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

Kenzo Ishikawa

Alisher S. Kadyrov

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.

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.

Domenico Finco

Tsin-Fu Jiang

Abstract We investigate the scattering of a wave packet by the Pöschl-Teller potential in momentum representation. The scattering dynamics of the wave packet for a long-time evolution is feasible in this representation. With the wave function in momentum space, we can construct the time-dependent phase space Wigner function. The corresponding density function in coordinate space is then calculated through the Wigner function. The reflectionless wave packet for integer ν and partially reflected for non-integer ν are demonstrated by analyzing the Wigner function.