Structure and properties of Hughston’s stochastic extension of the Schrödinger equation
Structure and properties of Hughston’s stochastic extension of the Schrödinger equation
复制标题
薛定谔方程的休斯顿随机扩展的结构和性质
DOI：
10.1063/1.533255
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发表时间：
1999
影响因子：
1.3
通讯作者：
L. Horwitz
中科院分区：
文献类型：
--
作者：
S. Adler;L. Horwitz
休斯顿最近提出了薛定谔方程的随机扩展，表示为射影希尔伯特空间上的随机微分方程。我们推导了新的投影希尔伯特空间恒等式，用它给出了休斯顿方程导致状态向量坍缩到能量本征态的一般证明，坍缩概率由初始状态计算的量子力学概率给出。讨论了休斯顿方程与早期关于保范随机方程的研究的关系，并证明了休斯顿方程可以明显地写成纯态密度矩阵的一元随机演化方程。我们讨论了作为独立子系统的直接产物构建的系统的行为，并简要地解决了基于能量的方法（如Hughston的方法）是否足以客观地解释量子力学中的测量过程的问题。
Hughston has recently proposed a stochastic extension of the Schrodinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston’s equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston’s equation to earlier work on norm-preserving stochastic equations, and show that Hughston’s equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston’s, suffices to give an objective interpretation of the measurement process in quantum mechanics.