arXiv quant-ph/04030052004-03quant-phmath-phmath.MPphysics.class-ph

经典动量路径的路径积分

Path integrals from classical momentum paths

John Hegseth

备注：: 17 pages, 3 figures, added a reference, corrected typos, added one paragraph

摘要：

量子力学的路径积分公式通过计算坐标空间中所有经典路径的作用量S来构造传播子。对应的动量路径积分也可以通过端点中的傅立叶变换来定义。虽然这些动量路径积分在几种特殊情况下特别简单，但据我所知，还没有人从动量空间中的所有经典路径中正式构造出它们。我证明这是可能的，因为存在另一个基于替代经典作用量R的经典力学。Hamilton正则方程是由S和R中的变分原理导出的。S使用固定的开始和结束空间点，而R使用固定的开始和结束动量点。这种替代作用的经典力学也包括一个哈密尔顿-雅可比方程。我还提出了一些重要的观点，关于应用规范变换所需的动作的开始和结束条件。这些性质解释了相空间路径积分中正则变换的失效。因此，路径积分可以用坐标表示中的S从经典位置路径构造，或者用动量表示中的R从经典动量路径构造。几个例子的计算，说明简化和实际的优势，可能通过这种更广泛的观点的路径积分。特别地，在不使用薛定谔方程的情况下求出了自由粒子的归一化振幅，简单而自然地导出了内自旋自由度，并计算了简谐振子。

英文摘要：

The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文