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arXiv:quant-ph/9601003

Particle Path Formulation of Quantum Mechanics

Shiv R. Vatsya

Published Jan 3, 1996• 0 citations
arXivHTMLPDF

Abstract

An extension of the classical action principle obtained in the framework of the gauge transformations, is used to describe the motion of a particle. This extension assigns many, but not all, paths to a particle. Properties of the particle paths are shown to impart wave like behaviour to a particle in motion and to imply various other assumptions and conjectures attributed to the formalism of Quantum Mechanics. The Klein-Gordon and other similar equations are derived by incorporating these properties in the path-integral formalism.

DOI
10.48550/arxiv.quant-ph/9601003

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An extension of the classical action principle obtained in the framework of the gauge transformations, is used to describe the motion of a particle. This extension assigns many, but not all, paths to a particle. Properties of the particle paths are shown to impart wave like behaviour to a particle in motion and to imply various other assumptions and conjectures attributed to the formalism of Quantum Mechanics. The Klein-Gordon and other similar equations are derived by incorporating these properties in the path-integral formalism.
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Abstract This chapter discusses the Feymann path-integral approach to quantum mechanics. First, it derives a path integral expression for the evolution operator. Next, it shows that the classical equations of motion, that is, those obtained from the principle of least action, are obtained from this path integral formulation in the limit where the variation in the action of the problem at hand is small as compared to the Planck’s constant h. Last, it discusses how the WKB approximation is derived from the path integral formulation.
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism. It is a generalization of the classical least action principle. The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state. Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral. We look in to the subject of two kinds of slit-experiments: The square slit, the single- and the double-Gaussian slit. Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics. It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral. Further applications of the path integral are discussed.