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Revista Mexicana de Física

Difracción en el tiempo asociada a una distribución de fuentes

M. Moshińsky

Published May 10, 2009• 1 citation
OpenAlex

Abstract

THE MEASURABILITY OF THE GRAVITATIONAL FIELD IS DISCUSSED FROM BOTH THE CLASSICALLY SUCH A MEASUREMENT CAN BE PERFORMED IF ONE USES TEST BODIES WITH VANISHING SMALL MASS. ON THE OTHER HAND, DUE TO THE HEISENBERG UNCERTAINTY PRINCIPLE, IT IS FOUND THAT ONE MUST USE TEST BODIES OF EXTREMELY LARGE MASS IN ORDER TO OBTAIN ACCURATE RESULTS. THUS IT APPEARS THAT AN ACCURATE DETERMINATION OF THE GRAVITATIONAL FIELD IS NOT POSSIBLE IF ONE TAKES INTO ACCOUNT THE QUANTUM RESTRICTIONS. CONSEQUENCES OF THIS RESULT WILL BE DISCUSSED.

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THE MEASURABILITY OF THE GRAVITATIONAL FIELD IS DISCUSSED FROM BOTH THE CLASSICALLY SUCH A MEASUREMENT CAN BE PERFORMED IF ONE USES TEST BODIES WITH VANISHING SMALL MASS. ON THE OTHER HAND, DUE TO THE HEISENBERG UNCERTAINTY PRINCIPLE, IT IS FOUND THAT ONE MUST USE TEST BODIES OF EXTREMELY LARGE MASS IN ORDER TO OBTAIN ACCURATE RESULTS. THUS IT APPEARS THAT AN ACCURATE DETERMINATION OF THE GRAVITATIONAL FIELD IS NOT POSSIBLE IF ONE TAKES INTO ACCOUNT THE QUANTUM RESTRICTIONS. CONSEQUENCES OF THIS RESULT WILL BE DISCUSSED.
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, ΔpΔq>h2π \Delta p\Delta q>\frac{h}{{2\pi}} ΔpΔq>2πh​ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
MOSHINSKY´S RESULTS ON ”DIFFRACTION IN TIME” ARE EXTENDED TO THE CASE OF A RECTANGULAR PULSE.BESIDES INTERFERENCE EFFECTS OF THE TYPE STUDIED BY MOSHINSKY, THERE APPEARS ALSO THE WELL KNOWN QUANTUM MECHANICAL SPREADING EFFECT. RISE TIME AND MONOCHROMATICITY REQUIREMENTS SEEM TO PRECLUDE THE POSSIBILITY OF EXPERIMENTAL OBSEVATION OF THE INTERFERENCE EFFECTS, SO THAT THEY NEED NOT BE TAKEN ACCOUNT IN NEUTRON VELOCITY SELECTORS. THE DETECTION OF THE SPREADING, WICH IS SUBJECT TO WEAKER REQUIREMENTS WOULD PROVIDE A DIRECT TEST OF THE UNCERTAINTY RELATION.
THIS WORK CONTAINSAN ANALYSIS OF SOME OF THE FUNDAMENTAL PRINCIPLES UNDERLAYING THE CONSTRUTION OF OPERATORS IN QUANTUM MECHANICS. FACH OPERATOR IS ASSOCIATED TO A PROJECTION OPERATOR AND THESE PROJECTION OPERATORS ARE SHOWS TO FULFIL A SET OF PROPERTIES FORMALLY INDENTICAL TO THOSE OF A PROBABILITY FIELD.IT IS SHOWN THAT, BY THE INTRODUCTION OFA MEASURE IN CONFIGURATION SPACE IT IS POSSIBLE TO OBTAIN COMPLETE SETS OF COMMUTING OBSERVABLES HAVING, WITHIN LIMITS, PREASIGNED SPECTRA. THE INTRODUCTION OF NEW OPERATORS BY CONSIDERATION OF THE GROUPS EXPRESSING THE SYMMETRIES OF THE SYSTEM, IS DISCUSSED.
THE PRESENT NOTE ANALYSES THE DYNAMICAL DESCRIPTION OF THE SCATTERING DUE TO A SHORT RANGE POTENTIAL. IN PARTICULAR, IT DEVELOPS A CLASS OF TRANSFORMS THAT ARE USEFUL FOR THIS DYNAMICAL DESCRIPTION. A TIME-DEPENDENT GREEN FUNCTION FOR THE SCATTERING PROCESS IS DISCUSSED, THAT TAKES US FROM THE INITIAL STATE (PLANE WAVE) TO THE FINAL STATE (PLANE PLUS SCATTERED WAVES). THIS GREEN FUNCTION CAN BE USED TO GIVER THE MOTION OF ANY WAVE PACKET THAT INITIALLY WAS OUTSIDE THE POTENTIAL. THE PRESENT PAPER COMPLEMENTS A PREVIOUS DYNAMICAL DESCRIPTION OF A SCATTERING PROCESS, IN WHICH THE INTERACTION WAS REPRESENTED BY APPROPRIATE BOUNDARY.