Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Aspects of Beam Physics

SINGLE-PARTICLE QUANTUM DYNAMICS IN A MAGNETIC LATTICE

Marco Venturini, Ronald D. Ruth

Published May 1, 2002• 0 citations
Publisher

Abstract

We study the quantum dynamics of a spinless charged-particle propagating through a magnetic lattice in a transport line or storage ring. Starting from the Klein-Gordon equation and by applying the paraxial approximation, we derive a Schrodinger-like equation for the betatron motion. A suitable unitary transformation reduces the problem to that of a simple harmonic oscillator. As a result we are able to nd an explicit expression for the particle wavefunction. Contributed to 18 th

DOI
10.1142/9789812777447_0045

Related papers

Algebra of quantum mechanics via classical phonons. I: The Schrodinger equation as the Newtonian equation of motion and quantum observables as classical averages

Emmanuel Giner

The Schrodinger equation for a single spinless particle is formally obtained via a classical phonon model, namely the Frenkel-Kontorova model. Starting from a one-dimensional lattice of coupled harmonic oscillators, we show that the continuous limit of the corresponding Newtonian equation of motion yields the Klein-Gordon equation for a real-valued field. By introducing a complex-valued change of variables mixing the real-valued displacement and velocity fields, and by separating fast and slow time scales, the Klein-Gordon equation is written as the Schrodinger equation within the non-relativistic limit. This complex change of variable also allows to rewrite classical global observables of the phonon field, such as the total energy or momentum, as the corresponding quantum observables. Additionally, we show that when a friction force is incorporated into the classical model, the corresponding Klein-Gordon equation can be rewritten as a Schrodinger equation with a non-Hermitian Hamiltonian. While the global approach is limited here to the non-relativistic regime and does not address the measurement problem, quantization or relativistic effects, it nonetheless illustrates how quantum algebra and complex-valued wave functions can be exactly reproduced using classical dynamics. The relativistic regime for a spinless particle and the link between commutators and Poisson brackets is addressed in the second part of this series.

quant-pharXiv:2607.03897•Jul 7, 2026•0 citations

Schrodinger Equation for a Half Spin Electron ¨ in a Time Dependent Magnetic Field

Hafeez Yusuf Hafeez, E. N. Chifu, Ibrahim Murtala Musa

PrPh•Jan 1, 2016•1 citation

Phase space formulations of the quantum electronic motion in a magnetic field

Thierry Champel

•Dec 5, 2025•0 citations

“Square-Root” Klein–Gordon Equation: The Harmonic and Morse Potentials

Luis A. Poveda, Bill Poirier, Arthur R. B. de Magalhães

Quantum relativistic solutions for a particle in a one-dimensional Morse potential are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultrarelativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior, and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method, and the so-called Klein-Gordon oscillator.

Atoms•Aug 1, 2026•0 citations

Spin dynamics in lattices of spinor atoms with quadratic Zeeman effect

Vyacheslav I. Yukalov, Elizaveta P. Yukalova

arXiv:1811.04271•The European Physical Journal D•Oct 30, 2018•3 citations

Constrained parameters and exact solution of Schrödinger equation of charged particle in a time-dependent electric field: A unitary transformation approach

Rachid Ahmim, Nabil Baouche, S. Askri

We present an exact analytical solution for the quantum dynamics of a charged particle subjected to both a time-dependent electric field and a static magnetic field aligned along the $$z$$ -direction. Using a systematic approach based on successive unitary transformations, we reduce the original three-dimensional problem to a two-dimensional system of decoupled, time-dependent harmonic oscillators. This technique produces free parameters that allow us to impose constrains to derive the exact solution of Schrödinger equation with the time-dependent Hamiltonian through the explicit derivation of the quantum propagator and shows the equivalence of this approach to established path integral methods for such systems. The developed framework provides new insights into quantum systems with time-dependent electromagnetic fields and offers analytical solutions.

Theoretical and Mathematical Physics•Aug 1, 2025•0 citations
We study the quantum dynamics of a spinless charged-particle propagating through a magnetic lattice in a transport line or storage ring. Starting from the Klein-Gordon equation and by applying the paraxial approximation, we derive a Schrodinger-like equation for the betatron motion. A suitable unitary transformation reduces the problem to that of a simple harmonic oscillator. As a result we are able to nd an explicit expression for the particle wavefunction. Contributed to 18 th
The Schrodinger equation for a single spinless particle is formally obtained via a classical phonon model, namely the Frenkel-Kontorova model. Starting from a one-dimensional lattice of coupled harmonic oscillators, we show that the continuous limit of the corresponding Newtonian equation of motion yields the Klein-Gordon equation for a real-valued field. By introducing a complex-valued change of variables mixing the real-valued displacement and velocity fields, and by separating fast and slow time scales, the Klein-Gordon equation is written as the Schrodinger equation within the non-relativistic limit. This complex change of variable also allows to rewrite classical global observables of the phonon field, such as the total energy or momentum, as the corresponding quantum observables. Additionally, we show that when a friction force is incorporated into the classical model, the corresponding Klein-Gordon equation can be rewritten as a Schrodinger equation with a non-Hermitian Hamiltonian. While the global approach is limited here to the non-relativistic regime and does not address the measurement problem, quantization or relativistic effects, it nonetheless illustrates how quantum algebra and complex-valued wave functions can be exactly reproduced using classical dynamics. The relativistic regime for a spinless particle and the link between commutators and Poisson brackets is addressed in the second part of this series.
Quantum relativistic solutions for a particle in a one-dimensional Morse potential are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultrarelativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior, and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method, and the so-called Klein-Gordon oscillator.
We present an exact analytical solution for the quantum dynamics of a charged particle subjected to both a time-dependent electric field and a static magnetic field aligned along the zzz -direction. Using a systematic approach based on successive unitary transformations, we reduce the original three-dimensional problem to a two-dimensional system of decoupled, time-dependent harmonic oscillators. This technique produces free parameters that allow us to impose constrains to derive the exact solution of Schrödinger equation with the time-dependent Hamiltonian through the explicit derivation of the quantum propagator and shows the equivalence of this approach to established path integral methods for such systems. The developed framework provides new insights into quantum systems with time-dependent electromagnetic fields and offers analytical solutions.