Lippmann–Schwinger equation
The Lippmann–Schwinger equation (named after Bernard Lippmann and Julian Schwinger) is one of the most used equations to describe particle collisions – or, more precisely, scattering – in quantum mechanics. It may be used in scattering of molecules, atoms, neutrons, photons or any other particles and is important mainly in atomic, molecular, and optical physics, nuclear physics and particle physics, but also for seismic scattering problems in geophysics. It relates the scattered wave function with the interaction that produces the scattering (the scattering potential) and therefore allows calculation of the relevant experimental parameters (scattering amplitude and cross sections ). The most fundamental equation to describe any quantum phenomenon, including scattering, is the Schrödinger equation. In physical problems, this differential equation must be solved with the input of an additional set of initial and/or boundary conditions for the specific physical system studied. The Lippmann–Schwinger equation is equivalent to the Schrödinger equation plus the typical boundary conditions for scattering problems. In order to embed the boundary conditions, the Lippmann–Schwinger equation must be written as an integral equation. For scattering problems, the Lippmann–Schwinger equation is often more convenient than the original Schrödinger equation. The Lippmann–Schwinger equation's general  form is (in reality, two equations are shown below, one for the formula_1 sign and other for the formula_2 sign): The potential energy formula_4 describes the interaction between the two colliding systems. The Hamiltonian formula_5 describes the situation in which the two systems are infinitely far apart and do not interact. Its eigenfunctions are formula_6 and its eigenvalues are the energies formula_7. Finally, formula_8 is a mathematical technicality necessary for the calculation of the integrals needed to solve the equation. It is a consequence of causality, ensuring that scattered waves consist only of outgoing waves. This is made rigorous by  the limiting absorption principle. Usage. The Lippmann–Schwinger equation is useful in a very large number of situations involving two-body scattering. For three or more colliding bodies it does not work well because of mathematical limitations; Faddeev equations may be used instead. However, there are approximations that can reduce a many-body problem to a set of two-body problems in a variety of cases. For example, in a collision between electrons and molecules, there may be tens or hundreds of particles involved. But the phenomenon may be reduced to a two-body problem by describing all the molecule  constituent particle potentials together with a pseudopotential. In these cases, the Lippmann–Schwinger equations may be used. Of course, the main motivations of these approaches are also the possibility of doing the calculations with much lower computational efforts. Derivation. We will assume that the Hamiltonian may be written as where is the free Hamiltonian (or more generally, a Hamiltonian with known eigenvectors). For example, in nonrelativistic quantum mechanics may be Intuitively is the interaction energy of the system. Let there be an eigenstate of : Now if we add the interaction formula_12 into the mix, the Schrödinger equation reads Now consider  the Hellmann–Feynman theorem, which requires the energy eigenvalues of the Hamiltonian to change continuously with continuous changes in the Hamiltonian. Therefore, we wish that formula_14 as formula_15. A naive solution to this equation would be where the notation denotes the inverse of . However is singular since is an eigenvalue of . As is described below, this singularity is eliminated in two distinct ways by making the denominator slightly complex, to give yourself a little wiggle room : By insertion of a complete set of free particle states, the Schrödinger equation is turned into an integral equation. The "in" and "out" states  are assumed to form bases too, in the distant past and distant future respectively having the appearance of free particle states, but being eigenfunctions of the complete Hamiltonian. Thus endowing them with an index, the equation becomes Methods of solution. From the mathematical point of view the Lippmann–Schwinger equation in coordinate representation is an integral equation of Fredholm type. It can be solved by discretization. Since it is equivalent to the differential time-independent Schrödinger equation with appropriate boundary conditions, it can also be solved by numerical methods for differential equations. In the case of the spherically symmetric potential  formula_20 it is usually solved by partial wave analysis. For high energies and/or weak potential it can also be solved perturbatively by means of Born series. The method convenient also in the case of many-body physics, like in description of atomic, nuclear or molecular collisions is the method of R-matrix of Wigner and Eisenbud. Another class of methods is based on separable expansion of the potential or Green's operator like the method of continued fractions of Horáček and Sasakawa. Very important class of methods is based on variational principles, for example the Schwinger-Lanczos method combining the  variational principle of Schwinger with Lanczos algorithm. Interpretation as in and out states. The "S"-matrix paradigm. In the S-matrix formulation of particle physics, which was pioneered by John Archibald Wheeler among others, all physical processes are modeled according to the following paradigm. One begins with a non-interacting multiparticle state in the distant past. Non-interacting does not mean that all of the forces have been turned off, in which case for example protons would fall apart, but rather that there exists an interaction-free Hamiltonian "H", for which the bound states have the same energy level spectrum as  the actual Hamiltonian . This initial state is referred to as the "in state". Intuitively, it consists of elementary particles or bound states that are sufficiently well