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

Physical Review A

Effects of scattering on wave-packet structure

Edward A. Olszewski

Published Jan 1, 1986• 0 citations
Publisher

Abstract

We consider a general reaction process in which two entrance-channel particles, the beam and target particles, interact off-resonance and produce two exit-channel particles. In the c.m. frame we derive a general expression relating the probability density of the exit-channel particles to that of the entrance-channel particles. Furthermore, we show that whenever the coherence length, ${w}_{B}$, of the beam particle is much larger than the coherence length, ${w}_{T}$, of the target particle, the wave function of each exit-channel particle may be expressed explicitly in terms of the wave function of the beam particle. We also derive expressions for the coherence time and coherence length of each exit-channel particle. In addition, whenever the speed of the target particle is much less than the speed ${v}_{i}$ (i=1,2) of an exit-channel particle, we show that a simple relationship exists between the coherence length, ${w}_{\mathrm{ci}}$, of the exit-channel particle and the beam particle: ${w}_{\mathrm{ci}}$=${v}_{i}$(${w}_{B}$/\ensuremath{\Vert}${\mathrm{v}}_{B}$${\mathrm{\ensuremath{-}}\mathrm{v}}_{T}$\ensuremath{\Vert}){1 ${\mathrm{\ensuremath{-}}[(\mathrm{v}}_{\mathrm{T}}$\ensuremath{\cdot}e${\mathrm{^}}_{3}$)/ ${v}_{i}$](costheta')}, where the velocity of the beam particle is ${\mathrm{v}}_{B}$ ${(\mathrm{v}}_{B}$=${v}_{B}$e${^}_{3}$), and the velocity of the target particle is ${\mathrm{v}}_{T}$ ${(\mathrm{v}}_{T}$=${v}_{T}$e${^}_{3}$). The angle theta' is the spherical polar angular coordinate, measured with respect to the unit vector e${^}_{3}$, of the point at which the coherence length ${w}_{\mathrm{ci}}$ is to be determined. The above relationship is applied in the case of photoionization of the hydrogen atom at low energies to obtain realistic estimates for the coherence lengths of the ionized electron and proton. The coherence length of the electron is shown to be of nearly macroscopic dimensions.

DOI
10.1103/physreva.33.131

Related papers

Loss of wave-packet coherence in ion-atom collisions

László Sarkadi, Ignacio Fabre, Francisco Navarrete +1 more

The projectile beam coherence effects occurring in ion-atom collisions are analyzed on the basis of the recent theory of Karlovets et al. [Phys. Rev. A 92, 052703 (2015)] developed for the elastic scattering of wave packets of particles off a potential field. This theory is generalized to estimate the loss of coherence for inelastically scattered projectiles in ionizing collisions. The results obtained by the suggested model are compared with experimental data for the ionization of hydrogen atoms and molecules by 75-keV proton impact. Significantly improved agreement is observed between the theory and experiment.

Physical Review A•Mar 3, 2016•34 citations

Quantum Beam Scattering—Beam’s Coherence Length, Which-Path Information and Weak Values

C. A. Chatzidimitriou‐Dreismann

The conventional theory of neutron beams interacting with many-body systems treats the beam as a classical system, i.e. with its dynamical variables appearing in the quantum dynamics of the scattering process not as operators but only as c-numbers. Moreover, neutrons are described with plane waves, i.e. the concept of neutron's (finite) coherence length is here irrelevant. The same holds for electron, atom or X-ray scattering. This simplification results in the full decoupling of the probe particle's dynamics from the quantum dynamics of the scatterer---a well-known fact also reflected in the standard formalism of time-correlation functions (see textbooks). Making contact with modern quantum theoretical approaches (e.g., quantum entanglement, 'which-path detection' versus interference, von Neumann measurement, Week Values, etc.) new observable effects may be exposed and/or predicted, for instance, a momentum-transfer deficit and an intensity deficit in neutron scattering from protons of hydrogen-containing samples. 7

Quantum Beam Science•Aug 15, 2023•1 citation

Wave Packets and Scattering Probabilities

Travis Norsen

We propose a simple, intuitive alternative method of deriving the rule for connecting asymptotic wave function amplitudes to scattering probabilities. This is illustrated using the standard example of a 1-D particle reflecting or transmitting from a potential step.

arXiv:0910.1513•Oct 8, 2009•0 citations

Shot noise and spin-orbit coherent control of entangled and spin-polarized electrons

