The **Lippmann–Schwinger equation** (named after Bernard Lippmann and Julian Schwinger ^{ [1] }) 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).

- Usage
- Derivation
- Methods of solution
- Interpretation as in and out states
- The S-matrix paradigm
- The connection to Lippmann–Schwinger
- Creating wavepackets
- A contour integral
- The complex denominator of Lippmann–Schwinger
- A formula for the S-matrix
- Homogenization
- See also
- References
- Bibliography
- Original publications

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.^{ [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):^{ [3] }

The potential energy describes the interaction between the two colliding systems. The Hamiltonian describes the situation in which the two systems are infinitely far apart and do not interact. Its eigenfunctions are and its eigenvalues are the energies . Finally, 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.

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.^{ [4] } 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 phenomenum may be reduced to a two-body problem by describing all the molecule constituent particle potentials together with a pseudopotential.^{ [5] } 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.

We will assume that the Hamiltonian may be written as

where *H*_{0} is the free Hamiltonian (or more generally, a Hamiltonian with known eigenvectors). For example, in nonrelativistic quantum mechanics *H*_{0} may be

- .

Intuitively *V* is the interaction energy of the system. Let there be an eigenstate of *H*_{0}:

- .

Now if we add the interaction 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 as . A naive solution to this equation would be

- .

where the notation 1/*A* denotes the inverse of *A*. However *E* − *H*_{0} is singular since *E* is an eigenvalue of *H*_{0}. 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

- .

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 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.

In the S-matrix formulation of particle physics, which was pioneered by John Archibald Wheeler among others,^{ [6] } all physical processes are modeled according to the following paradigm.^{ [7] }

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*_{0}, for which the bound states have the same energy level spectrum as the actual Hamiltonian *H*. 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 *H*, 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.

Intuitively, the slightly deformed eigenfunctions of the full Hamiltonian *H* are the in and out states. The are noninteracting states that resemble the **in** and **out** states in the infinite past and infinite future.

This intuitive picture is not quite right, because 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 and into wavepackets with some distribution of energies over a characteristic scale . The uncertainty principle now allows the interactions of the asymptotic states to occur over a timescale 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 and wavepackets is given by an integral over the energy *E*.

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 approach those of at time and so the corresponding wavepackets are equal at temporal infinity.

In fact, for very positive times *t* the factor in a Schrödinger picture state forces one to close the contour on the lower half-plane. The pole in the 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 , and so does not lie inside the integral contour and does not contribute to the integral. The remainder is equal to the wavepacket. Thus, at very late times , identifying as the asymptotic noninteracting **out** state.

Similarly one may integrate the wavepacket corresponding to 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 , which is in the lower half-plane. One then finds that the and wavepackets are equal in the asymptotic past, identifying as the asymptotic noninteracting **in** state.

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

The S-matrix *S* 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 and . As a result, the contour now does pick up the energy pole. This can be related to the 's if one uses the S-matrix to swap the two 's. Identifying the coefficients of the '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 with the corresponding eigenfunction of the free Hamiltonian *H*_{0}, 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 , which is equal to

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

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 the diagrams in 1948. The interaction of subatomic particles can be complex and difficult to understand; Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. According to David Kaiser, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." While the diagrams are applied primarily to quantum field theory, they can also be used in other fields, such as solid-state theory.

**Quantum decoherence** is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

In mathematics, a **self-adjoint operator** on a finite-dimensional complex vector space *V* with inner product is a linear map *A* that is its own adjoint: for all vectors v and w. If *V* is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of *A* is a Hermitian matrix, i.e., equal to its conjugate transpose *A*^{∗}. By the finite-dimensional spectral theorem, *V* has an orthonormal basis such that the matrix of *A* relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.

In quantum mechanics, the **spin–statistics theorem** relates the intrinsic spin of a particle to the particle statistics it obeys. In units of the reduced Planck constant *ħ*, all particles that move in 3 dimensions have either integer spin or half-integer spin.

In quantum mechanics, a **probability amplitude** is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.

In physics, the **S-matrix** or **scattering matrix** relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).

The **Wheeler–DeWitt equation** is a field equation. It is part of a theory that attempts to combine mathematically the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity. In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called 'problem of time'. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group".

In quantum field theory, the **LSZ reduction formula** is a method to calculate S-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.

In theoretical physics, the (one-dimensional) **nonlinear Schrödinger equation** (**NLSE**) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

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 after Max Born who proposed this approximation in early days of quantum theory development.

