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

arXiv:2603.09093 · APS Open Science

The helical quantum two-body problem and its wave packet dynamics

Peter Schmelcher

quant-phcond-mat.mes-hallSubmitted Mar 10, 2026• Announced Mar 11, 2026• 0 citations
arXivHTMLPDF

Abstract

We explore the helical quantum two-body problem i.e. two repulsively Coulomb interacting particles confined to move along a helix. The effective potential possesses a tunable number of potential wells superimposed on the repulsive Coulomb interaction that can be varied by changing the ratio of the pitch and radius of the helix. The anharmonicity of these wells depends crucially on this ratio and on the order of the well which can be seen also by analyzing the individual wells energy eigenvalue spacing. Our main focus is the investigation of the quantum dynamics of differently prepared wave packets that scatter from the multi-well potential landscape. We show that there exists a rich pattern forming transient evolution which depends also on the number of bound states of the individual wells. We demonstrate how the multiple wells leave their fingerprints in the dynamics leading, among others, to oscillatory structures on different spatial scales, the formation of beats and pulsed emission from single well localized wave packets due to their intrawell dynamics.

Related papers

Multilevel dynamics of matter waves scattering in finite potential wells

Miguel A. G. Mandujano, Jorge Villavicencio, Roberto Romo

Abstract We explore the dynamics of matter wave scattering in finite potential wells using analytical solutions of the Schrödinger equation within the framework of a quantum shutter model. We find that the incident wave interferes with the bound states of the quantum well, resulting in time-domain oscillations. These oscillations exhibit Rabi-type frequencies, characterised by the energy differences between the incident wave and the bound states of the quantum well. We show that in systems with double-bound states, the interference pattern is characterised by quantum beats in the time-dependent probability density. The period of these beatings depends on the energy difference between the bound states, which can be tuned by controlling the potential parameters. In the general case where bound, anti-bound, and resonant states coexist in the system spectrum, complex oscillations in the probability density arise from the interactions of the incident wave with different quantum states. We demonstrate that the bound states sector can effectively describe this complex behaviour, providing a simple and reliable analytical expression for the probability density in multilevel systems. This formula highlights the significant role of bound states, whose interaction with the incident wave dominates the transient probability density. This contrasts with conventional systems with potential barriers and wells, where resonances govern the wave dynamics.

Physica Scripta•Jul 22, 2024•1 citation

Computational Investigation of Wave Packet Scattering in the Complex Plane: Dynamics of Exact Quantum Trajectories

Róbert E. Wyatt, Brad A. Rowland

In two previous studies, the time-dependent scattering of a wave packet from a Gaussian barrier was investigated computationally in the complex z-plane. One of these involved the 'direct' propagation of the wave packet in the complex space, and the other used numerical analytic continuation techniques to generate the dynamics in the complex plane from the wave function computed on the real-axis. In the current study, the dynamics of exact quantum trajectories are analyzed for the same barrier scattering problem. Thousands of quantum trajectories were launched from positions near the center of the initial wave packet. These trajectories were computed by integrating equations-of-motion involving the quantum momentum function, which was obtained from the time-dependent wave function and its derivative. In order to analyze the dynamics, many trajectories were plotted on space-time diagrams. Particular emphasis was placed upon trajectories undergoing reflection in the barrier region. Some groups of strongly correlated trajectories form long-lived highly organized patterns, including helical wrappings around a series of stagnation filaments. These curves alternate with quasi-nodes where the amplitude of the wave function reaches low values. In addition, other trajectories for short times follow hyperbolic paths as they propagate near vorticity tubes surrounding these quasi-nodes.

Journal of Chemical Theory and Computation•Jan 29, 2009•4 citations

Dynamics of Bose-Einstein condensates in double-well potentials

M. Jääskeläinen, P. Meystre

We study the dynamics of Bose-Einstein condensates in symmetric double-well potentials following a sudden change of the potential from the Mott-insulator to the superfluid regime. We introduce a continuum approximation that maps that problem onto the wave-packet dynamics of a particle in an anharmonic effective potential. For repulsive two-body interactions the visibility of interference fringes that result from the superposition of the two condensates following a stage of ballistic expansion exhibits a collapse of coherent oscillations onto a background value whose magnitude depends on the amount of squeezing of the initial state. Strong attractive interactions are found to stabilize the relative number dynamics. We visualize the dynamics of the system in phase space using a quasiprobability distribution that allows for an intuitive interpretation of the various types of dynamics.

Physical Review A•Apr 11, 2005•47 citations

Wave-packet dynamics in nonlinear Schrödinger equations

Simon Moulieras, Alejandro Monastra, Marcos Saraceno +1 more

Coherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schr\"odinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: The center follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satisfies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape or of the confining potential. A rich variety of effects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the wavepacket is transferred toward incoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped toward the bottom of the well.

arXiv:1107.0259•Physical Review A•Jan 26, 2012•8 citations

Dynamics of Two-Body Systems

Twisha Choudhary

This study presents a comprehensive analysis of the two-body central-interaction problem in quantum mechanics. We investigate the Schrödinger equation for two particles interacting through a central potential, and derive the radial and angular parts of the wave function. The energy eigenvalues and Eigen functions are obtained for various central potentials, including the Coulomb, harmonic oscillator, and square well potentials.

