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Progress of Theoretical Physics

Quantum Dynamics and Non-Inertial Frames of Reference. I

Shin Takagi

Published Mar 1, 1991• 7 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.85.463

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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.
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 non-relativistic version of the multi-temporal quantization scheme of relativistic particles in a family of non-inertial frames (see hep-th/0502194) is defined. At the classical level the description of a family of non-rigid non-inertial frames, containing the standard rigidly linear accelereted and rotating ones, is given in the framework of parametrized Galilei theories. Then the multi-temporal quantization, in which the gauge variables, describing the non-inertial effects, are not quantized but considered as c-number generalized times, is applied to non relativistic particles. It is shown that with a suitable ordering there is unitary evolution in all times and that, after the separation of center of mass, it is still possible to identify the inertial bound states. The few existing results of quantization in rigid non-inertial frames are recovered as special cases.
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.
Beginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically motivated assumptions, these transformations are uniquely determined: they are exactly the group of time-dependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schrödinger equations. The general equation in this family, including terms considered by Kostin, by Bialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependent coefficients, can be obtained from the linear Schrödinger equation through gauge transformation and a subsequent process we call gauge generalization. We thus unify, on fundamental grounds, a rather diverse set of nonlinear time evolutions in quantum mechanics.