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Quantum Physics Letters

A Derivation of the Path-Integral Formalism of Quantum Mechanics from Probability Conservation along the Velocity Flows of Classical Particles

Frank Blume

Published Jul 29, 2013• 1 citation
Publisher

Abstract

Abstract: Denoting by v the velocity flow of a classical particle that is subject to a potential V, we demonstrate that the path-integral formalism of non-relativistic quantum mechanics can be obtained by superimposing wave functions that are solutions of a wave equation which in turn directly corresponds to the probability conservation equation div(ψ ∗ ψv)=0.

DOI
10.12785/qpl/020201

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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.
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.
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.
One of the important classic experiments that led to the fundamental difference between quantum and classical mechanics is the double slit experiment, which is interesting with respect to the path integral formalism since it leads to a conceptual motivation for introducing it. From Quantum Mechanics, the probability amplitude can also be determined from the Schrodinger equation as it describes the wave properties of the particles. From Quantum Mechanics, the probability amplitude (square of the probability function or wave function) can also be determined from the Schrodinger equation as it describes the wave properties of the particles. Here, the role of the trajectory of a point-like particle, formally is revived in a way compatible with the indetermination principle and is indeed the path integral approach to quantum mechanics.
Abstract We present Feynman’s path integral derivation of the Schrödinger equation. The van Hove formulation of non-relativistic scattering is presented. We then give a full quantum mechanical treatment of non-relativistic scattering. Both the interaction of the beams with the apparatus and the scattering process itself are treated quantum mechanically. In the spirit of Schrödinger, particle trajectories do not appear, with all of the physics being contained in the phases of the wave function.
In Feynman's space-time approach to nonrelativistic quantum mechanics, the wave function may be considered to be the sum of path integrals over Brownian-motion trajectories. This formalism is entirely equivalent to the Schr\"odinger-equation treatment with its intrinsic indeterminacy. Pursuing the analogy with Brownian motion further, one may introduce a hidden variable corresponding to particle velocity in order to represent the wave function as a sum of path integrals taken over phase-space trajectories. By means of a hidden parameter corresponding to the time scale of fictitious interactions of the particle with the vacuum, one can define generalized propagators which over very short times produce wave functions which are localizable in phase space and hence are deterministic. For long times the conventional Feynman formalism is recovered. This short-time causality does not violate the von Neumann injunction against hidden variables since the new velocity variable turns out to be non-Hermitian over the long time scales normally considered. In contrast to the models proposed by Bohm and others, the present Brownian model is linear and preserves the usual statistical interpretation of the wave function for sufficiently long times.