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arXiv · 2401.03464

Constrained Least Action and Quantum Mechanics

Winfried Lohmiller·Jean-Jacques Slotine
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Abstract

Recent work shows that the Schroedinger equation can be solved exactly based only on classical least action. The computation is based on solving a HamiltonJacobi equation for the action computing the classical density accordingly along all stationary action paths and finally constructing the exact wave function based on these classical quantities alone. The method requires that the action Laplacian or more generally the Laplacian of the propagated density be purely time-varying along stationary action paths. In the case of arbitrary nonlinear potentials this condition can still be verified without loss of generality by using a time rescaling. As the complexity of the wave computation is thus shifted to that of the action and the possible time rescaling this paper proposes a new analytical approach both to compute the action itself for a general nonlinear Hamilton-Jacobi pde and to concurrently construct a time rescaling as needed In contrast to solving the Schroedinger equation directly this computation of action and density extends naturally to systems with nonlinear potentials or position dependent inertia tensors. Hence in principle it can replace the approximations of quantum perturbation theory. For general nonlinear potentials, extending a result of Duru and Kleinert the time rescaling in a given metric is shown to correspond to a change of variables in the computed action density and wave with each eigenwave computed using that change of variables. We show that the approach makes it straightforward to construct the quantum wave for basic cases where no exact solution has been yet derived with the time rescaling unifying the computation. The approach first illustrated for the known three dimensional hyperbolic potential waves of the hydrogen atom is then used to compute the quantum waves for a quartic oscillator for which so far no direct exact solutions are known.

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Winfried Lohmiller, Jean-Jacques Slotine. 2024-01-07. Constrained Least Action and Quantum Mechanics. https://arxiv.org/abs/2401.03464

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