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Journal of Physics A Mathematical and General

Exact propagator for motion confined to a sector

Richard E. Crandall

Published Feb 21, 1983• 19 citations
Publisher

Abstract

In certain situations the Feynman path integral 'collapses' into a countable sum over classical paths. This effect is sought in motion confined to a sector (in polar coordinates: r>or=0, 0<or= phi <or= alpha ). The exact space-time propagator is derived as a sum over a general nonclassical image set, with resulting structure radically dependent on number-theoretic properties of pi / alpha . It is shown that collapse occurs if and only if this number is an integer.

DOI
10.1088/0305-4470/16/3/011

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In certain situations the Feynman path integral 'collapses' into a countable sum over classical paths. This effect is sought in motion confined to a sector (in polar coordinates: r>or=0, 0<or= phi <or= alpha ). The exact space-time propagator is derived as a sum over a general nonclassical image set, with resulting structure radically dependent on number-theoretic properties of pi / alpha . It is shown that collapse occurs if and only if this number is an integer.
The propagator relative to a particle constrained to move in a finite region of space is calculated in the framework of path integrals. This region of the three-dimensional space is delimited through a sector of opening angle α, and also through the action of two attractive harmonic potentials, one being central and located in the 0xy plan, and the other directed along the z axis, with respective pulsations ω and ω0. It is shown that for α=π/2 and π the propagator is the sum of propagators evaluated on classical paths. The important case of the edge (α=2π) is considered.
The Aharonov-Bohm effect is investigated in the Feynman path-integral formulation of quantum mechanics. We consider an idealized situation with an electron moving in a magnetic-field-free region outside a solenoid whose radius to length ratio is very small. The nonvanishing vector potential term in the Lagrangian is written as an angular-velocity-dependent potential. In order to account for a singularity due to the presence of the solenoid itself, a periodic constraint is imposed on the path integral. The propagator can then be evaluated using the polar-coordinate methods of Peak and Inomata. It is found that the propagator has the general form $K({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})=\ensuremath{\Sigma}{n}^{}{\ensuremath{\chi}}_{n}{K}_{n}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})$ where the sum is taken over all classes of homotopic paths and the ${\ensuremath{\chi}}_{n}$ are a one-dimensional representation of the homotopy group. This is the form of the propagator as conjectured by Schulman.
We will derive a rigorous real-time propagator for the non-relativistic quantum mechanical L 2 transition probability amplitude and for the non-relativistic wavefunction. The propagator will be given explicitly in terms of the time evolution operator. The derivation will be for all self-adjoint non-vector potential Hamiltonians. For systems with potentials that carry at most a finite number of singularity and discontinuities, we will show that our propagator can be written in the form of a rigorous real-time, time-sliced Feynman path integral via improper Riemann integrals. We will also derive the Feynman path integral in a non-standard analysis formulation. Finally, we will compute the propagator for the harmonic oscillator using the non-standard analysis Feynman path-integral formulation; we will compute the propagator without using any knowledge of the classical properties of the harmonic oscillator.
The mathematical similarities between non-relativistic wavefunction propagation in quantum mechanics and image propagation in scalar diffraction theory are used to develop a novel understanding of time and paths through spacetime as a whole. It is well known that Feynman’s original derivation of the path integral formulation of non-relativistic quantum mechanics uses time-slicing to calculate amplitudes as sums over all possible paths through space, but along a definite curve through time. Here, a 3+1D spacetime wave distribution and its 4-momentum dual are formally developed which have no external time parameter and therefore cannot change or evolve in the usual sense. Time is thus seen “from the outside”. A given 3+1D momentum representation of a system encodes complete dynamical information, describing the system’s spacetime behavior as a whole. A comparison is made to the mathematics of holograms, and properties of motion for simple systems are derived.
The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude.