The Schrödinger equation is a linear partial differential equation that describes the temporal evolution of the state of a quantum mechanical system. One of the pillars of quantum mechanics, the Schrödinger equation has been used to describe a wide variety of phenomena such as wave-particle duality, atomic orbitals, quantum oscillations, and even some fluid mechanics problems. Despite its established importance in the development of quantum mechanics, a solution of the Schrödinger equation for an arbitrary distribution of quantum mechanical potential or for an arbitrary number of space dimensions remains unknown. This work seeks to rectify this theoretical roadblock by providing an analytical solution through a recasting of the well-known Cole-Hopf transformation which is used to linearize a broad class of PDEs of which the Schrödinger equation becomes a member after an earlier change of variables. These two successive transformations render the general wave function proportional to the solution of a forced heat equation, allowing an explicit analytical solution applicable to any system in any number of space dimensions. Two practical applications to deriving the explicit wave functions of multielectronic atoms and to the forced Burgers equation of fluid mechanics are discussed, enabling greater mathematical insight into quantum entanglement and nonlinear fluid systems.
Abstract Using the simplest but fundamental example, the problem of the infinite potential well, this paper makes an ideological attempt (supported by rigorous mathematical proofs) to approach the issue of ≪understanding≫ the mechanism of quantum mechanics processes, despite the well-known examples of the EPR paradox type. The new exact solution of the Schrödinger equation is analyzed from the perspective of quantum mechanics in the phase space. It is the phase space, which has been extensively used recently in quantum computing, quantum informatics and communications, that is the bridge towards classical physics, where understanding of physical reality is still possible. In this paper, an interpretation of time-dependent processes of energy redistribution in a quantum system, probability waves, the temperature and entropy of a quantum system, and the transition to a time-independent ≪frozen state≫ is obtained, which is understandable from the point of view of classical physics. The material of the paper clearly illustrates the solution of the problem from the standpoint of continuum mechanics, statistical physics and, of course, quantum mechanics in the phase space.
The rigorous solutions of the stationary Schrodinger equation for hydrogen atom are solved with the wave-mechanics method within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The Fourier-like projection transformations of wave function from the phase space to position and momentum spaces are extended to three-dimensional systems. The eigenfunctions in general position and momentum spaces could be obtained through the transformations from eigenfunction in the phase space.