By A. G. Sitenko and P. J. Shepherd (Auth.)
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Extra info for Lectures in Scattering Theory
Example text
For large r, as is clear from eqn. 14) where βη is some constant. e. that the asymptotic form of v(r) as r -► oo has the form r +ε When this condition is fulfilled, the asymptotic form of the wave func tion ψι, which is a solution of eqn. 9), can be represented in the form of a linear combination of independent solutions of eqn. 10). In fact, at large distances from the centre, the potential v(r) varies suf ficiently smoothly, and therefore to find the asymptotic form of the wave function we can make use of the quasi-classical approximation.
5) 51 Lectures in Scattering Theory The radial part of the Laplace operator can be written in the equivalent forms d2 2d 1 S 2 M 1 d / 2 d\ 2 [r — ) = — + Ξ 0). 6) r2 dr \ drj or2 r dr r dr2 Since a central potential is characterized by spherical symmetry, the angular momentum is an integral of the motion. States corresponding to different values of the angular momentum take part independently in the scattering. 3), in the form of a superposition of partial waves cor responding to well-defined values of the angular momentum.
Let the frame K' move relative to the frame K with a constant velocity ». 81) where L is a unitary operator effecting the Lorentz transformation. 82) 29 Lectures in Scattering Theory The Lorentz invariance requirement for the scattering operator means that S' = S. 83) However, the matrix elements of S on going from K to K' can change in a way that depends on the choice of the initial and final states. Let the initial and final states ψΛ and ψβ in frame Kbc characterized by giving the energies and momenta of the particles <* = Ρι,Ε1\ρ2,Ε2\...
