论文标题
绝热近似崩溃的可解决模型
A solvable model of the breakdown of the adiabatic approximation
论文作者
论文摘要
令$ l \ geq0 $和$ 0 <\ varepsilon \ ll1 $。考虑具有缩放和翻译的谐波振荡器电位$ i \ varepsilon \ partial_t u _ {\ varepsilon} = - \ \ tfrac12 \ tfrac12 \ partial_x^2U _ {美元v（t，x）=（t-l）^2x^2/2 $，$ t> l $。 最初的价值问题可以从贝塞尔功能方面明确解决。我们使用明确的解决方案表明，绝热定理将其分解为$ \ varepsilon \至0 $。 对于情况，获得$ L = 0 $完整结果。基态的生存概率$π^{ - 1/4} \ exp（-x^2/2）在微观时间$ t = 1/\ varepsilon $ as $ 1/\ sqrt {2}+o（\ varepsilon）$。对于$ L> 0 $，给出了进一步计算的框架和初步结果。
Let $L\geq0$ and $0<\varepsilon\ll1$. Consider the following time-dependent family of $1D$ Schrödinger equations with scaled and translated harmonic oscillator potentials $ i\varepsilon\partial_t u_{\varepsilon}=-\tfrac12\partial_x^2u_{\varepsilon}+V(t,x)u_{\varepsilon}$, $u_{\varepsilon}(-L-1,x)=π^{-1/4}\exp(-x^2/2) $, where $ V(t,x)= (t+L)^2x^2/2$, $t<-L$, $ V(t,x)= 0$, $-L\leq t \leq L$, and $ V(t,x)=(t-L)^2x^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 $\varepsilon\to 0$. For the case $L=0$ complete results are obtained. The survival probability of the ground state $π^{-1/4}\exp(-x^2/2)$ at microscopic time $t=1/\varepsilon$ is $1/\sqrt{2}+O(\varepsilon)$. For $L>0$ the framework for further computations and preliminary results are given.
