# Scattering Matrix

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## Scattering matrix

An infinite-dimensional matrix or operator that expresses the state of a scattering system consisting of waves or particles or both in the far future in terms of its state in the remote past; also called the S matrix. In the case of electromagnetic (or acoustic) waves, it connects the intensity, phase, and polarization of the outgoing waves in the far field at various angles to the direction and polarization of the beam pointed toward an obstacle. It is used most prominently in the quantum-mechanical description of particle scattering, in which context it was invented in 1937 by J. A. Wheeler to describe nuclear reactions. Because an analog of the Schrödinger equation for the description of particle dynamics is lacking in the relativistic domain, W. Heisenberg proposed in 1943 that the S matrix rather than the hamiltonian or the lagrangian be regarded as the fundamental dynamical entity of quantum mechanics. This program played an important role in high-energy physics during the 1960s but is now largely abandoned. The physics of fundamental particles is now described primarily in terms of quantum gauge fields, and these are used to determine the S matrix and its elements for the collision and reaction processes observed in the laboratory. *See* Elementary particle, Nuclear reaction, Quantum mechanics, Relativistic quantum theory, Scattering experiments (atoms and molecules), Scattering experiments (nuclei)

The mathematical properties of the S matrix in nonrelativistic quantum mechanics have been thoroughly studied and are, for the most part, well understood. If the potential energy in the Schrödinger equation, or the scattering obstacle, is spherically symmetric, the eigenfunctions of the S matrix are spherical harmonics and its eigenvalues are of the form exp (2*i*δ_{l}), where the real number δ_{l} is the phase shift of angular momentum *l*. In the nonspherically symmetric case, analogous quantities are called the eigenphase shifts, and the eigenfunctions depend on both the energy and the dynamics. In the relativistic regime, without an underlying Schrödinger equation for the particles, the mathematical properties are not as well known. Causality arguments (no signal should propagate faster than light) lead to dispersion relations, which constitute experimentally verifiable consequences of very general assumptions on the properties of nature that are independent of the detailed dynamics. *See* Angular momentum, Causality, Dispersion relations, Eigenfunction

## Scattering Matrix

(S-matrix), a combination of quantities (a matrix) describing the process of transition of quantum-mechanical systems from some states to others upon interaction (scattering). The concept of a scattering matrix was introduced by W. Heisenberg in 1943.

During scattering, a system moves from one quantum state, the initial state (which may be related to an instant *t = — ∞*), to another state, the final state (*t = + ∞*). If the set of quantum numbers that characterize the initial state is designated as *i*, and the final state as *f _{l}* then the scattering amplitude (the square of whose absolute value determines the probability of scattering) may be written as

*S*. The combination of the scattering amplitudes forms a table with two inputs (

_{fi}*i*, the number of the row, and

*f*, the number of the column), which is called the scattering matrix

*S*. Each amplitude is a matrix element. Sets of the quantum numbers

*i*and

*f*may contain both continuous quantities (such as energy and the scattering angle) and discrete quantities (such as orbital quantum number, spin, isotopic spin, and mass). In the simplest case, a system of two spinless particles in non-relativistic quantum mechanics, the state is determined by the relative momentum p of the particles; then the scattering amplitude is a function of two variables, the energy

*E*and the scattering angle

*θ*:

*S _{fi} = F(E, θ)*

In the general case a scattering matrix contains elements that correspond both to elastic scattering and to the processes of particle conversion and production. The square of the absolute value of the matrix element *|S _{fi}|^{2}* determines the probability of the corresponding process (or its effective cross section).

The fundamental problem of quantum mechanics and quantum field theory is to find the scattering matrix. The scattering matrix contains all information concerning the behavior of the system if not only the numerical values but also the analytical properties of its elements are known; in particular, its poles determine the bound states of the system (and, consequently, its discrete energy levels). The most important property of the scattering matrix, its unitarity, follows from the fundamental principles of quantum theory. This is expressed in the form of the equation *SS ^{+}* = 1, where

*S*is the Hermitian conjugate of

^{+}*S*— that is, (

*S*, where the symbol * designates complex conjugation—or

^{+})_{fi}= S*_{if}and reflects the fact that the sum of the probabilities of scattering for all possible channels of the reaction must be equal to unity. The unitarity relation makes it possible to establish important relations among various processes and, in some cases, even to solve the problem completely. There is a trend in relativistic quantum mechanics in which the scattering matrix is considered to be a primary dynamic quantity; the requirements for the unitarity and analyticity of a scattering matrix should serve as the basis for constructing a complete system of equations that define the S-matrix.

V. B. BERESTETSKII