Quantum Mechanics In Phase Space
Quantum Mechanics In Phase Space
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相空间中的量子力学
DOI：
10.1142/9789810248604_0016
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
Jonathan Leach
中科院分区：
文献类型：
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作者：
Jonathan Leach
本论文的目的是描述与相空间中量子力学重新表述相关的基本概念。假设感兴趣的读者熟悉普通量子力学的原理和技术。在经典统计力学中，物理量的期望值被计算为相空间分布函数的平均值。在量子力学中可以获得类似的过程。然而，相空间是一个经典概念，由于算子 p 和 q 的不可交换性，因此没有量子力学等价物。在相空间中表达量子力学的一个合理的基本要求是量子算符和经典函数之间的线性一对一映射。这是通过 Weyl、Wigner 和 Moyal 的开创性工作实现的。他们设计的相空间我们称之为“伪相空间”。它是完全量子化的。伪相空间变量由动量 p 和位置 q 的交换 c 数表示；它们是分别将 Weyl 对应规则应用于算子 p 和 q 的结果。在某些方面，量子力学的伪相空间公式似乎让我们回到了“几乎”经典的领域，因为我们在纯代数方法中省去了非对易算子的微积分。量子力学的伪相空间公式是独立的。也就是说，在解决物理问题时，原则上没有必要切换到薛定谔或海森堡图，尽管这样做通常很方便。当考虑量子力学的半经典极限时，伪相空间方案的经典外观特别有用。事实上，对于高达并包括谐振子的势，方程代表经典运动。论文的结构引导读者逐步了解伪相空间理论的主要概念。所需的数学技术已在第 1 章的早期部分阐述过。这些技术贯穿整个论文。这样做的优点是，随着论文的进展，读者不必学习新的数学方法。量子力学的伪相空间公式在数学物理的所有领域都有应用。因此，有大量关于该主题的专业文献。希望本文能起到一个引言。
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. It is hoped that this thesis serves has an introduction.