Arrival times in quantum mechanics by Seidel D.

By Seidel D.

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7] by deriving properly constructed absorbing potentials and, more basically, by Werner [34]. Blanchard and Jadczyk developed their event- enhanced quantum theory that is based on the use of complex potentials in the time evolution to explain the occurrence of events and their timing [79, 80]. In contrast to a phenomenological introduction of absorbing potentials into the Schr¨odinger equation, Halliwell presented a more realistic two-channel detector model [22] in which the initially excited detector decays due to the presence of the particle.

Kinetic energy densities are present in the Thomas-Fermi theory [104], in density functional theory to determine the electronic structure of atoms, molecules, solids or fermionic gases [105], in partitioning molecular systems into fragments with well defined energies [106, 107] or to define intrinsic shapes of reactants, transition states, and products along the course of a chemical reaction [108]. In analogy to Eq. 3), the classical expression for the phase space variable related to the local kinetic energy density at x is K(x) = p2 δ(q − x).

88) Assuming that the delay problem is solved by this construction, one clearly attempts to simultaneously minimize reflections. This can be done by considering the limit γ → ∞ of Eq. 75). In that case, one has in leading order 1 W (ν) ∼ iνγ , Ω2 iΩ2 → 0, 2γ iγ ∼ − → ∞, 2 imΩ2 ∼ k+ → k, 2 kγ λ+ ∼ − λ− k+ k− ∼ q ∼ imγ , C− , R1 , R2 → 0, C+ → 1, (2) γ → ∞. 89) (2) Thus, γφk (x)φk (x) is exponentially damped for x ≤ 0, whereas for x ≥ 0 only the terms with k+ survive, leading to (2) (2) γφk (x)φk (x) ∼ Ω2 −i(k¯+ −k+ )x e Θ(x), 2πγ γ → ∞.

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