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In scattering theory, the S-matrix is an operator mapping free particle in-states to free particle out-states (scattering channels) in the Heisenberg picture. This is very useful because often we cannot describe the interaction (at least, not the most interesting ones) exactly.
The Kramers–Heisenberg dispersion formula is an expression for the cross section for scattering of a photon by an atomic electron.It was derived before the advent of quantum mechanics by Hendrik Kramers and Werner Heisenberg in 1925, [1] based on the correspondence principle applied to the classical dispersion formula for light.
In scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision (). The Hilbert space spanned by the states before collision (in states) is equal to the space spanned by the states after collision (out states) which are both Fock spaces if there is a mass gap .
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.
Any possible choice of parts will yield a valid interaction picture; but in order for the interaction picture to be useful in simplifying the analysis of a problem, the parts will typically be chosen so that H 0,S is well understood and exactly solvable, while H 1,S contains some harder-to-analyze perturbation to this system.
Crossing states that the same formula that determines the S-matrix elements and scattering amplitudes for particle to scatter with and produce particle and will also give the scattering amplitude for + ¯ + to go into , or for ¯ to scatter with to produce + ¯. The only difference is that the value of the energy is negative for the antiparticle.
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 .
The Korringa–Kohn–Rostoker (KKR) method is used to calculate the electronic band structure of periodic solids.In the derivation of the method using multiple scattering theory by Jan Korringa [1] and the derivation based on the Kohn and Rostoker variational method, [2] the muffin-tin approximation was used. [3]