{"title":"Classical Stochastic Representation of Quantum Mechanics","authors":"Mário J. de Oliveira","doi":"10.1007/s13538-024-01649-x","DOIUrl":null,"url":null,"abstract":"<div><p>We show that the dynamics of a quantum system can be represented by the dynamics of an underlying classical systems obeying the Hamilton equations of motion. This is achieved by transforming the phase space of dimension 2<i>n</i> into a Hilbert space of dimension <i>n</i> which is obtained by a peculiar canonical transformation that changes a pair of real canonical variables into a pair of complex canonical variables which are complex conjugate of each other. The probabilistic character of quantum mechanics is devised by treating the wave function as a stochastic variable. The dynamics of the underlying system is chosen so as to preserve the norm of the state vector.</p></div>","PeriodicalId":499,"journal":{"name":"Brazilian Journal of Physics","volume":"55 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Brazilian Journal of Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s13538-024-01649-x","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We show that the dynamics of a quantum system can be represented by the dynamics of an underlying classical systems obeying the Hamilton equations of motion. This is achieved by transforming the phase space of dimension 2n into a Hilbert space of dimension n which is obtained by a peculiar canonical transformation that changes a pair of real canonical variables into a pair of complex canonical variables which are complex conjugate of each other. The probabilistic character of quantum mechanics is devised by treating the wave function as a stochastic variable. The dynamics of the underlying system is chosen so as to preserve the norm of the state vector.
我们证明，量子系统的动力学可以用服从汉密尔顿运动方程的底层经典系统的动力学来表示。这是通过将维数为 2n 的相空间转换为维数为 n 的希尔伯特空间来实现的，而希尔伯特空间是通过一种奇特的规范变换得到的，这种变换将一对实规范变量转换为一对复规范变量，而这对复规范变量是互为复共轭的。量子力学的概率特性是通过将波函数视为随机变量而设计出来的。底层系统的动力学选择是为了保持状态向量的规范。
期刊介绍：
The Brazilian Journal of Physics is a peer-reviewed international journal published by the Brazilian Physical Society (SBF). The journal publishes new and original research results from all areas of physics, obtained in Brazil and from anywhere else in the world. Contents include theoretical, practical and experimental papers as well as high-quality review papers. Submissions should follow the generally accepted structure for journal articles with basic elements: title, abstract, introduction, results, conclusions, and references.
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