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American Journal of Physics

Transformation theory, accelerating frames, and two simple problems

Gary Bruno Schmid

Published Jul 1, 1977• 3 citations
Publisher

Abstract

An operator which transforms quantum state functions from inertial to certain noninertial reference frames is presented. This operator is applied to solve two well-known problems: (1) the stationary state wave functions for a particle in a uniform gravitational field; (2) the motion and spreading of a Gaussian wave packet in a uniform gravitational field. The salient features of both analyses are discussed.

DOI
10.1119/1.10804

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An operator which transforms quantum state functions from inertial to certain noninertial reference frames is presented. This operator is applied to solve two well-known problems: (1) the stationary state wave functions for a particle in a uniform gravitational field; (2) the motion and spreading of a Gaussian wave packet in a uniform gravitational field. The salient features of both analyses are discussed.
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.
We derive a useful expression for the time-development operator for quantum wave functions, exp(−iĤt/ℏ), for the case of a particle subject to a constant force. We apply it, in both position and momentum space, to construct wave packets that exhibit uniform acceleration. We use these results to generate quantum wave packets corresponding to the motion of a charged particle in a uniform electric field in two different gauges, explicitly demonstrating the change in phase of the wave function under a gauge transformation. The more standard description of the time dependence of accelerating wave packets using the formalism of propagators is briefly reviewed in an Appendix for comparison.
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.
The free relativistic particle, by definition, has to move in an inertial reference frame with uniform velocity less than the speed of light. The corresponding movement of a material quantum particle describes a wave packet, composed of matter waves—solutions of the Schr?dinger equation. The maximum of packet, corresponding to the largest probability to find the particle, has to move with the same uniform velocity, defined by the initial condition. It has been shown that the traditional definition of the quantum momentum operator i.e. taking it to correspond to the special relativity theory, relativistic momentum, cannot produce precise description of a relativistic matter particle. Different definitions are investigated and one that solves this issue is found. Obtained original expression of relativistic kinetic energy operator creates new possibilities for relativistic quantum systems theory.
Mathjax On | Off Abstract In the preceding chapter, we showed how one-particle quantum states can be obtained from unitary irreducible representations of the Galilei group, the transformation group of Newton relativity. Since the Galilei group defines transformations amongst inertial reference frames, this description of one-particle quantum states is limited to inertial observers. In particular, observers who are accelerating are excluded in this description. This motivates the question of whether it is possible to provide a description of one-particle quantum states that holds for non-inertial observers by expanding the notion of Newton relativity to include non-inertial reference frames.