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American Journal of Physics

Fundamental quantum mechanics—a graphic presentation

M. Norton Wise, Tim G. Kelley

Published Apr 1, 1977• 2 citations
Publisher

Abstract

A presentation of basic quantum mechanics for non-physical science majors is described which, by relying on computer-generated graphic display, circumvents the usual mathematical limitations and allows a detailed treatment of free particle motion in a wave picture. The development is based on a computer-simulated, single-slit, particle diffraction experiment and culminates in a complete description and graphic display of the progress of the diffraction pattern from a square pulse at the slit to the Fraunhofer pattern at infinity.

DOI
10.1119/1.10851

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A presentation of basic quantum mechanics for non-physical science majors is described which, by relying on computer-generated graphic display, circumvents the usual mathematical limitations and allows a detailed treatment of free particle motion in a wave picture. The development is based on a computer-simulated, single-slit, particle diffraction experiment and culminates in a complete description and graphic display of the progress of the diffraction pattern from a square pulse at the slit to the Fraunhofer pattern at infinity.
The years of the third decade of the present century were heady times. Old social orders were being swept away, monarchies becoming republics and the United States of America was emerging as a world power. In physics too, revolutions were taking place, none more potent than that which resulted in the emergence of quantum mechanics as a model for the behaviour of sub-atomic particles. It was Newton who first suggested that light was particulate but during the nineteenth century this view had fallen into neglect. Indeed a number of experiments, for example the classic Young's slits experiment, demonstrated quite conclusively that light was a wave motion. How else could interference fringes occur? However, the experiments on the photoelectric effect demonstrated just as conclusively that light energy was carried in packets, or quanta, and that a continuous wave description was not applicable. This ‘wave or particle’ dilemma was not unique to the behaviour of light. J. J. Thompson showed that cathode rays were charged, had a well defined mass and had all the properties expected of a beam of particles. Nevertheless, Davisson and Germer showed that diffraction of electrons could take place, an effect made visually much more dramatic by G. P. Thompson's transmission electron diffraction patterns. Nowadays, electron diffraction is used as a routine analytical tool in all advanced metallurgical and materials science laboratories (Fig. 2.1). Clearly new axioms were required.
This paper provides a complete physical model associated with mathematical analysis and formulation that is based on Schrödinger equation and Feynman path integral. This model answers to single Photon (electron) double slit experiment with high degree of accuracy. It also gives a new method of analysis and formulation for Fresnel and Fraunhofer diffractions that are completely according to the Schrödinger equation and Feynman path concept. The results of these methods are precisely in agreement with the near field and far field diffraction experiments. In other words the new formulation in all above mentioned phenomena i.e. single photon (electron) double slit experiment, Fresnel and Fraunhofer diffractions are completely confirmed by lab experiments(measurements). Lab experiments mean common intensity and pattern experiments resulting from Fresnel (near field) diffraction, Fraunhofer (far field) diffraction and patterns resulting in single photon (electron) lab experiment. The resultant formulas are sketched and calculated by matlab program, the resultant graphs and values are compared with lab experiments in each section. This analysis substitutes wave function and probability density function concepts instead of interference of light beams emitted from infinitesimal pinholes or classical interpretation of near field diffraction and far field diffraction phenomena. The ultimate goal of this paper is to give a very accurate answer to the single particle double slit experiment and also to unify diffraction concept of classical optics in to quantum mechanics, deleting the traditional concept of phases in classical optics and substituting the concept of deflection of Feynman paths. The above mentioned analysis may be a glad tiding to a reliable method to unify classical optics in to quantum mechanics.
This paper provides a complete physical model associated with mathematical analysis and formulation that is based on Schrödinger equation and Feynman path integral. This model answers to single Photon (electron) double slit experiment with high degree of accuracy. It also gives a new method of analysis and formulation for Fresnel and Fraunhofer diffractions that are completely according to the Schrödinger equation and Feynman path concept. The results of these methods are precisely in agreement with the near field and far field diffraction experiments. In other words the new formulation in all above mentioned phenomena i.e. single photon (electron) double slit experiment, Fresnel and Fraunhofer diffractions are completely confirmed by lab experiments(measurements). Lab experiments mean common intensity and pattern experiments resulting from Fresnel (near field) diffraction, Fraunhofer (far field) diffraction and patterns resulting in single photon (electron) lab experiment. The resultant formulas are sketched and calculated by matlab program, the resultant graphs and values are compared with lab experiments in each section. This analysis substitutes wave function and probability density function concepts instead of interference of light beams emitted from infinitesimal pinholes or classical interpretation of near field diffraction and far field diffraction phenomena. The ultimate goal of this paper is to give a very accurate answer to the single particle double slit experiment and also to unify diffraction concept of classical optics in to quantum mechanics, deleting the traditional concept of phases in classical optics and substituting the concept of deflection of Feynman paths. The above mentioned analysis may be a glad tiding to a reliable method to unify classical optics in to quantum mechanics.
This innovative new text presents quantum mechanics in a manner that directly reflects the methods used in modern physics research—making the material more approachable and preparing students more thoroughly for real research. Most texts in this area start with a bit of history and then move directly to wave-particle problems with accompanying heavy mathematical analysis; Quantum Mechanics provides a foundation in experimental phenomena and uses a more approachable, less intimidating, more powerful mathematical matrix model. Beginning with the Stern-Gerlach experiments and the discussion of spin measurements, and using bra-ket notation, the authors introduce an important notational system that is used throughout quantum mechanics. This non-traditional presentation is designed to enhance students’ understanding and strengthen their intuitive grasp of the subject.
This article is a ‘‘sampler,’’ which shows how quantum mechanics may be presented to students in a way that makes apparent how natural quantum mechanics is as a description of the world. The mathematical machinery of Hilbert space, the idea of representing observables by operators, the Schrödinger equation, and the position-momentum uncertainty relation all follow from natural assumptions that students can readily accept. The basic ideas of quantum mechanics are developed from intuitive first principles to the point where one can connect with more traditional treatments of quantum mechanics.