Potential Scatterings in $L^2$ Space: (2) Rigorous Scattering Probability of Wave PacketsPotential Scatterings in $L^2$ Space: (2) Rigorous Scattering Probability of Wave Packets

Kenzo Ishikawa

Jan 1, 20230 citations

A two-Hilbert-space formulation of multi-channel scattering theory

Colston Chandler, Archie G. Gibson

Springer eBooksApr 6, 20085 citations

Modulation of electron wave packets by scattering on time-harmonic potentials

Mads Brøndum Carlsen, Lars Bojer Madsen

The coherent interaction between free electrons and optical near-fields enables the active modulation of electron wave packets, a mechanism central to photon-induced near-field electron microscopy (PINEM). While existing theories effectively describe these interactions at high kinetic energies, the growing interest in low-energy ultrafast electron microscopy demands frameworks that explicitly account for finite wave packet geometries and recoil effects. In this paper, we develop a rigorous 3D quantum scattering theory for electron wave packets interacting with time-periodic potentials, capturing the case of optical near-field interaction. By mapping the time-dependent dynamics into an extended Floquet space, we formally connect the modulation process to time-independent multi-channel scattering. We evaluate the resulting scattering amplitudes using both an exact R-matrix approach and a multi-channel eikonal approximation. The latter analytical approach recovers PINEM-like probabilities weighted by the wave packet's transverse profile. Application of the theory to an oscillating potential demonstrates the generation of distinct energy sidebands, revealing that the modulation strength is sensitive to the transverse focusing of the incident electron pulse, underlining the importance of a fully 3D treatment.The coherent interaction between free electrons and optical near-fields enables the active modulation of electron wave packets, a mechanism central to photon-induced near-field electron microscopy (PINEM). While existing theories effectively describe these interactions at high kinetic energies, the growing interest in low-energy ultrafast electron microscopy demands frameworks that explicitly account for finite wave packet geometries and recoil effects. In this paper, we develop a rigorous 3D quantum scattering theory for electron wave packets interacting with time-periodic potentials, capturing the case of optical near-field interaction. By mapping the time-dependent dynamics into an extended Floquet space, we formally connect the modulation process to time-independent multi-channel scattering. We evaluate the resulting scattering amplitudes using both an exact R-matrix approach and a multi-channel eikonal approximation. The latter analytical approach recovers PINEM-like probabilities weighted by the wave packet's transverse profile. Application of the theory to an oscillating potential demonstrates the generation of distinct energy sidebands, revealing that the modulation strength is sensitive to the transverse focusing of the incident electron pulse, underlining the importance of a fully 3D treatment.

quant-pharXiv:2605.10126May 12, 2026

Rigorous Derivation of the Phase Shift Formula for the Hilbert Space Scattering Operator of a Single Particle

T. A. Green, Oscar E. Lanford

For a single nonrelativistic particle moving in a spherically symmetric potential, the existence of the Hilbert space wave operators and S operator is proved and phase shift formulas for these operators are deduced. The probability, P(Ω), for scattering into the solid angle Ω is obtained from the time dependent theory. The relation between P(Ω) and the R matrix of the standard plane wave formulation of scattering theory is established. For collimated incoming packets, it is shown that P(Ω) can be expressed as an energy average of the differential cross section.For a single nonrelativistic particle moving in a spherically symmetric potential, the existence of the Hilbert space wave operators and S operator is proved and phase shift formulas for these operators are deduced. The probability, P(Ω), for scattering into the solid angle Ω is obtained from the time dependent theory. The relation between P(Ω) and the R matrix of the standard plane wave formulation of scattering theory is established. For collimated incoming packets, it is shown that P(Ω) can be expressed as an energy average of the differential cross section.

Journal of Mathematical PhysicsMar 1, 196047 citations

Scattering theory for arbitrary potentials (7 pages)

Alisher S. Kadyrov

Physical Review AJan 1, 20050 citations

Rigorous Solution of Quantum Scattering Theory

Chinedu E. Ekuma

Research Journal of PhysicsJan 1, 20071 citation

Potential Scatterings in

L^2

Space: (2) Rigorous Scattering Probability of Wave Packets