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Lippmann-Schwinger Equation and Application to Envelope-Modulated Potential

Anna Roland, Preet Sharma

Published Aug 28, 2026• 0 citations
Publisher

Abstract

The Lippmann-Schwinger equation is one of the most important concepts to understand a scattering process in quantum physics. The applications are extended to solid-state physics, atomic and molecular physics, engineering and even in current fields such as quantum wires. In this study we have applied the Lippmann-Schwinger equation to an Envelope- Modulated Sinusoidal Potential. These types of potentials are used actively in fields such as seismology, and various other fields in engineering. Here we have shown that Envelope- Modulated Sinusoidal Potential through the Lippmann-Schwinger equation can be a very important tool.

DOI
10.14293/pr2199.004316.v1

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arXiv:2204.03651•Apr 8, 2022•1 citation
The Lippmann-Schwinger equation is one of the most important concepts to understand a scattering process in quantum physics. The applications are extended to solid-state physics, atomic and molecular physics, engineering and even in current fields such as quantum wires. In this study we have applied the Lippmann-Schwinger equation to an Envelope- Modulated Sinusoidal Potential. These types of potentials are used actively in fields such as seismology, and various other fields in engineering. Here we have shown that Envelope- Modulated Sinusoidal Potential through the Lippmann-Schwinger equation can be a very important tool.
The purpose of these lectures is to give an accessible and self contained introduction to quantum scattering theory in one dimension. Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time domain (Asymptotic Condition, in- and out- states, scattering operator S^\hat{S}S^). The aim of Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain. This amounts to introduce the Lippmann-Schwinger equation for the stationary scattering states (denoted as ∣ψE(±1)±⟩| ψ_{E(\pm 1)}^\pm \rangle∣ψE(±1)±​⟩), to discuss fundamental properties of ∣ψE(±1)±⟩| ψ_{E(\pm 1)}^\pm \rangle∣ψE(±1)±​⟩, and subsequently to construct S^\hat{S}S^ and T^\hat{T}T^ operators in terms of ∣ψE(±1)±⟩| ψ_{E(\pm 1)}^\pm \rangle∣ψE(±1)±​⟩. Physical contents of the S^\hat{S}S^ and T^\hat{T}T^ operators is then illuminated by deriving explicit formulas for the probability of transmission/reflection of our quantum particle through/from the interaction region of the potential. An illustrative numerical example is given, which also highlights an existence of scattering resonances. Finally, Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism), which offers an extremely powerful tool suitable for clear cut understanding of the resonance phenomena.