Propagator and multiple scattering approach to the time of arrival problem

Abstract

The propagator approach combined with the multiple-scattering theory is applied to the particle time of arrival (TOA) problem. This approach allows us to naturally include in the consideration the components of the particle initial wavefunction (defined at t = t0) corresponding to the positive (forward-moving term) and negative (backward-moving term) momenta. For a freely moving particle it is shown that the Allcock definition of the ideal total TOA probability disregards the backward-moving and interference terms entirely. In the presence of a measuring apparatus modeled by an imaginary step potential with the amplitude V0, the general expression for the TOA rate is obtained, the forward-moving component of which coincides with that obtained by Allcock. It is shown that when the initial particle wavefunction is well separated from the point of arrival and has a well-defined average momentum, the contribution of the backward-moving and interference terms are small and can be neglected. For a small V0, except the well-known convolution result by Allcock–Kijowski, the exponential form of the TOA rate follows at the double limit condition V0 → 0, t − t0 ∼ ℏ/2V0 → ∞ (2V0(t − t0)/ℏ is finite) while the backward-moving and interference terms vanish. We show that the Allcock result for the TOA rate is valid in the entire range of V0 including the Zeno case (V0 → ∞) and the normalized TOA rate can be introduced for all values of V0 as a probability distribution. The latter is illustrated for the Gaussian wave packet.

Time-Dependent Scattering by an Asymmetric Spin-Dependent Rectangular Potential in Nanostructures

“…From Eqs. (2), (16) and (17) we see that the dependence of the Green's function on E and k  comes in the combination…”

“…, the imaginary parts of the Green's functions vanish in all spatial regions, as is seen from definitions (16) and 17 ( ,…”

“…An important advantage of our approach to propagator calculation is also that it allows for a natural decomposition of the general initial wave function evolution in time into both the forward-(p  ħk  0) and backward-moving (negative momentum p  0) components and an analysis of the contribution of both these terms and their interference to particle reflection and transmission. This approach has been further applied to the analysis of the time-dependent properties of the scattering by the imaginary step [16] (related to calculation of the particle time of arrival) and rectangular symmetric barrier/well potentials [17,18].…”

An Exact Solution of the Time-Dependent Schrödinger Equation with a Rectangular Potential for Real and Imaginary Times

“…where A(x, x ′ ; E) is given by (13). Thus, the density matrix ρ(x, x ′ ; β) follows from the propagator (12) by the substitution t → −i β (β = 1/k B T ).…”

“…Thus, the density matrix ρ(x, x ′ ; β) follows from the propagator (12) by the substitution t → −i β (β = 1/k B T ). From the properties (13) we see that the density matrix (29) is self-adjoint. The density operator (28) satisfies the Bloch equation (5).…”