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arXiv:1608.02501 · Journal of Physics A Mathematical and Theoretical

WKB propagators in position and momentum space for a linear potential with a ‘ceiling’ boundary

Todd Zapata, S. A. Fulling

Published Jan 5, 2017• 1 citation
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Abstract

Abstract As a model for the semiclassical analysis of quantum-mechanical systems with both potentials and boundary conditions, we construct the WKB propagator for a linear potential sloping away from an impenetrable boundary. First, we find all classical paths from point y to point x in time t and calculate the corresponding action and amplitude functions. A large part of space-time turns out to be classically inaccessible, and the boundary of this region is a caustic of an unusual type, where the amplitude vanishes instead of diverging. We show that this curve is the limit of caustics in the usual sense when the reflecting boundary is approximated by steeply rising smooth potentials. Then, to improve the WKB approximation we construct the propagator for initial data in momentum space; this requires classifying the interesting variety of classical paths with initial momentum p arriving at x after time t . The two approximate propagators are compared by applying them to Gaussian initial packets by numerical integration; the results show physically expected behavior, with advantages to the momentum-based propagator in the classically forbidden regime (large t ).

DOI
10.1088/1751-8121/aa56f7

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Abstract As a model for the semiclassical analysis of quantum-mechanical systems with both potentials and boundary conditions, we construct the WKB propagator for a linear potential sloping away from an impenetrable boundary. First, we find all classical paths from point y to point x in time t and calculate the corresponding action and amplitude functions. A large part of space-time turns out to be classically inaccessible, and the boundary of this region is a caustic of an unusual type, where the amplitude vanishes instead of diverging. We show that this curve is the limit of caustics in the usual sense when the reflecting boundary is approximated by steeply rising smooth potentials. Then, to improve the WKB approximation we construct the propagator for initial data in momentum space; this requires classifying the interesting variety of classical paths with initial momentum p arriving at x after time t . The two approximate propagators are compared by applying them to Gaussian initial packets by numerical integration; the results show physically expected behavior, with advantages to the momentum-based propagator in the classically forbidden regime (large t ).
The physical problem this thesis deals with is a quantum system with linear\npotential driving a particle away from a ceiling (impenetrable barrier). This thesis\nwill construct the WKB approximation of the quantum mechanical propagator. The\napplication of the approximation will be for propagators corresponding to both initial\nmomentum data and initial position data.\nAlthough the analytic solution for the propagator exists, it is an indefinite integral of Airy functions and di±cult to use in obtaining probability densities by numerical integration or other schemes considered by the author. The WKB construction\nis less problematic because it is representable in exact form, and integration schemes\n(both numerical and analytic) to obtain probability densities are straightforward to\nimplement. Another purpose of this thesis is to be a starting point for the construction of WKB propagators with general potentials but the same type of boundary,\nimpenetrable barrier.\nResearch pertaining to this thesis includes determining all classical paths and\nconstraints for the one-dimensional linear potential with ceiling, and using these\nequations to construct the classical action, and hence the WKB approximation. Also,\nevaluation of final quantum wave functions using numerical integration to check and\nbetter understand the approximation is part of the research.\nThe results indicate that the validity of the WKB approximation depends on the\ntype of classical paths (i.e. the initial data of the path) used in the construction. Specifically, the presence of the ceiling may cause the semi-classical wave packets\nto become vanishingly small in one representation of initial classical data, while not\neffecting the packets in another. The conclusion of this phenomenon is that the\nrepresentation where the packets are not annihilated is the correct representation.
We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential.The models studied in detail have potentials proportional to x α for x > 0; the limit α → ∞ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases α = 1 and 2, for which exact solutions in terms of well known functions are available for comparison.We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action S, the amplitude function A and the Laplacian of A. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.)The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion.We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential.Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.
We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential. The models studied in detail have potentials proportional to xαx^αxα for x>0x>0x>0; the limit α→∞α\to\inftyα→∞ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases α=1α=1α=1 and 2, for which exact solutions in terms of well known functions are available for comparison. We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action SSS, the amplitude function AAA and the Laplacian of AAA. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.) The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion. We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.
The time-dependent propagator for the potential V(x)=1/2m(${\ensuremath{\omega}}^{2}$x2{x}^{2}x2+${\ensuremath{\rho}}^{2}$/x2{x}^{2}x2), x\ensuremath{\in}scrR+{\mathrm{scrR}}^{+}scrR+ is known. It is used in this article to see how certain general ideas about the semiclassical approximation for the propagator are in fact realized. The phenomena demonstrated include caustics at classical focal points, analytic structure of the classical action, complex trajectories for forbidden regions (not forbidden in this instance by energy), and the Langer correction for approximating the centrifugal potential. We also provide an unusual derivation of the solution to a classical problem from its quantum version.
The semiclassical approximation for a quantum amplitude is given by the sum of contributions from intersections of the appropriate manifolds in classical phase space. The intersection overlaps are just the Van Vleck determinants multiplied by a phase given by a classical action. Here we consider two nonstandard instances of this semiclassical prescription which would appear to be on shaky ground, yet the corresponding physical situations are not unusual. The first case involves momentum-space WKB theory for scattering potentials; the second is a propagator for the whisker map that arises in generic two-dimensional systems. In the former case two manifolds become asymptotically tangent, and the semiclassical formula needs to be uniformized in order to give a meaningful wave function. We give a uniformization procedure. In the latter case, there are an infinite number of intersections in phase space within a zone with the area of Planck's constant (the limit of resolution for quantum mechanics), yet the semiclassical sum over all contributions is shown to be correct.
The general method of path integrals in quantum mechanics introduced by R.P.Feynman is applied in this paper to discuss the amplitude of transition probability in quantum mechanics, that is, the subject of the propagator.Mauriel M.Mizrahi's semiclassical (WKB) approximation method has been used in studying the general expansion in detail for the propagator corresponding to an arbitrary Hamiltonian operator H. ?? where S_c is the action functional evaluated at the classical path (q_c, p_c) , ρ is a constant, ?? Maurice M.Mizrahi's solution of the K_(WKB) expansion has been generalized, and another linear independent particular solution is obtained.This is the solution when 0~2 = det M equals zero.The K_(WKB) can be applied to all systems corresponding to an arbitrary Hamiltonian operator.