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Canadian Journal of Physics

Classical orbits from the wave function in the large-quantum-number limit

James D Bonnar, Jeffrey Robert Schmidt

Published Jul 1, 2003• 0 citations
Publisher

Abstract

Classical trajectories for the Coulomb potential are obtained from the large principle quantum-number limit of solutions to the nonrelativistic Schrödinger equation, by use of integral equations satisfied by the radial probability density function. These trajectories are found to be in excellent agreement with those computed directly from classical mechanics, in accordance with a statement of the Bohr Correspondence principle, except in a region very close to the center of force. PACS No.: 05.45.Mt

DOI
10.1139/p03-065

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Classical trajectories for the Coulomb potential are obtained from the large principle quantum-number limit of solutions to the nonrelativistic Schrödinger equation, by use of integral equations satisfied by the radial probability density function. These trajectories are found to be in excellent agreement with those computed directly from classical mechanics, in accordance with a statement of the Bohr Correspondence principle, except in a region very close to the center of force. PACS No.: 05.45.Mt
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The trajectory of motion of a scattering electron in the Coulomb potential from the wave function of the Schroedinger equation is presented in two ways, spherical polar coordinates and Temple coordinates, and is compared with each other and with the corresponding motion of classical mechanics. A good correspondence among dynamics by wave functions and the classical dynamics has been acknowledged by comparing computed examples.Detailed computing examples discriminate the optimal dynamics of the wave function that should be verified by an experiment.
Abstract We show that the quantum central‐force problems can be modeled and solved exactly by quantum Hamilton–Jacobi formulation, from which the quantum operators L^\hat{L}L^ z, L^\hat{L}L^ 2, and H^\hat{H}H^ can be derived without using the quantization principle p → (ℏ\hbarℏ /i)∂/∂x. Quantum conservation laws expressed by the Poisson bracket show that the eigenvalues of these quantum operators are just equal to the constants of motion along the eigen‐trajectories defined in a complex domain. The shell structure observed in bound systems, such as the hydrogen atom, is found to stem from the structure of the quantum potential, by which the quantum forces acting on the electron can be uniquely determined, the stability of atomic configuration can be justified, and the quantum trajectories of the electron can be obtained by integrating the related quantum Lagrange equations. On solving the quantum equations of motion, the solution of the Schrödinger equation serves as the first integration of the second‐order quantum Lagrange equations. The stable equilibrium points of the derived first‐order nonlinear quantum dynamics are shown to be identical to the positions with maximum probability predicted by standard quantum mechanics. The internal mechanism of how the quantum dynamics evolve continuously to classical dynamics and of how the quantum conservation laws transit continuously to the classical conservation laws as n → ∞ are analyzed in detail. The construction of the quantum scattering trajectory by searching for an unbound solution for the Schrödinger equation is investigated. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006
The quantum trajectories in the de Broglie-Bohm formulation of quantum mechanics depend on an additional quantum potential derived from the full wave solution of Schrödinger's equation. The task of supplying collectively all the correct quantum results strongly alters the characteristics of the corresponding classical trajectories, which underlie semiclassical approximations to the evolving wave function. Both classical and quantum trajectories are here considered to be conservative with no influence of an external environment, even though this is the source of eventual classicality in quantum systems, that is, decoherence. The concept of integrability, closely correspondent in classical and quantum mechanics, is not preserved by the quantum trajectories. General systems, in which classical chaotic motion participates, are much harder to treat semiclassically, but quantum trajectories can be chaotic even for integrable systems. This discrepancy between the character of classical and quantum trajectories in the de Broglie-Bohm interpretation does not clarify the singular classical-quantum transition.