Derivation of the Schrödinger Equation

Olavo L. Silva Filho, Marcello Ferreira

In this paper, our main objective is to derive the Caldirola-Kanai equation from first principles and present some analytical solutions to it. To do so, we will extend the axioms of the derivation of the Schrödinger equation and, thus, present the derivation of the usual Schrödinger equation, as a means of comparison to its extension. Next, we will present the derivation of the Caldirola-Kanai equation from the same axioms already used to derive the usual Schrödinger equation, now extended to include dissipation; right away, we will give some examples of the application of the Caldirola-Kanai equation (those that are analytically solvable).

H. N. Ioannidou

M. Roncadelli

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.

Doug Jensen

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.