Before going further to discuss the one–dimensional problems in quantum mechanics, let us first describe the general properties of the one–dimensional Schrödinger equation.
A duality between an electrostatic problem in a three dimensional world and a quantum mechanical problem in a one dimensional world which allows one to obtain the ground state solution of the Schrödinger equation by using electrostatic results is generalized to three dimensions. Here, it is demonstrated that the same transformation technique is also applicable to the s-wave Schrödinger equation in three dimensions for central potentials. This approach leads to a general relationship between the electrostatic potential and the s-wave function and the electric energy density to the quantum mechanical energy.
On the basis of Nelson’s stochastic mechanics derivation of the Schrödinger equation, a formal mathematical structure of non-relativistic quantum mechanics equivalent to the one in classical analytical mechanics has been established in the literature. We recently were able to augment this structure by deriving quantum Hamilton equations of motion by finding the Nash equilibrium of a stochastic optimal control problem, which is the generalization of Hamilton’s principle of classical mechanics to quantum systems. We showed that these equations allow a description and numerical determination of the ground state of quantum problems without using the Schrödinger equation. We extend this approach here to deliver the complete discrete energy spectrum and related eigenfunctions for bound states of one-dimensional stationary quantum systems. We exemplify this analytically for the one-dimensional harmonic oscillator and numerically by analyzing a quartic double-well potential, a model of broad importance in many areas of physics. We furthermore point out a relation between the tunnel splitting of such models and mean first passage time concepts applied to Nelson’s diffusion paths in the ground state.