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

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 Schrödinger 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 $\widetildeγ(z)=\partial_z f(z)/[2f(z)]$ and a $z$-dependent oscillator frequency $\widetildeω(z)=p_0Ω(z)/\sqrt{f(z)}$. 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.

Figures (2)

• Figure 1: Evolution of the beam width (top panel) and the total relativistic energy (bottom panel) for a twisted electron during acceleration in a system of sequential solenoids and accelerating sections. The red curve corresponds to an initial wave packet size of $b(0)=2$ and $b'(0)=-1$. The energy increases from $2~\mathrm{MeV}$ to $8.45~\mathrm{MeV}$. The maximal magnetic-field amplitude is $0.5~\mathrm{T}$ and the maximal electric-field amplitude is $-10~\mathrm{MV/m}$. The longitudinal coordinate $z$ is measured in units of $z_0 = p_0 \rho^2_H/\hbar \approx 0.025~\mathrm{m}$.
• Figure 2: Evolution of the twisted electron beam width in a homogeneous electric field with zero magnetic field. The parameters are $w_0 = 0.2 \; \mu m$, $\partial_{\tilde{z}} w_0 = - 2/(p_0 w_0)$, $p_0 = 0.02 \; MeV$, $q = - |e|$. Blue and dashed black curves correspond to the electric field $E_z = -1.4 \; MV/m$, purple and dashed rose curves correspond to $E_z = -3 \; MV/m$. The longitudinal coordinate $z$ is measured in units of $z_0 = p_0 \rho^2_H/\hbar \approx 0.025$ m.