量子哈密顿-雅可比理论
Quantum Hamilton-Jacobi Theory
摘要：
自量子理论诞生以来，量子正则变换就引起了人们的兴趣。基于它们的经典类似物，人们会期望它们提供一个强大的量子工具。然而，求解非线性算子偏微分方程（如量子哈密顿-雅可比方程（QHJE））的困难阻碍了沿着这条有前途的途径的进展。我们克服了这个困难。我们表明，解决方案的QHJE可以构造一个简单的处方从传播相关联的薛定谔方程。我们的结果为量子Hamilton-Jacobi理论的实际应用提供了可能。作为一个应用程序，我们开发了一个令人惊讶的操作程序之间的关系和密度的路径周围的半经典轨迹。
英文摘要：
Quantum canonical transformations have attracted interest since the beginning of quantum theory. Based on their classical analogues, one would expect them to provide a powerful quantum tool. However, the difficulty of solving a nonlinear operator partial differential equation such as the quantum Hamilton-Jacobi equation (QHJE) has hindered progress along this otherwise promising avenue. We overcome this difficulty. We show that solutions to the QHJE can be constructed by a simple prescription starting from the propagator of the associated Schroedinger equation. Our result opens the possibility of practical use of quantum Hamilton-Jacobi theory. As an application we develop a surprising relation between operator ordering and the density of paths around a semiclassical trajectory.
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