Lippmann–Schwinger Equation
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The Lippmann–Schwinger equation (named after Bernard Lippmann and
Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize-winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
) is one of the most used equations to describe particle collisions – or, more precisely,
scattering In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
 – in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. 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 Atomic, molecular, and optical physics (AMO) is the study of matter–matter and light–matter interactions, at the scale of one or a few atoms and energy scales around several electron volts. The three areas are closely interrelated. AMO th ...
,
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies th ...
and
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, but also for seismic scattering problems in
geophysics Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
. 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 In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. Formulation Scattering in quantum mechanics begins with a p ...
and
cross section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture and engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **A ...
s). The most fundamental equation to describe any quantum phenomenon, including scattering, is the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
. In physical problems, this differential equation must be solved with the input of an additional set of initial and/or
boundary condition In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
s 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 In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2 ...
. 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 + \, sign and other for the - \, sign): , \psi^ \rangle = , \phi \rangle + \frac V , \psi^ \rangle. \, The potential energy V describes the interaction between the two colliding systems. The
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
H_0 describes the situation in which the two systems are infinitely far apart and do not interact. Its
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s are , \phi \rangle \, and its
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s are the energies E \,. Finally, i \epsilon \, 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 The Faddeev equations, named after their discoverer Ludvig Faddeev, describe, at once, all the possible exchanges/ interactions in a system of three particles in a fully quantum mechanical formulation. They can be solved iteratively. In gener ...
may be used instead. However, there are approximations that can reduce a
many-body problem The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Terminology ''Microscopic'' here implies that quantum mechanics has to be ...
to a set of
two-body problem In classical mechanics, the two-body problem is to calculate and predict the motion of two massive bodies that are orbiting each other in space. The problem assumes that the two bodies are point particles that interact only with one another; th ...
s 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 physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering. The pseudopotential approximation was first introduce ...
. 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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
may be written as H = H_0 + V where is the free Hamiltonian (or more generally, a Hamiltonian with known eigenvectors). For example, in nonrelativistic quantum mechanics may be H_0 = \frac. Intuitively is the interaction energy of the system. Let there be an
eigenstate In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
of : H_0 , \phi \rangle = E , \phi \rangle. Now if we add the interaction V into the mix, the Schrödinger equation reads \left( H_0 + V \right) , \psi \rangle = E , \psi \rangle. Now consider the
Hellmann–Feynman theorem In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem ...
, which requires the energy eigenvalues of the Hamiltonian to change continuously with continuous changes in the Hamiltonian. Therefore, we wish that , \psi \rangle \to , \phi \rangle as V \to 0. A naive solution to this equation would be , \psi \rangle = , \phi \rangle + \frac V , \psi \rangle. where the notation denotes the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse, the inverse of a number that, when added to the ...
of . However is
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singula ...
since is an eigenvalue of . As is described below, this singularity is eliminated in two distinct ways by making the denominator slightly complex: , \psi^ \rangle = , \phi \rangle + \frac V , \psi^ \rangle. By insertion of a complete set of free particle states, , \psi^ \rangle = , \phi \rangle + \int d\beta\frac \langle \phi_\beta , V, \psi^ \rangle, \quad H_0 , \phi_\beta\rangle = E_\beta, \phi_\beta\rangle, 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 , \psi^_\alpha \rangle = , \phi_\alpha \rangle + \int d\beta\frac, \quad T^_ = \langle \phi_\beta , V, \psi^_\alpha \rangle.


Methods of solution

From the mathematical point of view the Lippmann–Schwinger equation in coordinate representation is an
integral equation In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2 ...
of Fredholm type. It can be solved by
discretization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...
. Since it is equivalent to the differential time-independent
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
with appropriate boundary conditions, it can also be solved by numerical methods for differential equations. In the case of the spherically symmetric potential V 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 The method of continued fractions is a method developed specifically for solution of integral equations of quantum scattering theory like Lippmann–Schwinger equation or Faddeev equations. It was invented by Jiří Horáček (physicist), Horáček ...
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 The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power iteration, power methods to find the m "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an n \times n ...
.


