标题： 粒子在带有测量的双缝实验中的动力学 显示英文标题

标题： Dynamics of a particle in the double-slit experiment with measurement

摘要： 自发坍缩模型通过非线性随机修改薛定谔方程来抑制测量可观测量本征态的叠加，并将状态驱动到一个本征态。最近的研究表明，可以使用由高斯酉系综中的随机矩阵表示的哈密顿量的线性薛定谔方程来模拟跃迁概率的玻恩规则。在观察期间不同时间点表示哈密顿量的矩阵被假定为独立的。与抑制叠加不同，这种薛定谔演化使状态在状态的射影空间上进行各向同性的随机游走。在随机游走中到达任意可观测量不同本征态的相对频率被证明满足玻恩规则。在这里，我们将这种方法应用于双缝实验中的粒子行为研究。我们的分析表明，在这个基本情况下，可以通过状态空间二维子流形上的随机游走有效地捕捉粒子状态的演化。这种随机游走再现了在粒子到达其中一个缝的情况下，找到粒子靠近缝的概率的玻恩规则。为了确保满足这一条件，我们引入了一个代表位置可观测量方差变化的漂移项。还考虑了一个基于探测器无法区分的状态等价类的无漂移模型。无论是否有漂移，所产生的随机游走都为描述实验中从初始状态到测量可观测量本征态的转变提供了一个合适的模型，为理解其潜在机制提供了新的见解。 显示英文摘要

摘要： Spontaneous collapse models use non-linear stochastic modifications of the Schroedinger equation to suppress superpositions of eigenstates of the measured observable and drive the state to an eigenstate. It was recently demonstrated that the Born rule for transition probabilities can be modeled using the linear Schroedinger equation with a Hamiltonian represented by a random matrix from the Gaussian unitary ensemble. The matrices representing the Hamiltonian at different time points throughout the observation period are assumed to be independent. Instead of suppressing superpositions, such Schroedinger evolution makes the state perform an isotropic random walk on the projective space of states. The relative frequency of reaching different eigenstates of an arbitrary observable in the random walk is shown to satisfy the Born rule. Here, we apply this methodology to investigate the behavior of a particle in the context of the double-slit experiment with measurement. Our analysis shows that, in this basic case, the evolution of the particle's state can be effectively captured through a random walk on a two-dimensional submanifold of the state space. This random walk reproduces the Born rule for the probability of finding the particle near the slits, conditioned on its arrival at one of them. To ensure that this condition is satisfied, we introduce a drift term representing a change in the variance of the position observable for the state. A drift-free model, based on equivalence classes of states indistinguishable by the detector, is also considered. The resulting random walk, with or without drift, serves as a suitable model for describing the transition from the initial state to an eigenstate of the measured observable in the experiment, offering new insights into its potential underlying mechanisms.