Nonrelativistic Perturbation Theory for a Discrete Spectrum

Kaya Güven Akbaş, Fatih Erman, O. Teoman Turgut

Evgenii Viktorovich Vybornyi

A. L. Kitanin

Victor Galitski, Boris Mikhailovich Karnakov, Vladimir I Kogan +1 more

Abstract This chapter analyzes problems that deal with stationary perturbation theory (discrete spectrum); variational method; stationary perturbation theory (continuous spectrum); non-stationary perturbation theory (transitions in continuous spectrum); sudden perturbations; and adiabatic approximation.

C. L. Hammer, T. A. Weber

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.