Exact computation of quantum wave functions for nonlinear potentials

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.