Toy Model Sample Clauses

Toy Model. Scattering on the Half-Line‌ The concepts of time delay and CPA can be illustrated by the simple example of scattering on the half-line x ≥ 0 with a potential V (x) supported in [0, a] such that V (0) = ∞ (this corresponds to the boundary condition ψ(0) = 0). The scattering solutions of the Schr¨odinger equation at x > a with incoming boundary condition are 1 −ik(x−a) ik(x−a) Ψk(x) = √|k| e − S(k)e , (27) ∫ which is a sum of an incoming and outgoing part. The ratio of the amplitudes of the outgoing and incoming parts of Ψk is the S matrix, which in one dimension is simply a phase S(k) = e2iθ(k). If we consider the time evolution of a wave packet Ψ(x) = b(k)Ψk(x)dk where the profile b(k) is narrowly peaked around k0, it can be split into the evolution of an incoming part Ψinc(x, t) = ∞ 1 ∫ ∫−ik t −∞ √|k|b(k)e −ik t−ik(x−a)dk dk and an outgoing part 2 ∞ 1 √ ' e 0 b(k)e −∞ |k| −ik(x−a+v0t) dk, ∫−ik t−2iθ(k )0 Ψout(x, t) = −∞ √|k|b(k)e ∫ ∞ 1 −ik t+ik(x−a)+2iθ(k) where
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