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The WKB approximation for a linear potential and ceiling

Todd Zapata

Published May 15, 2009• 3 citations
OpenAlex

Abstract

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.

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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 ).

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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.

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Construction of an approximate solution of the Wigner equation by uniformization of WKB functions

Konstantina-Stavroula Giannopoulou

In this thesis, we construct an approximate series solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in two-dimensional phase space. These curves are defined by the eigenvalues and the Hamiltonian function of the associated one-dimensional Schrödinger operator, and they play a crucial role in the quantum interference mechanism in phase space. We assume that the potential of the Schrödinger operator is a single potential well, such that the spectrum is discrete. The construction starts from an eigenfunction series expansion of the solution, which is derived here for first time in a systematic way, by combining the elementary technique of separation of variables with involved spectral results for the Moyal star exponential operator. The eigenfunctions of the Wigner equation are the Wigner transforms of the Schrödinger eigenfunctions, and they are approximated in terms of Airy functions by a uniform stationary phase approximation of the Wigner transforms of the WKB expansions of the Schrödinger eigenfunctions. The approximation of the eigenfunction series is an approximated solution of the Wigner equation, which by projection onto the configuration space provides an approximate wave amplitude, free of turning point singularities. It is generally expected that, the derived wave amplitude is bounded, and correctly scaled, even on caustics, since only finite terms of the approximate terms are significant for WKB initial wave functions with finite energy. The details of the calculations are presented for the simple potential of the harmonic oscillator, in order to be able to check our approximations analytically. But, the same construction can be applied to any potential well, which behaves like the harmonic oscillator near the bottom of the well.

arXiv:1706.03111•Jun 9, 2017•0 citations
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.
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 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.
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 time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
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.
In this thesis, we construct an approximate series solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in two-dimensional phase space. These curves are defined by the eigenvalues and the Hamiltonian function of the associated one-dimensional Schrödinger operator, and they play a crucial role in the quantum interference mechanism in phase space. We assume that the potential of the Schrödinger operator is a single potential well, such that the spectrum is discrete. The construction starts from an eigenfunction series expansion of the solution, which is derived here for first time in a systematic way, by combining the elementary technique of separation of variables with involved spectral results for the Moyal star exponential operator. The eigenfunctions of the Wigner equation are the Wigner transforms of the Schrödinger eigenfunctions, and they are approximated in terms of Airy functions by a uniform stationary phase approximation of the Wigner transforms of the WKB expansions of the Schrödinger eigenfunctions. The approximation of the eigenfunction series is an approximated solution of the Wigner equation, which by projection onto the configuration space provides an approximate wave amplitude, free of turning point singularities. It is generally expected that, the derived wave amplitude is bounded, and correctly scaled, even on caustics, since only finite terms of the approximate terms are significant for WKB initial wave functions with finite energy. The details of the calculations are presented for the simple potential of the harmonic oscillator, in order to be able to check our approximations analytically. But, the same construction can be applied to any potential well, which behaves like the harmonic oscillator near the bottom of the well.