{2}E/m$. At exact resonance there is no bounded periodic response without damping; a finite switch-on produces secular growth rather than a finite steady-state value of the pole expression.

The Kubo formula expresses a linear response through an equilibrium correlation function. For a perturbation $H^{\prime }(t)=-f(t)B$ and observable $A$, one convention gives $\delta ⟨A(t)⟩={\int }_{-\infty }^t{\chi }_{AB}(t-t^{\prime })f(t^{\prime })d{t}^{\prime }$, withThe step function enforces causality, and the commutator records the quantum response.

The sign depends on the declared perturbing-Hamiltonian convention. Applying the framework to currents or polarization relates transport coefficients to fluctuations in an unperturbed system, forming a foundation of linear response theory and the fluctuation-dissipation theorem.^[ref]

Linear response theory approximates a small change in an observable by a linear function of an applied perturbation around a specified reference state. A weak force can produce a proportional displacement, and a weak applied magnetic field can produce proportional magnetization. The response coefficient depends on the material and state; linearity is an approximation to test by changing the drive size.

For a causal time-dependent response, writeThe kernel $\chi$ describes the delayed effect of the perturbation $f$. A sinusoidal input probes the frequency-dependent response: its in-phase part and out-of-phase part determine the response magnitude and phase lag. Causality links the real and imaginary parts through Kramers-Kronig relations under the appropriate convergence assumptions. Comparing the response at several small drive sizes tests linearity; a linear-looking response at one size alone cannot establish the range of validity.

A theoretical physicist who developed a general formulation of linear response theory. His 1957 treatment expresses electrical and magnetic response through the time-dependent correlations of microscopic variables. This connects the response to an applied disturbance with fluctuations already present at equilibrium, giving an atomistic route to quantities such as electrical conductivity and magnetic susceptibility.^[ref]