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Revista Brasileira de Ensino de Física

Derivation of a dissipative Schrödinger equation VI: the Caldirola-Kanai equation

Olavo L. Silva Filho, Marcello Ferreira

Published Jan 1, 2025• 0 citations
Publisher

Abstract

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).

DOI
10.1590/1806-9126-rbef-2025-0207

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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).
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms.
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.
Quantization processes generally assume the Hamiltonian formulation of classical mechanics. The notion of dissipation does not fit easily into the Hamiltonian structure. There appeared in the literature a quantization process that circumvents the use of the Hamiltonian approach and derives the Schr\"odinger equation from first principles. Thus, the usual approach of assuming a dissipative force of the type fd=k(t)v{f}_{d}=k(t)vfd​=k(t)v, where k(t)k(t)k(t) is a scalar depending only on the time and vvv is the velocity, can be approached using this quantization process to mathematically derive a Schr\"odinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as fi(x,t)=Kij(x,t)vj{f}_{i}(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t){v}_{j}fi​(x,t)=Kij​(x,t)vj​, where K(x,t)\mathcal{K}(\mathbf{x},t)K(x,t) is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schr\"odinger-like equations already known from the literature.