Journal of Research in Science and Engineering•Mar 27, 2026•0 citations

Formation of classical crystals of dipolar particles in a helical geometry

J K Pedersen, D V Fedorov, A S Jensen +1 more

We consider crystal formation of particles with dipole-dipole interactions that are confined to move in a one-dimensional helical geometry with their dipole moments oriented along the symmetry axis of the confining helix. The stable classical lowest energy configurations are found to be chain structures for a large range of pitch-to-radius ratios for relatively low density of dipoles and a moderate total number of particles. The classical normal mode spectra support the chain interpretation both through structure and the distinct degeneracies depending discretely on the number of dipoles per revolution. A larger total number of dipoles leads to a clusterization where the dipolar chains move closer to each other. This implies a change in the local density and the emergence of two length scales, one for the cluster size and one for the inter-cluster distance along the helix. Starting from three dipoles per revolution, this implies a breaking of the initial periodicity to form a cluster of two chains close together and a third chain removed from the cluster. This is driven by the competition between in-chain and out-of-chain interactions, or alternatively the side-by-side repulsion and the head-to-tail attraction in the system. The speed of sound propagates along the chains. It is independent of the number of chains although depending on geometry.

arXiv:1402.4335•Journal of Physics B Atomic Molecular and Optical Physics•Aug 8, 2014•15 citations
We explore the helical quantum two-body problem i.e. two repulsively Coulomb interacting particles confined to move along a helix. The effective potential possesses a tunable number of potential wells superimposed on the repulsive Coulomb interaction that can be varied by changing the ratio of the pitch and radius of the helix. The anharmonicity of these wells depends crucially on this ratio and on the order of the well which can be seen also by analyzing the individual wells energy eigenvalue spacing. Our main focus is the investigation of the quantum dynamics of differently prepared wave packets that scatter from the multi-well potential landscape. We show that there exists a rich pattern forming transient evolution which depends also on the number of bound states of the individual wells. We demonstrate how the multiple wells leave their fingerprints in the dynamics leading, among others, to oscillatory structures on different spatial scales, the formation of beats and pulsed emission from single well localized wave packets due to their intrawell dynamics.
Abstract We explore the dynamics of matter wave scattering in finite potential wells using analytical solutions of the Schrödinger equation within the framework of a quantum shutter model. We find that the incident wave interferes with the bound states of the quantum well, resulting in time-domain oscillations. These oscillations exhibit Rabi-type frequencies, characterised by the energy differences between the incident wave and the bound states of the quantum well. We show that in systems with double-bound states, the interference pattern is characterised by quantum beats in the time-dependent probability density. The period of these beatings depends on the energy difference between the bound states, which can be tuned by controlling the potential parameters. In the general case where bound, anti-bound, and resonant states coexist in the system spectrum, complex oscillations in the probability density arise from the interactions of the incident wave with different quantum states. We demonstrate that the bound states sector can effectively describe this complex behaviour, providing a simple and reliable analytical expression for the probability density in multilevel systems. This formula highlights the significant role of bound states, whose interaction with the incident wave dominates the transient probability density. This contrasts with conventional systems with potential barriers and wells, where resonances govern the wave dynamics.
In two previous studies, the time-dependent scattering of a wave packet from a Gaussian barrier was investigated computationally in the complex z-plane. One of these involved the 'direct' propagation of the wave packet in the complex space, and the other used numerical analytic continuation techniques to generate the dynamics in the complex plane from the wave function computed on the real-axis. In the current study, the dynamics of exact quantum trajectories are analyzed for the same barrier scattering problem. Thousands of quantum trajectories were launched from positions near the center of the initial wave packet. These trajectories were computed by integrating equations-of-motion involving the quantum momentum function, which was obtained from the time-dependent wave function and its derivative. In order to analyze the dynamics, many trajectories were plotted on space-time diagrams. Particular emphasis was placed upon trajectories undergoing reflection in the barrier region. Some groups of strongly correlated trajectories form long-lived highly organized patterns, including helical wrappings around a series of stagnation filaments. These curves alternate with quasi-nodes where the amplitude of the wave function reaches low values. In addition, other trajectories for short times follow hyperbolic paths as they propagate near vorticity tubes surrounding these quasi-nodes.
We study the dynamics of Bose-Einstein condensates in symmetric double-well potentials following a sudden change of the potential from the Mott-insulator to the superfluid regime. We introduce a continuum approximation that maps that problem onto the wave-packet dynamics of a particle in an anharmonic effective potential. For repulsive two-body interactions the visibility of interference fringes that result from the superposition of the two condensates following a stage of ballistic expansion exhibits a collapse of coherent oscillations onto a background value whose magnitude depends on the amount of squeezing of the initial state. Strong attractive interactions are found to stabilize the relative number dynamics. We visualize the dynamics of the system in phase space using a quasiprobability distribution that allows for an intuitive interpretation of the various types of dynamics.
Coherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schr\"odinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: The center follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satisfies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape or of the confining potential. A rich variety of effects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the wavepacket is transferred toward incoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped toward the bottom of the well.
We consider crystal formation of particles with dipole-dipole interactions that are confined to move in a one-dimensional helical geometry with their dipole moments oriented along the symmetry axis of the confining helix. The stable classical lowest energy configurations are found to be chain structures for a large range of pitch-to-radius ratios for relatively low density of dipoles and a moderate total number of particles. The classical normal mode spectra support the chain interpretation both through structure and the distinct degeneracies depending discretely on the number of dipoles per revolution. A larger total number of dipoles leads to a clusterization where the dipolar chains move closer to each other. This implies a change in the local density and the emergence of two length scales, one for the cluster size and one for the inter-cluster distance along the helix. Starting from three dipoles per revolution, this implies a breaking of the initial periodicity to form a cluster of two chains close together and a third chain removed from the cluster. This is driven by the competition between in-chain and out-of-chain interactions, or alternatively the side-by-side repulsion and the head-to-tail attraction in the system. The speed of sound propagates along the chains. It is independent of the number of chains although depending on geometry.