Interpretation as in and out states


The ''S''-matrix paradigm

In the
S-matrix In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...
formulation of
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, which was pioneered by
John Archibald Wheeler John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr to e ...
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
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
s would fall apart, but rather that there exists an interaction-free
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
''H''0, 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 In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
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
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. In the 1960s, the S-matrix paradigm was elevated by many physicists to a fundamental law of nature. In
S-matrix theory ''S''-matrix theory was a proposal for replacing local quantum field theory as the basic principle of elementary particle physics. It avoided the notion of space and time by replacing it with abstract mathematical properties of the ''S''-matrix ...
, 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 In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
restricted to the
mass-shell In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called on the mass shell (on shell); while those that do not are called off the mass shell (off shell). In quantu ...
, and led to the construction of dual resonance models. But it was very controversial, because it denied the validity of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
based on local fields and Hamiltonians.


The connection to Lippmann–Schwinger

Intuitively, the slightly deformed eigenfunctions \psi^ of the full Hamiltonian ''H'' are the in and out states. The \phi 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 \psi^ 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 cannot become noninteracting. This problem is easily circumvented by assembling \psi^ and \phi into wavepackets with some distribution g(E) of energies E over a characteristic scale \Delta E. The
uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
now allows the interactions of the asymptotic states to occur over a timescale \hbar/\Delta E 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 \psi^_g(t)=\int dE\, e^ g(E)\psi^ and \phi_g(t)=\int dE\, e^ g(E)\phi of the wavepackets we see that, at a given time, the difference between the \psi_g(t) and \phi_g(t) wavepackets is given by an integral over the energy .


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 In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in t ...
, as a sum of the residues at the various poles. We will now argue that the residues of \psi^ approach those of \phi at time t \to \mp\infty and so the corresponding wavepackets are equal at temporal infinity. In fact, for very positive times ''t'' the e^ factor in a
Schrödinger picture In physics, the Schrödinger picture or Schrödinger representation is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exceptio ...
state forces one to close the contour on the lower half-plane. The pole in the (\phi ,V \psi^) 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 \psi^, and so does not lie inside the integral contour and does not contribute to the \psi^ integral. The remainder is equal to the \phi wavepacket. Thus, at very late times \psi^=\phi, identifying \psi^ as the asymptotic noninteracting out state. Similarly one may integrate the wavepacket corresponding to \psi^ 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 \psi^, which is in the lower half-plane. One then finds that the \psi^ and \phi wavepackets are equal in the asymptotic past, identifying \psi^ as the asymptotic noninteracting in state.


The complex denominator of Lippmann–Schwinger

This identification of the \psi's as asymptotic states is the justification for the \pm\epsilon in the denominator of the Lippmann–Schwinger equations.


A formula for the ''S''-matrix

The ''S''-matrix is defined to be the inner product S_ = (\psi^-_a, \psi^+_b) of the ''a''th and ''b''th
Heisenberg picture In physics, the Heisenberg picture or Heisenberg representation is a Dynamical pictures, formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which observables incorporate a dependency on time, but the quantum state, st ...
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 \psi^+ and \psi^-. As a result, the contour now does pick up the energy pole. This can be related to the \phi's if one uses the S-matrix to swap the two \psi's. Identifying the coefficients of the \phi's on both sides of the equation one finds the desired formula relating ''S'' to the potential S_ = \delta(a-b) - 2i\pi \delta(E_a-E_b) (\phi_a, V\psi^+_b). In the
Born approximation Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named ...
, corresponding to first order
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
, one replaces this last \psi^+ with the corresponding eigenfunction \phi of the free Hamiltonian , yielding S_=\delta(a-b)-2i\pi\delta(E_a-E_b)(\phi_a,V\phi_b)\, 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 b \to a, which is equal to , S_ - \delta_, ^2.


Homogenization

With the use of Green's function, the Lippmann–Schwinger equation has counterparts in homogenization theory (e.g. mechanics, conductivity, permittivity).


See also

* Bethe–Salpeter equation


References


Bibliography

* * *


Original publications

* * {{DEFAULTSORT:Lippmann-Schwinger equation Scattering