Time-Dependent Perturbation Theory

Abstract

A time‐dependent perturbation theory which is based upon the U‐matrix approach is presented using the interaction e^‐αt(1 + e^‐βt)⁻¹V(x) with β real, α complex and β > Re α > 0. Thus the adiabatic or time‐independent approximation (β → 0) and the sudden approximation (β → ∞) can be obtained using just one formalism. The usual series for the U matrix is derived and shown to converge for a semi infinite range if the interaction is of the form |t|^‐δ, δ > 1, for large times t. Two interesting results are (1) The derivation of the ``golden rule'' from the discrete‐state time‐dependent perturbation theory presented in most text books of quantum mechanics leads to erroneous physical interpretations because the energy‐level shift is ignored. (2) When considering scattering between states in the continuum, it is found that a characteristic feature of time‐dependent interactions is a discrete momentum spectrum of final states which, in the relativistic case, leads to a mass spectrum. These spectra cannot be obtained using the S matrix.