The Schrödinger equation

Sutee Boonchui, Virulh Sa‐Yakanit, Wichit Sritrakool

The clocked Schr\"odinger equation was proposed by Sokolovski using the Feynman path integral with constraint (Phys. Rev. A 52, R2, 1995). Sokolovski pointed out that the clocked Schr\"odinger equation cannot be derived directly from the Schr\"odinger equation. In this paper, we show that the clocked Schr\"odinger equation can be derived by starting from the normal Schr\"odinger equation for a composite system, composed of the observed system and the measuring device, as defined by von Neumann. Details of the derivation and the physical meaning of the clocked Schr\"odinger equation are given.

Walter Greiner

Robert Wieser, Chan‐Ho Yang

Abstract The missing derivation of the time-dependent Schrödinger equation following Schrödinger’s original description of the time-independent Schrödinger equation. Also, this description is extended to derive the Caldirola-Kanai, the Schuch-Schrödinger, and the Gisin-Schrödinger equation. In the second part, the Gisin-Schrödinger equation will be derived once more using the Ito formalism of stochastic differential equations. Furthermore, we discuss the extension to larger spin-system using the cluster mean-field theory.

Masao Nagasawa

Schrödinger Equations and Diffusion Theory addresses the question "What is the Schrödinger equation?" in terms of diffusion processes, and shows that the Schrödinger equation and diffusion equations in duality are equivalent. In turn, Schrödinger's conjecture of 1931 is solved. The theory of diffusion processes for the Schrödinger equation tell us that we must go further into the theory of systems of (infinitely) many interacting quantum (diffusion) particles. The method of relative entropy and the theory of transformations enable us to construct severely singular diffusion processes which appear to be equivalent to Schrödinger equations. The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schrödinger equation, namely, quantum mechanics. The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level.

Rolf E. Hummel