Lippmann–Schwinger equation

Lippmann–Schwinger equation
Name	Lippmann–Schwinger equation
Field	Theoretical physics
Introduced	1950
Associated	Walter Lippmann; Julian Schwinger
Related	Schrödinger equation; Green's function; scattering theory

Lippmann–Schwinger equation The Lippmann–Schwinger equation is an integral equation formulation of quantum scattering theory that expresses scattering states in terms of an incident state and a resolvent operator built from a free Hamiltonian and an interaction potential. It provides a bridge between the Schrödinger equation and practical computations of transition amplitudes used in analyses by researchers associated with institutions such as Institute for Advanced Study and in collaborations involving figures like Julian Schwinger and contemporaries at Harvard University. The equation underlies many methods applied in contexts ranging from experiments at CERN to models developed at Bell Labs.

Introduction

The Lippmann–Schwinger equation recasts the time-independent Schrödinger equation for scattering into an integral form using a Green's function for the free Hamiltonian. Historically introduced in the context of postwar developments in quantum mechanics alongside work at Columbia University and Massachusetts Institute of Technology, it became a standard tool in treatments found in texts by authors affiliated with Princeton University and Cambridge University. The formulation is central to computing S-matrix elements and resonances studied in settings such as Los Alamos National Laboratory and applied in analyses referencing experiments at Brookhaven National Laboratory.

Derivation

Starting from the time-independent Schrödinger equation Hψ = Eψ with H = H0 + V, one isolates the interacting state by inverting the operator (E − H0 ± iε). The formal inversion uses the resolvent (E − H0 ± iε)^{-1}, constructed originally in operator-theoretic work linked to mathematicians at Institute for Advanced Study and physicists at University of Chicago. Introducing the free Green's function G0(E ± i0) yields the integral form ψ = φ + G0Vψ, where φ is an incoming eigenstate of H0; analogous derivations appear in lecture notes from California Institute of Technology and reviews associated with Stanford University.

Scattering states and boundary conditions

Physical scattering states satisfy outgoing or incoming boundary conditions set by the ±iε prescription, a detail emphasized in analyses by Julian Schwinger and contemporaries at Yale University. The +i0 choice corresponds to outgoing waves used when comparing to asymptotic formulations from work at Imperial College London, while the −i0 choice gives incoming boundary conditions common in treatments at ETH Zurich. These conditions ensure the proper selection of the retarded or advanced resolvent required when connecting to S-matrix elements evaluated in collaborations involving groups from CERN and SLAC National Accelerator Laboratory.

Integral equation formulations and Green's functions

The core object G0(E ± i0) is the free Green's function for H0, often derived using Fourier transforms and contour integration techniques developed in seminars at Sorbonne University and University of Cambridge. Alternative formulations replace G0 with full Green's operators or T-operators leading to the operator equation T = V + V G0 T, a structure exploited in computational programs at Los Alamos National Laboratory and algorithms implemented by researchers at Max Planck Institute for Physics. Multiple scattering series, or Born series, arise by iterating the integral equation, linking to perturbative expansions used in analyses at SLAC National Accelerator Laboratory and in textbooks from Oxford University Press.

Applications and examples

The Lippmann–Schwinger equation is used to compute scattering amplitudes in low-energy nuclear physics studied at Brookhaven National Laboratory, in atomic collision problems researched at Lawrence Berkeley National Laboratory, and in condensed matter scattering handled in work at IBM Research. Concrete examples include potential scattering by short-range potentials modeled in calculations at Argonne National Laboratory, and Coulomb scattering where modifications connect to treatments developed by researchers at University of Tokyo. It also underpins inverse scattering approaches employed in geophysical imaging projects coordinated with institutions like Sandia National Laboratories.

Mathematical properties and solutions

Mathematically, existence and uniqueness of solutions relate to spectral properties of H0 and V studied in operator theory circles at Princeton University and Institut Henri Poincaré. Compactness criteria and Fredholm theory, topics advanced through collaborations involving Mathematical Sciences Research Institute, determine when the Born series converges; resolvent estimates used in scattering theory appear in monographs from Cambridge University Press. Analytic continuation of the resolvent links to resonance theory investigated in studies at Weizmann Institute of Science.

Extensions and generalizations

Generalizations include multichannel Lippmann–Schwinger systems appearing in coupled-channel analyses used by teams at Fermilab and extensions to relativistic scattering where forms compatible with Dirac equation frameworks are developed in work at CERN. Time-dependent generalizations and formulations in many-body contexts connect to methods created in collaborations involving Argonne National Laboratory and Rutherford Appleton Laboratory. Non-Hermitian and open-system variants used in nuclear reaction theory and photonics are explored by groups at KAIST and University of Illinois Urbana–Champaign.

Category:Quantum mechanics