自由粒子在μ空间的运动描述

On description of a free particle in μ-space

对一维自由粒子在μ空间的运动,经典力学和量子力学各自提供了看似不同、实则等价的两种描述.一些经典教材对这个问题的表述有误或者不完整.本文首先通过分析经典力学中自由粒子在一维盒子中和在全空间的运动,画出了μ空间的轨迹;然后通过分析量子力学中自由粒子在无限深势阱中和在全空间周期性边界条件下的运动,画出μ空间每个量子态占据的面积元.最后本文讨论了周期性边界条件下动量为零的态存在性,一个量子态在相空间中的体积元和不确定性关系之间的区别等问题.

Inμspace,classical mechanics and quantum mechanics provide,respectivley,two seemingly different and actually equivalent descriptions for one-dimensional free particles.There are some errorous presentations in some classic textbooks,and completed results are elucidated.In this paper,the trajectories ofμspace are plotted with analyzing the motion of free particles in a one-dimensional box and in full space in classical mechanics,respectivley.Then,with examining particle motion in quantum mechanics under the condition of periodic boundary in full space and in an infinite depth well respectivley,we show that the same area element is occupied by each quantum state inμspace,irrespective of the different methods taken.Two related problems,the existence of the quantum state of zero-momnentum and relationship between the Heisenberg uncertainty principle and the area element for a quantum state to accommodate,are re-visited.