Tutorials on Angular Momentum in Quantum Mechanics

V. T. Davis

K. Heyde

Fungamwango Hashel Mwansa, Phillippe Kabwita

This study aimed at investigating the problems students face in understanding the concepts of angular momentum in classical and quantum mechanics, finding an effective way of improving understanding and learning the concepts of angular momentum in quantum mechanics, lastly document the challenges investigated and suggest some solutions to help students understand angular momentum in both classical and quantum mechanics. This study summarizes a series of investigations into how undergraduate Physics students perceive angular momentum in quantum mechanics at the University of Zambia. We investigated undergraduate physics student's perception of angular momentum in quantum mechanics by administering written questions (open-ended) to two different groups of classes. Out of these investigations we developed a Quantum Interactive Learning Tutorials (Qu I L Ts) and peer instruction tools to assist the students build up a considerable knowledge structure of quantum mechanics in general and angular momentum in specific, through a guided approach technique [1]. Eleven questions were administered to a group of N = 15 students, statistics for answered questions are as follows; question one 4 out of 15 representing a percentage of 26.67%, question two only 6 out of 15 student representing 40%, question three 11 out of 15 representing a percentage of 73.33%, for question four only 1 out of 15 representing a percentage of 6.67%, question five none of the students answered, question six only 1 out of 15 students representing a percentage of 6.67% managed to answer, question seven 1 out of 15 representing a percentage of 6.67% managed to answer, question eight 11 out of 15 representing a percentage of 73.33 % got the correct answer, question nine 6 out of 15 representing a percentage 40% got the right answer, question ten only three students making a percentage of 20% answered correctly, question eleven no student managed to get it right. Preliminary results show that the undergraduate physics students perception of quantum mechanics is improved after using the research-based learning tools in the lower class level quantum mechanics courses [1]. We also build up a common conceptual survey that can help lecturers better investigate undergraduate physics students understand the concept of angular momentum in particular and quantum mechanics courses in general.

Marcis P. Auzinsh

A standard description of the angular momentum in atomic or molecular physics is based on quantum mechanics. However, especially at large angular-momentum limit in molecules, sometimes a classical approach greatly simplifies problems that are extremely complicated from a quantum-mechanical viewpoint. The aim of this paper is to examine the relationship between the classical and quantum descriptions of the angular-momentum distribution in an ensemble of particles. At first glance, quantum and classical approaches appear to be absolutely different and have very little in common. In this paper this relationship is analyzed. It is shown that similarities can be found between the precession of angular momentum in the external field and the influence of this field upon the wave-function phase of a microparticle. The importance of wave function phases is stressed even for stationary conditions, and some examples of coherent superposition of quantum states are presented. The general method of how to pass from the quantum approach to a classical one is formulated. Results are visualized in the form of 3D distribution functions. This approach allows a graphical interpretation and thus results in a better understanding, sometimes at first glance, of counterintuitive quantum results. As an example the phenomenon of alignment- orientation transition is considered. Rather than presenting only formal derivations, the emphasis is on the instructional aspects, and on illustrating the general approach.

Randall B. Stafford, Michel Louis Lauzon, Mohammad Sabati +2 more

The purpose of this tutorial is to derive the precessional characteristics of the magnetic moments of hydrogen nuclei in the presence of a constant external magnetic field using the Dirac bra-ket formulation of quantum mechanics (QM). This behaviour has many applications, most notably in nuclear magnetic resonance (NMR) and magnetic resonance (MR) imaging. Many NMR and MR imaging textbooks claim that the QM expectation value of the magnetic moment of a proton in a magnetic field reduces to the classical picture of a precessing magnetic dipole. This paper validates this conclusion by reducing the cumbersome QM integrals using Dirac notation and matrix algebra, and then comparing this result with the classical picture. It illustrates the connections between the quantum and classical pictures and demonstrates quantitatively how they differ and how they are similar. This tutorial targets students and researchers interested in the fundamental physics behind the MR phenomenon and assumes that the reader has a basic understanding of QM. This may be helpful for undergraduate QM students learning about spin angular momentum and Dirac notation.