arXiv 0711.22302007-11-14quant-phmath-phmath.MP

一维两谐振子的联合激发几率与Mott问题

Joint excitation probability for two harmonic oscillators in dimension one and the Mott problem

Gianfausto Dell'Antonio, Rodolfo Figari, Alessandro Teta

备注：: 26 pages

摘要：

我们分析了一个一维量子系统中一个测试粒子与两个谐振子的相互作用，其中两个谐振子的位置分别为$a_1$，$a_2$，其中$a_1 >0$，$|a_2|>a_1$，在两种可能的情况下：$a_2>0$和$a_2 <0$ .在零时刻，谐振子处于基态，测试粒子处于两个波包的叠加态，这两个波包以原点为中心，具有相反的平均动量。% $\pm M v_0$。在对模型的物理参数作适当假设的情况下，考虑了波函数的时间演化，计算了概率$\mathcal{P}^{-}_{n_1 n_2} (t)$（分别为. $\mathcal{P}^{+}_{n_1 n_2} (t)$），两个振荡器在时间$t > |a_2| v_0^{-1}$处于由$n_1$、$n_2 >0$标记的激发态，当$a_2 <0$（分别为$a_2 >0$）。我们证明，$\mathcal{P}_{n_1 n_2}^- (t)$是可以忽略的$\mathcal{P}_{n_1 n_2}^+ (t)$，高达二阶含时微扰理论。我们考虑的系统是一个简化的，一维版本的原始模型的云室介绍莫特在\cite{m}，在那里的结果被认为是使用euristic参数的框架内的时间无关微扰理论的固定薛定谔方程。证明的方法是完全初等的，它基本上是基于一个固定相位的论点。我们还注意到，所有的计算都涉及三粒子系统的薛定谔方程，而没有涉及波包坍缩假设。

英文摘要：

We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions $a_1$, $a_2$, with $a_1 >0$, $|a_2|>a_1$, in the two possible situations: $a_2>0$ and $a_2 <0$. At time zero the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. %$\pm M v_0$. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability $\mathcal{P}^{-}_{n_1 n_2} (t)$ (resp. $\mathcal{P}^{+}_{n_1 n_2} (t)$) that both oscillators are in the excited states labelled by $n_1$, $n_2 >0$ at time $t > |a_2| v_0^{-1}$ when $a_2 <0$ (resp. $a_2 >0$). We prove that $\mathcal{P}_{n_1 n_2}^- (t)$ is negligible with respect to $\mathcal{P}_{n_1 n_2}^+ (t)$, up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott in \cite{m}, where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schrödinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schrödinger equation for the three-particle system, with no reference to the wave packet collapse postulate.

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