Fresnel operator, squeezed state and Wigner function for Caldirola-Kanai Hamiltonian

TL;DR

This paper introduces a Fresnel operator approach to transform the Caldirola-Kanai Hamiltonian into a time-independent form, enabling exact solutions for the wavefunction and Wigner function of a squeezed state, applicable to other time-dependent oscillators.

Contribution

It presents a novel method using the Fresnel operator and IWOP technique to solve the Schrödinger equation for the Caldirola-Kanai Hamiltonian, producing exact wavefunctions and Wigner functions.

Findings

Exact wavefunction as a squeezed number state

Derivation of the Wigner function using Weyl ordering

Method applicable to other time-dependent oscillators

Abstract

Based on the technique of integration within an ordered product (IWOP) of operators we introduce the Fresnel operator for converting Caldirola-Kanai Hamiltonian into time-independent 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 partial differential equations set up in the above mentioned converting process. In this way the exact wavefunction solution of the Schr\"odinger 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. The method used here can be suitable for solving Schr\"odinger equation of other time-dependent oscillators.