arXiv:2609.14480

Exact computation of quantum wave functions for nonlinear potentials

Winfried Loghmiller, Jean-Jacques Slotine

quant-phSubmitted Sep 13, 2026• Announced Sep 15, 2026• 0 citations

Abstract

Recent work shows that the Schroedinger equation can be solved exactly based only on classical least action rspa.2025.0413. The computation is based on first solving a Hamilton-Jacobi equation for the action, computing the classical propagated density along all stationary action paths, and finally constructing the exact kernel and wave function based on these classical quantities alone. The method requires that the propagated classical square root density along each stationary action path is harmonic (where a sufficient condition is that the Laplacian of the propagated action is purely time-varying), as is the case for basic examples such as the double-slit, quantum tunneling, the Einstein-Podolsky-Rosen experiment, or also as discussed here the relativistic propagator of a Higgs boson. This density is fundamentally different from using the Madelung density, which is the norm of an existing overall quantum wave. In the case of arbitrary nonlinear potentials, this condition can still be verified without loss of generality, by using a harmonic coordinate transformation and an associated time scaling for the propagated classical density and quantum wave, as this paper details. The resulting quantum wave of the time-scaled Schroedinger equation is equivalent to the quantum wave of the original Schroedinger equation, extending a standard result of Duru and Kleinert. Hence, in principle it can replace the approximations of quantum perturbation theory. For 1 to 3-dimensional systems, this harmonic coordinate change transforms a nonlinear potential into the quadratic potential of a linear oscillator, for which an exact analytic action and wave solution are already known. This makes the construction of the quantum wave beyond the hydrogen wave straightforward for basic cases where no exact solution has been derived, such as the quartic potential or the nonlinear pendulum.

Comments: This paper computes exact wave functions for nonlinear potentials where they have not been analytically derived, using the action based construction from rspa.2025.0413. Examples include the quartic potential and the nonlinear pendulum

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Time scaling in "Computing quantum waves exactly from classical action"

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In contrast to his earlier posting arXiv:2605.02621 [5] commenting on the articlerspa.2025.0413 [4] and answered in arXiv:2605.20443 [3], the same author of the newposting arXiv:2606.05197 [6] does not seem to dispute any longer that, as detailedin [3], (i) Assuming in [4] that, along each extremal action path, the action Laplacianor more generally the propagated density is space-independent, implies that theBohm-like potential terms are exactly zero – a condition which is directly verified inall examples of [4] (double slit, Aharonov-Bohm, potential well, harmonic oscillator,tunneling, EPR, QED) as well as in its relativistic Dirac and Maxwell extensions, withthe exception of the hydrogen atom example, and (ii) Just as in Feynman’s well-knownresults, the standard polynomials in the treatment in [4] of the harmonic oscillator andthe hydrogen atom appear naturally from the Taylor expansion of the kernel, and thusthere is no ”circularity”.He does not appear to dispute either that Bohm-like terms along individual stationaryaction paths are indeed very different from the standard Bohm-Madelung potentialon the overall wave. This is illustrated, e.g., in the detailed double slit example in [3].which can be solved very simply using the construction in [4], and whose Bohm-liketerms are exactly zero while the usual Bohm-Madelung potential can be very large.Also, in that example the action behind the slits is a conic function, so that exactFeynman or Van Vleck computations based on quadratic actions do not apply.His new posting [6] therefore concentrates on the claim in [3] that the spaceindependenceassumption on the Laplacian of the action, if not directly verified, canbe achieved through time-rescaling, which is a fair discussion topic. We all agree thatthe Schr¨odinger equation has to be fulfilled in the original x, t coordinates. To thiseffect, as in the hydrogen atom example of the original paper [4], we use a general resultof Duru and Kleinert, which the author of [5] and [6] may be unaware of. There,independence of the constructed eigenwave on the rescaled time is key to the exactbackmapping to the original x, t coordinates.Keywords − Response to arXiv:2606.05197, in relation to rspa.2025.0413 ”On computingquantum waves exactly from classical action”, Feynman path integrals, Duru-Kleinert time rescaling, Bohm-Madelung potential.

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