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Physical Review A

Transformation theory and translation factors in inelastic atomic collisions

G. Bruno Schmid

Published Apr 1, 1977• 24 citations
Publisher

Abstract

It is shown through the use of transformation theory that unique semiclassical atomic scattering states which obey the asymptotic conditions of formal scattering theory can be derived by transforming "nontraveling" atomic states, i.e., states whose coordinate variables are referred to a stationary origin, to frames at rest with respect to the incoming or outgoing particles. An overview of the problem of properly defining such scattering states is presented. The operator which carries out the necessary transformation from inertial to noninertial frames is derived and its properties are discussed. The relation of this transformation operator to the "translation factor" discussed in the literature is presented. The application of this operator to transform the time-dependent Schr\"odinger equation from an inertial to a noninertial frame is presented and shown to introduce new terms in the resulting equation. The implications of these new terms to scattering problems are discussed.

DOI
10.1103/physreva.15.1459

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Journal of Physics B Atomic Molecular and Optical Physics•Oct 10, 2006•6 citations
It is shown through the use of transformation theory that unique semiclassical atomic scattering states which obey the asymptotic conditions of formal scattering theory can be derived by transforming "nontraveling" atomic states, i.e., states whose coordinate variables are referred to a stationary origin, to frames at rest with respect to the incoming or outgoing particles. An overview of the problem of properly defining such scattering states is presented. The operator which carries out the necessary transformation from inertial to noninertial frames is derived and its properties are discussed. The relation of this transformation operator to the "translation factor" discussed in the literature is presented. The application of this operator to transform the time-dependent Schr"odinger equation from an inertial to a noninertial frame is presented and shown to introduce new terms in the resulting equation. The implications of these new terms to scattering problems are discussed.
The general framework of a time-dependent scale transformation as applied to non-relativistic time-dependent Schrödinger equations is described. The nature of the transformation is clarified by comparing it with a time-dependent translation and rotation. Particular attention is paid to the associated gauge transformation and the resulting harmonic inertial force. Corresponding classical results are also displayed. This work intends to pave the way for various applications to be presented in the subsequent papers of this series.
The general framework of a time-dependent scale transformation as applied to non-relativistic time-dependent Schrödinger equations is described. The nature of the transformation is clarified by comparing it with a time-dependent translation and rotation. Particular attention is paid to the associated gauge transformation and the resulting harmonic inertial force. Corresponding classical results are also displayed. This work intends to pave the way for various applications to be presented in the subsequent papers of this series.
A formalism, previously presented, for describing atom-atom scattering in the adiabatic representation to higher orders in mM\frac{m}{M}Mm​ is generalized to describe the problem of different isotopes and different atoms. This is accomplished by a sequence of nonlinear coordinate transformations which introduce a "switching function" which is defined only in the asymptotic limit of separated atoms. It is found that the effective potential for elastic scattering contains energy-dependent terms in order mM\frac{m}{M}Mm​ and that it also depends upon the switching function in this order so that the potential is not unique. It is also found that in this order the definition of adiabatic states is not a trivial problem. A variational choice is made for the switching function and explicit calculations are presented for the He-He interaction. The coupled equations describing transitions are then derived and it is shown that terms are omitted from the usual coupling operator which are of the same order of magnitude as the one usually kept. This leads to a modification of the usual Landau-Zener transition formula.
The Galilean transformation carries one inertial frame into another with a different velocity. The extended Galilean transformation carries an inertial frame into a rigid frame with uniform spatial, but arbitrary translational acceleration x′=x+ξ (t). Besides being useful in discussing the equivalence principle, this transformation provides a physical interpretation for the theorem in nonrelativistic quantum mechanics, that we cannot coherently combine states of different mass. This restriction comes about because we cannot distinguish between coordinate and proper time in nonrelativistic physics.
Radiative atomic processes, in which charged particles moving with velocities much less than the speed of light emit a photon, are normally considered by using the Schrödinger equation. By analysing the transformation properties of the radiative transition amplitudes under the change of a reference frame we establish that the total cross sections and decay rates, calculated with the Schrödinger equation, are Galilean invariant and that this invariance holds irrespective of the accuracy of wavefunctions describing the charged particles. By considering, within the scope of one reference frame, the radiative processes for two identical atomic systems which have different translational (centre-of-mass) velocities we show that the total cross sections and decay rates in general depend on this velocity if approximate wavefunctions are employed and that this is the direct consequence of the problem of gauge dependence. We also present a detailed discussion of the intimate interrelation between Galilean and gauge transformations which, if overlooked, can easily lead to the misinterpretation of the problem of gauge dependence as the 'problem of Galilean non-invariance'.