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Solved Problems in Quantum Physics - angular momentum and spin

Jaroslav Kafka

Published Jan 1, 2022• 0 citations
OpenAlex

Abstract

Title: Solved Problems in Quantum Physics - angular momentum and spin Author: Jaroslav Kafka Department: Department of Physics Education Supervisor: RNDr. Zdeňka Koupilová, Ph.D., Department of Physics Education Abstract: The aim of this bachelor thesis was to create a set of solved problems in quantum physics for the electronic Collection of Solved Problems in Physics. Within this thesis 18 problems were created, which belong to the chapter Microworld Physics, mostly at the undergraduate level. There is described a brief history of the electronic collection and its technical background in the context of the user and the administrator. Part of the thesis includes description of the tasks in the matter of physical issues and its interconnection with other tasks of the Collection of Solved Problems in Physics. In the appendix there are presented selected solved problems with structured sollution. All tasks can be found on the website http://reseneulohy.cz/ or on the attached CD. Keywords: electronic collection, solved problems, quantum physics 1

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Title: Solved Problems in Quantum Physics - angular momentum and spin Author: Jaroslav Kafka Department: Department of Physics Education Supervisor: RNDr. Zdeňka Koupilová, Ph.D., Department of Physics Education Abstract: The aim of this bachelor thesis was to create a set of solved problems in quantum physics for the electronic Collection of Solved Problems in Physics. Within this thesis 18 problems were created, which belong to the chapter Microworld Physics, mostly at the undergraduate level. There is described a brief history of the electronic collection and its technical background in the context of the user and the administrator. Part of the thesis includes description of the tasks in the matter of physical issues and its interconnection with other tasks of the Collection of Solved Problems in Physics. In the appendix there are presented selected solved problems with structured sollution. All tasks can be found on the website http://reseneulohy.cz/ or on the attached CD. Keywords: electronic collection, solved problems, quantum physics 1
The main aim of this thesis was to create a set of problems in Quantum Mechanics, which would be included in a collection of solved problems in Physics available online on the website http://www.fyzikalniulohy.cz. As a result, thirty nine problems were completed, covering seven thematic areas partially of high school and mostly of introductory college Quantum Mechanics course. In general, the collection is mainly designed for college students as an exercise in basic quantum-mechanical problems and also for high school students, who have a desire to deal with some special problems. A detailed description of content, structure, and technical execution of this collection can be found in chapter 1 of this thesis. Another task was to review existing online collections of applets dealing with Quantum Mechanics. A selection of them is introduced in chapter 2 of this thesis.
Abstract This book introduces the quantum theory of angular momentum to students who are unfamiliar with it and develops it to a stage useful for research. The first part contains the basic theory of rotations and angular momentum. As the book aims to emphasize applications, mathematical details are avoided and difficult theorems stated without proof. The second part contains examples of applications to a wide range of physical phenomena and presents a collection of results helpful in solving problems.
The objective of this bachelor thesis is to prepare a study text on selected topics of quantum mechanics, which would be appropriate for the se- cond year students of education of Physics at FMP CU attending the Quantum mechanics course. This thesis consists of two topics - Multiparticle Systems and Chemical Bond. The study text should help students understand the mathema- tical methods used in the field of multiparticle systems, and solve elementary problems. The text also extends secondary-school chemistry knowledge about atom orbitals and chemical bond and provides an explanation of the structure of simple molecules using the valence bond theory. The thesis is based on author's own experience of the Quantum mechanics course, the main sources are the fore- ign textbooks of quantum mechanics and author's own notes from lectures held on FMP CU. 1
The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.
Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors.
In this chapter, we develop the background needed to move our quantum mechanical formalism fully into three dimensions. We discuss angular momentum, which is the motional complement to spin angular momentum, that we developed earlier. There are some critical differences between orbital and spin angular momentum, and we need to carefully discuss how they relate and how they differ. Why is angular momentum so important? Because in problems where we have central forces (forces that depend only on the distance between two particles and act along the line connecting the two particles), the angular momentum is conserved. So, it becomes an integral part of how we solve the problem. The most important problem we have to solve, the hydrogen atom, is a central force problem; hence, we need this work before we can move forward to some really interesting physics that will come in Chap. 11 . We cover the angular momentum operators, eigenvalues, and eigenstates, their matrix representations, how to combine angular momentum, and spherical harmonics. We revisit the EPR experiment using our new formalism. We also discuss some unique representations of angular momentum by mapping to two-dimensional harmonic oscillators, as developed by Schwinger.