薛定谔方程引出过程中存在的问题及解决方案  

PROBLEMS AND SOLUTIONS IN DERIVATION OF SCHRODINGER EQUATION

作　　者：蔡志东[1] CAI Zhidong(Zhenjiang College,Zhenjiang,Jiangsu 212300)

机构地区：[1]镇江高等专科学校,江苏镇江212300

出　　处：《物理与工程》2022年第4期115-124,共10页Physics and Engineering

摘　　要：本文介绍了部分量子力学教材中薛定谔方程的引出过程,并指出其中存在的问题,即能量E和动量P的物理意义前后不一致:在讨论平面电磁波和算符与物理量之间的对应关系时,认为能量E和动量P是相对论性的,而在讨论能量和动量的关系时,却认为能量E和动量P是非相对论性的,即认为在低速条件下,物体的能量包含了经典动能和势能,而经典动能等于动量的平方除以静止质量的两倍。有人可能会产生一个疑惑,既然能量E和动量P都是非相对论性的,那么为什么我们仍然可以用能量算符和动量算符来分别对应能量和动量?要知道,这种对应关系只有在相对论性条件下才能成立。本文将对这一问题作一个合理的解释,以表明三种近似方法的结果是一致的。一种方法就是传统的做法,虽然这会导致E、P前后物理意义不一致的问题,但不影响最终结果。第二种方法是从爱因斯坦能量—动量公式出发,保留E和P的相对论性,级数展开后,把能量算符和动量算符代入展开式,引出级数形式的波动方程,然后通过适当的近似,引出常见的薛定谔方程。这样做的好处是,可以避免E、P物理意义前后不一致而引发的问题,即在经典能量和动量情况下,算符和物理量之间的对应关系到底是否可用的问题,从而消除了学生的疑虑。第三种方法就是直接从相对论性的克莱因—高登方程出发,通过近似处理得出薛定谔方程,但是,这种方法虽然严密但较为复杂。In this paper, the derivation process of Schrodinger equation in some quantum mechanics textbooks is introduced, and the existing problems are pointed out. That is the physical significance of energy E and momentum P is inconsistent: when discussing the corresponding relationship between the plane electromagnetic wave sum operator and the physical quantity, energy E and momentum P are considered in relativistic way, while when discussing the relationship between energy and momentum, energy E and momentum P are treated in non-relativistic way. That is to say, at low speed, the energy of the object includes classical kinetic energy and potential energy, and the classical kinetic energy is equal to the square of momentum divided by twice the static mass. One may wonder that since energy E and momentum P are both non-relativistic, why can we still use energy operator and momentum operator to correspond to energy and momentum respectively? It should be known that this correspondence can only be established under the condition of relativity. This paper will give a reasonable explanation to this problem to show that the results of the three approximation methods are consistent. One way is the traditional way, although this will lead to the inconsistency of the physical meaning of E and P, it does not affect the final result. The second method is based on Einstein’s energy momentum formula and retain the relativity of E and P. After the series expansion, the energy operator and momentum operator are substituted into the expansion, and the wave equation in series form is derived. Then, through appropriate approximation, the common Schrodinger equation is derived. The advantage of this is that it can avoid the problems caused by the inconsistency of the physical meaning of E and P, that is, in the case of classical energy and momentum, whether the corresponding relationship between operator and physical quantity is available or not, so as to eliminate the students’ doubts. The third method is to obtain the Schrodinger

关 键 词：薛定谔方程 引出过程 问题 解决方案 克莱因-高登方程 规范场论 

分 类 号：O175.2[理学—数学] O413.1[理学—基础数学]