By L. A. Shepp (ed.)
During this quantity, the gathering of articles through Shepp, Helgason, Radon, and others, supplies mathematicians unexpected with utilized arithmetic a slightly complete spectrum of versions of computed tomography. integrated are great difficulties either proper and of intrinisic curiosity instructed through all of the papers
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Extra info for Computed Tomography
231) and also the background phase shift δbg can be regarded as constants over the whole width of the resonance. For a broader resonance, however, the unique deﬁnition of its position and width can become a diﬃcult problem (see also Sect. 3). The derivative of the phase shift with respect to energy is also a measure for the strength of the closed-channel component in the solution of the coupledchannel equations. 229). The 48 1 Review of Quantum Mechanics strength of the closed-channel admixture is quantitatively given by the square of the amplitude A cos δ in front of the (bound) wave function φ0 , which is normalized to unity.
Then the motion of the particle approaches that of a free particle asymptotically, provided the energy is large enough, E > V+ , E > V− . A particle incident from the left and travelling in the direction of the positive x-axis with a welldeﬁned energy E > V− can be described by a monochromatic wave function, x→−∞ ¯ k = 2µ(E − V− ). The solution of the time-independent ψ ∝ eikx , with h Schr¨ odinger equation may also contain a leftward travelling contribution describing a part of the wave function reﬂected through the inﬂuence of the x→−∞ −ikx ∝ e .
5) G(r, r ) = −π φreg (r)φirr (r ) for r ≤ r , φreg (r )φirr (r) for r ≤ r . 220) is always smaller than r, because φ0 (r ) is a bound wave function so that the integrand vanishes for large r . 227) for G(r, r ) and perform the integration over r . 224) this leads to the following asymptotic form of φ1 (r): φ1 (r) = φreg (r) + tan δ φirr (r) 2µ sin(kr + δbg + δ) , π¯ h2 k and the angle δ ist given by = 1 cos δ tan δ = −π | φ0 |V2,1 |φreg |2 . 229) Being solutions of a homogeneous system of diﬀerential equations, the two-channel wave functions are determined only to within multiplication by a common arbitrary constant.