Principal solution of the Schrödinger equation with Caldirola-Kanai Hamiltonian

I. C. Moeeira

Roohollah Daneshmand, Mohammad Kazem Tavassoly

Jeong Ryeol Choi

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).

Kálmán Varga, Joseph A. Driscoll

Shuai Wang, Hong‐Chun Yuan, Hong-Yi Fan

Based on the technique of integration within an ordered product (IWOP) of operators, we introduce the Fresnel operator for converting a kind of time-dependent Hamiltonian into the standard harmonic oscillator Hamiltonian. The Fresnel operator with the parameters A, B, C, D corresponds to classical optical Fresnel transformation, these parameters are the solution to a set of coupled partial differential equations set up in the above-mentioned converting process. In this way, the Caldirola–Kanai Hamiltonian has been easily converted into the standard harmonic oscillator Hamiltonian. And then the exact wave function solution of the Schrödinger equation governed by the Caldirola–Kanai Hamiltonian is obtained, which represents a squeezed number state. The corresponding Wigner function is derived by virtue of the Weyl ordered form of the Wigner operator and the order-invariance of Weyl ordered operators under similar transformations.