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WKB Approximation to the Power Wall

Fernando Daniel Mera, S. A. Fulling, J D Bouas, Krishna Thapa

Published Jan 1, 2013• 1 citation
OpenAlex

Abstract

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.

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WKB approximation to the power wall

F D Mera, S A Fulling, J D Bouas +1 more

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 $α\to\infty$ 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.

arXiv:1301.1066•Journal of Physics A Mathematical and Theoretical•Apr 25, 2013•2 citations

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

Todd Zapata, S. A. Fulling

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 ).

arXiv:1608.02501•Journal of Physics A Mathematical and Theoretical•Jan 5, 2017•1 citation

Semiclassical approximation to the propagator of the Wigner function for particles in confined spaces

Rodríguez Villalba, Óscar Eduardo

Abstract: This work constructs the exact Wigner propagator for particles confined by one or two infinitely high potential walls. It takes advantage of the limit of polynomial potentials so as to obtain the semiclassical approximation for smooth potentials close to the box with two walls

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The semiclassical expansion of the anharmonic-oscillator propagator

Maurice M. Mizrahi

This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum-mechanical propagator corresponding to the quartic anharmonic-oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt-Morette, which does not entail the usual time-slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small-disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.

Journal of Mathematical Physics•May 1, 1979•29 citations

The WKB approximation for a linear potential and ceiling

Todd Zapata

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.

•May 15, 2009•3 citations

Semiclassical approximations in wave mechanics

Michael Berry, Keith Mount

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.

Reports on Progress in Physics•Jan 1, 1972•1,424 citations
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.
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 ).
This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum-mechanical propagator corresponding to the quartic anharmonic-oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt-Morette, which does not entail the usual time-slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small-disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.
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 review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.