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It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More formally, in the context of QFT, the S -matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states ) in the Hilbert space of physical states.
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 ...
In regular quantum mechanics, which includes quantum chemistry, the relevant equation is the Schrödinger equation, although equivalent formulations, such as the Lippmann-Schwinger equation and the Faddeev equations, are also largely used. The solutions of interest describe the long-term motion of free atoms, molecules, photons, electrons, and ...
The following description follows the canonical way of introducing elementary scattering theory. A steady beam of particles scatters off a spherically symmetric potential V ( r ) {\displaystyle V(r)} , which is short-ranged, so that for large distances r → ∞ {\displaystyle r\to \infty } , the particles behave like free particles.
The equation was later extended to quantum scattering theory by several individuals, and came to be known as the Bohr–Peierls–Placzek relation after a 1939 paper. It was first referred to as the "optical theorem" in print in 1955 by Hans Bethe and Frederic de Hoffmann , after it had been known as a "well known theorem of optics" for some time.
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 the early days of quantum theory development. [1]
Scattering theory is the theory of scattering events which can occur as well in quantum mechanics, classical electrodynamics or acoustics. The associated general mathematical frame bears the same name though its range of application may be larger.
Levinson's theorem is an important theorem of scattering theory. In non-relativistic quantum mechanics, it relates the number of bound states in channels with a definite orbital momentum to the difference in phase of a scattered wave at infinite and zero momenta.