A solvable model of the breakdown of the adiabatic approximation

Abstract

Let $L\ge 0$ and $0<\epsilon \ll 1$. Consider the following time-dependent family of $1D$ Schrödinger equations with scaled and translated harmonic oscillator potentials $i\epsilon {\partial }_t{u}_{\epsilon }=-\frac{1}{2}{\partial }_x^2{u}_{\epsilon }+V(t,x)u_{\epsilon }$, $u_{\epsilon }(-L-1,x)={\pi }^{-1/4}\exp (-x^2/2)$, where $V(t,x)=(t+L{)}^2{x}^2/2$, $t<-L$, $V(t,x)=0$, $-L\le t\le L$, and $V(t,x)=(t-L{)}^2{x}^2/2$, $t>L$. 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 $\epsilon \to 0$. For the case $L=0$ complete results are obtained. The survival probability of the ground state ${\pi }^{-1/4}\exp (-x^2/2)$ at microscopic time $t=1/\epsilon$ is $1/\sqrt{2}+O(\epsilon )$. For $L>0$ the framework for further computations and preliminary results are given.

18 pages, revised version