separated that their interactions with each other are ignored. The idea is that whatever physical process one is trying to study may be modeled as a scattering process of these well separated bound states. This process is described by the full Hamiltonian , but once it's over, all of the new elementary particles and new bound states separate again and one finds a new noninteracting state called the "out state". The S-matrix is  more symmetric under relativity than the Hamiltonian, because it does not require a choice of time slices to define. This paradigm allows one to calculate the probabilities of all of the processes that we have observed in 70 years of particle collider experiments with remarkable accuracy. But many interesting physical phenomena do not obviously fit into this paradigm. For example, if one wishes to consider the dynamics inside of a neutron star sometimes one wants to know more than what it will finally decay into. In other words, one may be interested in measurements that are not in the asymptotic  future. Sometimes an asymptotic past or future is not even available. For example, it is very possible that there is no past before the Big Bang. In the 1960s, the S-matrix paradigm was elevated by many physicists to a fundamental law of nature. In S-matrix theory, it was stated that any quantity that one could measure should be found in the S-matrix for some process. This idea was inspired by the physical interpretation that S-matrix techniques could give to Feynman diagrams restricted to the mass-shell, and led to the construction of dual resonance models. But  it was very controversial, because it denied the validity of quantum field theory based on local fields and Hamiltonians. The connection to Lippmann–Schwinger. Intuitively, the slightly deformed eigenfunctions formula_21 of the full Hamiltonian "H" are the in and out states. The formula_22 are noninteracting states that resemble the in and out states in the infinite past and infinite future. Creating wavepackets. This intuitive picture is not quite right, because formula_21 is an eigenfunction of the Hamiltonian and so at different times only differs by a phase. Thus, in particular, the physical state does not evolve and so it can not become noninteracting. This problem is easily circumvented by assembling formula_21 and formula_25 into wavepackets with some distribution formula_26 of energies formula_27 over a characteristic scale formula_28. The uncertainty principle now allows the interactions of the asymptotic states to occur over a timescale formula_29 and in particular it is no longer inconceivable that the interactions may turn off outside of this interval. The following argument suggests that this is indeed the case. Plugging the Lippmann–Schwinger equations into the definitions and of the wavepackets we see that, at a given time, the difference between the formula_32 and formula_33 wavepackets is  given by an integral over the energy "E". A contour integral. This integral may be evaluated by defining the wave function over the complex "E" plane and closing the "E" contour using a semicircle on which the wavefunctions vanish. The integral over the closed contour may then be evaluated, using the Cauchy integral theorem, as a sum of the residues at the various poles. We will now argue that the residues of formula_21 approach those of formula_25 at time formula_36 and so the corresponding wavepackets are equal at temporal infinity. In fact, for very positive times "t" the formula_37 factor  in a Schrödinger picture state forces one to close the contour on the lower half-plane. The pole in the formula_38 from the Lippmann–Schwinger equation reflects the time-uncertainty of the interaction, while that in the wavepackets weight function reflects the duration of the interaction. Both of these varieties of poles occur at finite imaginary energies and so are suppressed at very large times. The pole in the energy difference in the denominator is on the upper half-plane in the case of formula_39, and so does not lie inside the integral contour and does not contribute to the  formula_39 integral. The remainder is equal to the formula_22 wavepacket. Thus, at very late times formula_42, identifying formula_39 as the asymptotic noninteracting out state. Similarly one may integrate the wavepacket corresponding to formula_44 at very negative times. In this case the contour needs to be closed over the upper half-plane, which therefore misses the energy pole of formula_44, which is in the lower half-plane. One then finds that the formula_44 and formula_25 wavepackets are equal in the asymptotic past, identifying formula_44 as the asymptotic noninteracting in state. The complex denominator of Lippmann–Schwinger. This identification of the formula_49 's as asymptotic states is the justification for the formula_50 in the denominator of the Lippmann–Schwinger equations. A formula for the "S"-matrix. The "S"-matrix is defined to be the inner product of the "a"th and "b"th Heisenberg picture asymptotic states. One may obtain a formula relating the "S"-matrix to the potential "V" using the above contour integral strategy, but this time switching the roles of formula_52 and formula_53. As a result, the contour now does pick up the energy pole. This can be related to the formula_22's if one uses the S-matrix to swap the two formula_49 's. Identifying the coefficients of the formula_22's on both sides of the equation one finds the desired formula relating "S" to the potential In the Born approximation, corresponding to first order perturbation theory, one replaces this last formula_52 with the corresponding eigenfunction formula_25 of the free Hamiltonian "H", yielding which expresses the S-matrix entirely in terms of "V" and free Hamiltonian eigenfunctions. These formulas may in turn be used to calculate the reaction rate of the process formula_61, which is equal to formula_62 Homogenization. With the use of Green's function, the Lippmann–Schwinger equation has counterparts in  homogenization theory (e.g. mechanics, conductivity, permittivity).