arXiv quant-ph/04100612004-10quant-ph

量子力学

Quantum Mechanics

Hitoshi Kitada

备注：: 168 pages, LaTeX, descriptions in section 6.1 improved
发表信息：: Lectures in Mathematical Sciences vol. 23, The University of Tokyo, 2005, ISSN 0919-8180, ISBN 1-000-01896-2

摘要：

我在这本书中考虑了量子力学的一个公式。通常QM是基于时间和空间的概念，这两者都被认为是先验的给定量或概念。然而，当我们试图定义速度或动量的概念时，我们遇到了一个困难，我们将在第1章中看到。问题是，如果时间的概念是先验的，那么速度在给定位置时就确定了，这与海森堡的不确定原理相矛盾。然后，我们将量子力学的基础建立在位置和动量算符的概念上，如第2章所述。然后，局部系统的时间被近似地定义为空间坐标$x$和速度$v$之间的比率$|x|/|v|$。在量子力学的这个公式中，我们可以保持不确定原理，时间是一个没有精确值的量，不像通常假设的时间概念那样。局域时的特征是它是每个局域系统的固有时间，局域系统被定义为量子力学粒子的有限集合。我们现在有无限多的本地时间，它们对每个本地系统都是唯一的和适当的。基于局部时的概念，局部系统内部的运动由通常的薛定谔方程描述。我们在第二部分中研究了给定局部系统中的这种运动。这是一个常见的量子力学。在对局部运动的研究作了一些偏离之后，我们在第三部分考虑了复数局部系统之间的相对关系或运动。在最后的第四部分中，我们将证明至少存在一个宇宙波函数$ϕ$，其中所有的局部系统都有局部运动，从而有局部时。这就结束了我们的量子力学公式。

英文摘要：

I consider in this book a formulation of Quantum Mechanics. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio $|x|/|v|$ between the space coordinate $x$ and the velocity $v$. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrödinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. In the final part IV, we will prove that there is at least one Universe wave function $ϕ$ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.

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