Time as an operator/observable in nonrelativistic quantum mechanics

نویسنده

  • G. E. Hahne
چکیده

The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional space-time entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation. Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle’s crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate. In conventional formulations the hyperplane is always spacelike, i.e., is a time=constant hyperplane in Galilean relativity, and the result is then trivial. A nontrivial result is obtained if the plane is not of this type. When the space-time coordinates are (t, x, y, z), the paper analyzes in detail the case that the hyperplane is of the type z=constant. Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface. Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory. A detailed formalism for computing average crossing times on a z=constant hyperplane, and average dwell times and delay times for a zone of interaction contained between a pair of z=constant hyperplanes, is presented. ∗email: [email protected]

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Time as a Quantum Observable, Canonically Conjugated to Energy, and Foundations of Self-Consistent Time Analysis of Quantum Processes

Recent developments are reviewed and some new results are presented in the study of time in quantum mechanics and quantum electrodynamics as an observable, canonically conjugate to energy. This paper deals with the maximal Hermitian but nonself-adjoint operator for time which appears in nonrelativistic quantum mechanics and in quantum electrodynamics for systems with continuous energy spectra a...

متن کامل

Interference in Time Lawrence

I discuss the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory, used by the authors in their analysis, does not have the property of coherence in time, and hence cannot account for the results found. Therefore, this experiment has fundamental importance beyond the technical advances it represents. Some...

متن کامل

Interference in Time: a Comment

I comment on the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory, used by the authors in their analysis, cannot account for the results found, and therefore that this experiment has fundamental importance beyond the technical advances it represents. The recent experiment of Lindner, et al, clearly sho...

متن کامل

Observables, Measurements and Phase Operators from a Bohmian Perspective

Bohmian mechanics is a deterministic theory of point particles in motion. While avoid. ing all the paradoxes of nonrelativistic quantum mechanics, it, yields the quantum formalism itself—especially the role of self-adjoint operators--as a macroscopic measurement, formal­ ism. As an "application" it is shown that much of the confusion connected with the phase operator for the electromagnetic f...

متن کامل

Fibre Bundle Formulation of Nonrelativistic Quantum Mechanics 0. Preliminary Considerations: Quantum Mechanics from a Geometric-observer's Viewpoint

We propose a version of the non-relativistic quantum mechanics in which the pure states of a quantum system are described as sections of a Hilbert (generally infinitely-dimensional) fibre bundle over the space-time. There evolution is governed via (a kind of) a parallel transport in this bundle. Some problems concerning observables are considered. There are derived the equations of motion for t...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2008