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Physical Review A

Bohr’s correspondence principle: The cases for which it is exact

Adam J. Makowski, K. Górska

Published Dec 6, 2002• 18 citations
Publisher

Abstract

Two-dimensional central potentials leading to the identical classical and quantum motions are derived and their properties are discussed. Some of zero-energy states in the potentials are shown to cancel the quantum correction $Q=(\ensuremath{-}{\ensuremath{\Elzxh}}^{2}/2m)\ensuremath{\Delta}R/R$ to the classical Hamilton-Jacobi equation. The Bohr's correspondence principle is thus fulfilled exactly without taking the limits of high quantum numbers, of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0,$ or of the like. In this exact limit of $Q=0,$ classical trajectories are found and classified. Interestingly, many of them are represented by closed curves. Applications of the found potentials in many areas of physics are briefly commented.

DOI
10.1103/physreva.66.062103

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Two-dimensional central potentials leading to the identical classical and quantum motions are derived and their properties are discussed. Some of zero-energy states in the potentials are shown to cancel the quantum correction Q=(\ensuremath−\ensuremath\Elzxh2/2m)\ensuremathΔR/RQ=(\ensuremath{-}{\ensuremath{\Elzxh}}^{2}/2m)\ensuremath{\Delta}R/RQ=(\ensuremath−\ensuremath\Elzxh2/2m)\ensuremathΔR/R to the classical Hamilton-Jacobi equation. The Bohr's correspondence principle is thus fulfilled exactly without taking the limits of high quantum numbers, of \ensuremath\Elzxh\ensuremath→0,\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0,\ensuremath\Elzxh\ensuremath→​0, or of the like. In this exact limit of Q=0,Q=0,Q=0, classical trajectories are found and classified. Interestingly, many of them are represented by closed curves. Applications of the found potentials in many areas of physics are briefly commented.
Two-dimensional noncentral potentials are derived in the exact classical limit of quantum mechanics where the quantum correction to the classical Hamilton-Jacobi equation is canceled. It is then shown explicitly that there exist Ermakov-type invariants I for a large class of such potentials. Some of the invariants are found to be positive definite, I>0,I>0,I>0, and for the other, we have found I=0.I=0.I=0.
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
Planck formulated the correspondence principle between quantum and classical mechanics as the limit in which the Planck constant h goes to zero. Bohr formulated the correspondence principle to be the limit of large quantum numbers. For three common quantum mechanical systems it is shown that in order for eigenvalues of quantum mechanical observables to have meaningful classical limits, it is necessary to take the double limit as both the Planck constant goes to zero and the quantum number goes to infinity, subject to the constraint that their product is equal to an appropriate classical action. This synthesis of the Bohr and Planck formulations of the correspondence principle is also used to show that the quantum mechanical transition frequency between adjacent levels approaches the corresponding classical frequency. The features these systems have in common in their classical limit are explained by general considerations of the classical limit of the Schrödinger equation.
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