The **Hückel method** or **Hückel molecular orbital theory**, proposed by Erich Hückel in 1930, is a very simple linear combination of atomic orbitals molecular orbitals method for the determination of energies of molecular orbitals of π-electrons in π-delocalized molecules, such as ethylene, benzene, butadiene, and pyridine. It is the theoretical basis for Hückel's rule for the aromaticity of π-electron cyclic, planar systems. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon, known in this context as heteroatoms. A more dramatic extension of the method to include σ-electrons, known as the extended Hückel method (EHM), was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general and was used to provide computational justification for the Woodward–Hoffmann rules. To distinguish the original approach from Hoffmann's extension, the Hückel method is also known as the **simple Hückel method** (SHM).

The **time-evolving block decimation** (**TEBD**) algorithm is a numerical scheme used to simulate one-dimensional quantum many-body systems, characterized by at most nearest-neighbour interactions. It is dubbed Time-evolving Block Decimation because it dynamically identifies the relevant low-dimensional Hilbert subspaces of an exponentially larger original Hilbert space. The algorithm, based on the Matrix Product States formalism, is highly efficient when the amount of entanglement in the system is limited, a requirement fulfilled by a large class of quantum many-body systems in one dimension.

In quantum mechanics, the **Pauli equation** or **Schrödinger–Pauli equation** is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.

**Schwinger variational principle** is a variational principle which expresses the scattering T-matrix as a functional depending on two unknown wave functions. The functional attains stationary value equal to actual scattering T-matrix. The functional is stationary if and only if the two functions satisfy the Lippmann-Schwinger equation. The development of the variational formulation of the scattering theory can be traced to works of L. Hultén and J. Schwinger in 1940s.

In quantum mechanics and quantum field theory, a **Schrödinger field**, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation. While any situation described by a Schrödinger field can also be described by a many-body Schrödinger equation for identical particles, the field theory is more suitable for situations where the particle number changes.

The **fractional Schrödinger equation** is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term *fractional Schrödinger equation* was coined by Nick Laskin.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

The **Born series** is the expansion of different scattering quantities in quantum scattering theory in the powers of the interaction potential . It is closely related to Born approximation, which is the first order term of the Born series. The series can formally be understood as power series introducing the coupling constant by substitution . The speed of convergence and radius of convergence of the Born series are related to eigenvalues of the operator . In general the first few terms of the Born series are good approximation to the expanded quantity for "weak" interaction and large collision energy.

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 Horáček and Sasakawa in 1983. The goal of the method is to solve the integral equation

**Multiple scattering theory** (MST) is the mathematical formalism that is used to describe the propagation of a wave through a collection of scatterers. Examples are acoustical waves traveling through porous media, light scattering from water droplets in a cloud, or x-rays scattering from a crystal. A more recent application is to the propagation of quantum matter waves like electrons or neutrons through a solid.

- ↑ Lippmann & Schwinger 1950 , p. 469
- ↑ Joachain 1983 , p. 112
- ↑ Weinberg 2002 , p. 111
- ↑ Joachain 1983 , p. 517
- ↑ Joachain 1983 , p. 576
- ↑ Wheeler 1937 , pp. 1107
- ↑ Weinberg 2002 , Section 3.1.

- Joachain, C. J. (1983).
*Quantum collision theory*. North Holland. ISBN 978-0-7204-0294-0.CS1 maint: ref=harv (link) - Sakurai, J. J. (1994).
*Modern Quantum Mechanics*. Addison Wesley. ISBN 978-0-201-53929-5.CS1 maint: ref=harv (link) - Weinberg, S. (2002) [1995].
*Foundations*. The Quantum Theory of Fields.**1**. Cambridge: Cambridge University Press. ISBN 978-0-521-55001-7.CS1 maint: ref=harv (link)

- Lippmann, B. A.; Schwinger, J. (1950). "Variational Principles for Scattering Processes. I".
*Phys. Rev. Lett*.**79**(3): 469–480. Bibcode:1950PhRv...79..469L. doi:10.1103/PhysRev.79.469.CS1 maint: ref=harv (link) - Wheeler, J. A. (1937). "On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure".
*Phys. Rev*.**52**(11): 1107–1122. Bibcode:1937PhRv...52.1107W. doi:10.1103/PhysRev.52.1107.CS1 maint: ref=harv (link)

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