A solvable model of the breakdown of the adiabatic approximation

Artbazar Galtbayar, Arne Jensen*, Kenji Yajima

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Abstrakt

Let L ≥ 0 and 0 < ϵ ≪ 1. Consider the following time-dependent family of 1D Schrödinger equations with scaled harmonic oscillator potentials iϵαtuϵ=-12αx2uϵ+V(t,x)uϵ, uϵ(-L - 1, x) = π-1/4 exp(-x2/2), where V(t, x) = (t + L)2x2/2, t < - L, V(t, x) = 0, - L ≤ t ≤ L, and V(t, x) = (t - L)2x2/2, t > L. The initial value problem is explicitly solvable in terms of Bessel functions. Using the explicit solutions, we show that the adiabatic theorem breaks down as ϵ → 0. For the case L = 0, complete results are obtained. The survival probability of the ground state π-1/4 exp(-x2/2) at microscopic time t = 1/ϵ is 1/2+O(ϵ). For L > 0, the framework for further computations and preliminary results are given.

OriginalsprogEngelsk
Artikelnummer092105
TidsskriftJournal of Mathematical Physics
Vol/bind61
Udgave nummer9
Antal sider12
ISSN0022-2488
DOI
StatusUdgivet - sep. 2020

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