从Hamilton-Jacobi观点看量子力学
Quantum Mechanics from the Hamilton-Jacobi Point of View
摘要：
本文在量子力学的哈密顿-雅可比理论的框架下发展了量子力学。我们将表明，薛定谔的观点和汉密尔顿-雅可比的观点是完全等价的，在他们的描述物理系统，但在他们的描述方式不同。作为主要结果，Hamilton-Jacobi理论中的波函数不仅可以渐近地分解为空间任意点的行波。著名的WKB理论将是更一般的理论的一个特殊结果，我们将在下面展开。以线性势和谐振子为例，从哈密顿-雅可比观点讨论量子力学。软边值问题可以作为联络问题精确求解。量子化能量和Maslov指数可以直接计算，而无需正交归一化波函数。此外，我们将集中在轨道的主题，这与薛定谔的观点，自然遵循量子力学的作用函数。
英文摘要：
In this article, we develop quantum mechanics upon the framework of the quantum mechanical Hamilton-Jacobi theory. We will show, that the Schroedinger point of view and the Hamilton-Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result, a wave function in Hamilton-Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. The well known WKB-theory will be a special result of the more general theory, we will develop below. By the example of the linear potential and the harmonic oscillator, we will discuss quantum mechanics from the Hamilton-Jacobi point of view. Soft boundary value problems as the connection problem can be solved exactely. Quantizised energies and Maslov-indices can be calculated directely without orthonormalizing wave-functions. Also, we will focus on trajectory themes, which, in contrast to the Schroedinger point of view, follow naturally from the quantum mechanical action function.
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