Effective Caldirola-Kanai Model for Accelerating Twisted Dirac States in Nonuniform Axial Fields

TL;DR

This paper develops an effective Caldirola-Kanai model to describe the evolution of twisted Dirac states in nonuniform axial fields, providing a closed-form solution that generalizes previous models.

Contribution

It introduces a novel Caldirola-Kanai Hamiltonian approach for relativistic twisted states in inhomogeneous fields, enabling analytical solutions via Ermakov mapping.

Findings

Derived a nonstationary Schrödinger equation governing transverse dynamics.

Obtained a closed-form wave function using Ermakov mapping and Landau basis.

Unified previous models for uniform acceleration and solenoid focusing cases.

Abstract

We study relativistic twisted (orbital-angular-momentum) states of a massive charged particle propagating through an axially symmetric, longitudinally inhomogeneous solenoid field and a co-directed accelerating or decelerating electric field. Starting from the Dirac equation and using controlled spinless and paraxial approximations, we show that the transverse envelope obeys an effective nonstationary Schr\"odinger equation governed by a Caldirola--Kanai Hamiltonian. The longitudinal energy gain or loss encoded in $f(z)=[E_0-V(z)]^2-m^2$ generates an effective gain or damping rate $\widetilde{\gamma}(z)=\partial_z f(z)/[2f(z)]$ and a $z$-dependent oscillator frequency $\widetilde{\omega}(z)=p_0\Omega(z)/\sqrt{f(z)}$. Exploiting the Ermakov mapping (unitary equivalence of Caldirola--Kanai systems), we obtain a closed-form propagated twisted wave function by transforming the stationary Landau basis. The transverse evolution is controlled by a single scaling function $b(z)$ that satisfies a generalized Ermakov--Pinney equation with coefficients determined by $E_z(z)$ and $B_z(z)$. In the limiting cases of uniform acceleration with $B_z=0$ and of solenoid focusing with negligible acceleration, our solution reduces to previously known analytic results, providing a direct bridge to established models.