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American Journal of Physics

Natural relationship between classical orbits and quantum Hamiltonians for the Kepler problem

Jason D. Tran, Leanne Doughty, James K. Freericks

Published Apr 22, 2025• 1 citation
Publisher

Abstract

Central force problems involve coupled linear differential equations of motion. By decoupling the differential equations, we show how to efficiently solve for the orbits of the Kepler problem without any complicated integrals. Introducing quantization provides a deep connection between the solution for these classical orbits and the quantum Hamiltonian of hydrogen for states that have definite total angular momentum.

DOI
10.1119/5.0220799

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Central force problems involve coupled linear differential equations of motion. By decoupling the differential equations, we show how to efficiently solve for the orbits of the Kepler problem without any complicated integrals. Introducing quantization provides a deep connection between the solution for these classical orbits and the quantum Hamiltonian of hydrogen for states that have definite total angular momentum.
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We investigate the semiclassical Kepler/Coulomb problem using the classical constants of the motion in the framework of Nelson's stochastic mechanics. This is done by considering the eigenvalue relations for a family of coherent states (known as the atomic elliptic states) whose wave functions are concentrated on the elliptical orbit corresponding to the associated classical problem. We show that these eigenvalue relations lead to identities for the semiclassical energy, angular momentum and Hamilton–Lenz–Runge vectors in the elliptical case. These identities are then extended to include the cases of circular, parabolic and hyperbolic motions. We show that in all cases the semiclassical wave function is determined by our identities and so our identities can be seen as defining a semiclassical Kepler/Coulomb problem. The results are interpreted in terms of two dynamical systems: one a complex valued solution to the classical mechanics for a Coulomb potential and the other the drift field for a semiclassical Nelson diffusion.
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
One of the most important developments of modern science was the emergence of the concept of energy quantization which forced its way into our previous classical understanding of matter in nature. Modern day view of matter constituents and present understanding of atomic structure basic to quantum chemistry arose in those first decades of last century. In this line, a key feature for the understanding of microscopic atomic structures and their stability was the concept of angular momentum quantization. To capture the beauty and the power of these concepts is therefore extremely relevant for apprehending the contemporary view of matter, energy and interactions in nature. In this work we present a possible alternative approach of quantizing the atomic orbit for the hydrogen's atom model proposed by Bohr without using his hypothesis of angular momentum quantization from the start. In our approach here we show that when we identify the number of photons and the quantum number for the orbit, we are led to Bohr's hypothesis as a consequence of the Planck's energy quantization.
Solutions of Schrödinger's equation for Rydberg states of hydrogen are shown to display patterns that can be understood semi-classically within the framework of Bohr-Sommerfeld planetary orbits and de Broglie's concept of matter waves. Using n = 10 and n = 100 quantum levels we show how de Broglie-wave-analyzed Kepler orbits can be matched to excited-state wavefunctions and charge distributions. Making the connection between the old and new quantum theories can help beginning students develop an intuition about quantum mechanics and allow them to develop an appreciation of how Schrödinger's wave mechanics was stimulated by the theories of his predecessors.