J. Carlos Egues, Guido Burkard, Daniel S. Saraga +2 more

We extend our previous work on shot noise for entangled and spin polarized electrons in a beam-splitter geometry with spin-orbit (SO) interaction in one of the incoming leads (lead 1). In addition to accounting for both the Dresselhaus and the Rashba spin-orbit terms, we present general formulas for the shot noise of singlet and triplets states derived within the scattering approach. We determine the full scattering matrix of the system for the case of leads with two orbital channels coupled via weak SO interactions inducing channel anticrossings. We show that this interband coupling coherently transfers electrons between the channels and gives rise to an additional modulation angle---dependent on both the Rashba and Dresselhaus interaction strengths---which allows for further independent coherent control of the electrons traversing the incoming leads. We derive explicit shot noise formulas for a variety of correlated pairs (e.g., Bell states) and lead spin polarizations. Interestingly, the singlet and each of the triplets defined along the quantization axis perpendicular to lead 1 (with the local SO interaction) and in the plane of the beam splitter display distinctive shot noise for injection energies near the channel anticrossings; hence, one can tell apart all the triplets, in addition to the singlet, through noise measurements. We also find that spin-orbit induced backscattering within lead 1 reduces the visibility of the noise oscillations, due to the additional partition noise in this lead. Finally, we consider injection of two-particle wavepackets into leads with multiple discrete states and find that two-particle entanglement can still be observed via noise bunching and antibunching.

arXiv:cond-mat/0509038•Physical Review B•Dec 20, 2005•51 citations

Wave Packet Sizes in Quantum Mechanical Scatterings: New Perspective

Kenzo Ishikawa, O. Jinnouchi

Particles exist with a probability of 1 and are represented by wave packets in quantum mechanics. These wave packets satisfy the Schrödinger equation and are classified as either bound states or continuum states with finite coherence lengths. They undergo transitions according to unambiguous absolute probabilities, the magnitudes of which depend on the sizes of the wave packets. Because these sizes are not universal constants but depend on the environment and experimental conditions, their accurate estimation is essential. The sizes of propagating wave packets are determined by their environmental coherence lengths, whereas those of bound states are determined by their wave functions. Here, we present a systematic analysis of wave-packet sizes.

arXiv:2509.04539•Sep 4, 2025•0 citations

REAL TIME QUANTUM TUNNELING

GEORGE J. PAPADOPOULOS

The time evolution of the probability and current densities stemming from an initial state of a particle in the form of a wave packet on the left hand side of a square barrier is studied. In particular, the space distribution at a given time of the above densities is given. Furthermore, the evolution in time at the entrance and exit of the barrier is given. The numerical results show repeated reversal in the current density at the barrier entrance, while being unidirectional at the exit. However, the probability entering the barrier over a long time equals the corresponding probability exiting the barrier. Owing to the fact that the wave packet expands on both of its sides, as time goes by, it is possible to have transmission even if the particle’s initial momentum points away from the barrier. The effect, in question, becomes evident in a diagram for the transmission coefficient in terms of the particle initial momentum, over a relatively small region.

•Jan 1, 2015•1 citation
We consider a general reaction process in which two entrance-channel particles, the beam and target particles, interact off-resonance and produce two exit-channel particles. In the c.m. frame we derive a general expression relating the probability density of the exit-channel particles to that of the entrance-channel particles. Furthermore, we show that whenever the coherence length, wB{w}_{B}wB​, of the beam particle is much larger than the coherence length, wT{w}_{T}wT​, of the target particle, the wave function of each exit-channel particle may be expressed explicitly in terms of the wave function of the beam particle. We also derive expressions for the coherence time and coherence length of each exit-channel particle. In addition, whenever the speed of the target particle is much less than the speed vi{v}_{i}vi​ (i=1,2) of an exit-channel particle, we show that a simple relationship exists between the coherence length, wci{w}_{\mathrm{ci}}wci​, of the exit-channel particle and the beam particle: wci{w}_{\mathrm{ci}}wci​=vi{v}_{i}vi​(wB{w}_{B}wB​/\ensuremath{\Vert}{\mathrm{v}}_{B}$${\mathrm{\ensuremath{-}}\mathrm{v}}_{T}\ensuremath{\Vert}){1 \ensuremath−[(vT{\mathrm{\ensuremath{-}}[(\mathrm{v}}_{\mathrm{T}}\ensuremath−[(vT​\ensuremath{\cdot}e{\mathrm{^}}_{3})/ vi{v}_{i}vi​](costheta')}, where the velocity of the beam particle is vB{\mathrm{v}}_{B}vB​ (vB{(\mathrm{v}}_{B}(vB​=vB{v}_{B}vB​e{^}_{3}), and the velocity of the target particle is vT{\mathrm{v}}_{T}vT​ (vT{(\mathrm{v}}_{T}(vT​=vT{v}_{T}vT​e{^}_{3}). The angle theta' is the spherical polar angular coordinate, measured with respect to the unit vector e{^}_{3}, of the point at which the coherence length wci{w}_{\mathrm{ci}}wci​ is to be determined. The above relationship is applied in the case of photoionization of the hydrogen atom at low energies to obtain realistic estimates for the coherence lengths of the ionized electron and proton. The coherence length of the electron is shown to be of nearly macroscopic dimensions.
The projectile beam coherence effects occurring in ion-atom collisions are analyzed on the basis of the recent theory of Karlovets et al. [Phys. Rev. A 92, 052703 (2015)] developed for the elastic scattering of wave packets of particles off a potential field. This theory is generalized to estimate the loss of coherence for inelastically scattered projectiles in ionizing collisions. The results obtained by the suggested model are compared with experimental data for the ionization of hydrogen atoms and molecules by 75-keV proton impact. Significantly improved agreement is observed between the theory and experiment.
The conventional theory of neutron beams interacting with many-body systems treats the beam as a classical system, i.e. with its dynamical variables appearing in the quantum dynamics of the scattering process not as operators but only as c-numbers. Moreover, neutrons are described with plane waves, i.e. the concept of neutron's (finite) coherence length is here irrelevant. The same holds for electron, atom or X-ray scattering. This simplification results in the full decoupling of the probe particle's dynamics from the quantum dynamics of the scatterer---a well-known fact also reflected in the standard formalism of time-correlation functions (see textbooks). Making contact with modern quantum theoretical approaches (e.g., quantum entanglement, 'which-path detection' versus interference, von Neumann measurement, Week Values, etc.) new observable effects may be exposed and/or predicted, for instance, a momentum-transfer deficit and an intensity deficit in neutron scattering from protons of hydrogen-containing samples. 7
We extend our previous work on shot noise for entangled and spin polarized electrons in a beam-splitter geometry with spin-orbit (SO) interaction in one of the incoming leads (lead 1). In addition to accounting for both the Dresselhaus and the Rashba spin-orbit terms, we present general formulas for the shot noise of singlet and triplets states derived within the scattering approach. We determine the full scattering matrix of the system for the case of leads with two orbital channels coupled via weak SO interactions inducing channel anticrossings. We show that this interband coupling coherently transfers electrons between the channels and gives rise to an additional modulation angle---dependent on both the Rashba and Dresselhaus interaction strengths---which allows for further independent coherent control of the electrons traversing the incoming leads. We derive explicit shot noise formulas for a variety of correlated pairs (e.g., Bell states) and lead spin polarizations. Interestingly, the singlet and each of the triplets defined along the quantization axis perpendicular to lead 1 (with the local SO interaction) and in the plane of the beam splitter display distinctive shot noise for injection energies near the channel anticrossings; hence, one can tell apart all the triplets, in addition to the singlet, through noise measurements. We also find that spin-orbit induced backscattering within lead 1 reduces the visibility of the noise oscillations, due to the additional partition noise in this lead. Finally, we consider injection of two-particle wavepackets into leads with multiple discrete states and find that two-particle entanglement can still be observed via noise bunching and antibunching.
Particles exist with a probability of 1 and are represented by wave packets in quantum mechanics. These wave packets satisfy the Schrödinger equation and are classified as either bound states or continuum states with finite coherence lengths. They undergo transitions according to unambiguous absolute probabilities, the magnitudes of which depend on the sizes of the wave packets. Because these sizes are not universal constants but depend on the environment and experimental conditions, their accurate estimation is essential. The sizes of propagating wave packets are determined by their environmental coherence lengths, whereas those of bound states are determined by their wave functions. Here, we present a systematic analysis of wave-packet sizes.
The time evolution of the probability and current densities stemming from an initial state of a particle in the form of a wave packet on the left hand side of a square barrier is studied. In particular, the space distribution at a given time of the above densities is given. Furthermore, the evolution in time at the entrance and exit of the barrier is given. The numerical results show repeated reversal in the current density at the barrier entrance, while being unidirectional at the exit. However, the probability entering the barrier over a long time equals the corresponding probability exiting the barrier. Owing to the fact that the wave packet expands on both of its sides, as time goes by, it is possible to have transmission even if the particle’s initial momentum points away from the barrier. The effect, in question, becomes evident in a diagram for the transmission coefficient in terms of the particle initial momentum, over a relatively small region.