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American Journal of Physics

Pedagogical aspects of time-dependent rotation operators

C. Leubner

Published Jul 1, 1980• 1 citation
Publisher

Abstract

For tutorial purposes, the simple problem of a classical magnetic moment interacting with various magnetic field configuration is reformulated in terms of coordinate-free, time-dependent rotation operators. In this approach, the corresponding equation of motion assumes the form of an operator equation, identical with the Schrödinger equation of a spin-1 interacting with the same field configurations. It is solved utilizing deliberately the concepts of a Hermitian Hamiltonian and a unitary time evolution operator to familiarize students early in their curriculum with these important ingredients of the quantum-mechanical formalism. It turns out that besides furnishing useful exercises for the manipulation of three-dimensional rotation operators, being an end in itself, the approach also provides simple but nontrivial examples for a number of theorems for which the students will have frequent use in their quantum-mechanics course.

DOI
10.1119/1.12063

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For tutorial purposes, the simple problem of a classical magnetic moment interacting with various magnetic field configuration is reformulated in terms of coordinate-free, time-dependent rotation operators. In this approach, the corresponding equation of motion assumes the form of an operator equation, identical with the Schrödinger equation of a spin-1 interacting with the same field configurations. It is solved utilizing deliberately the concepts of a Hermitian Hamiltonian and a unitary time evolution operator to familiarize students early in their curriculum with these important ingredients of the quantum-mechanical formalism. It turns out that besides furnishing useful exercises for the manipulation of three-dimensional rotation operators, being an end in itself, the approach also provides simple but nontrivial examples for a number of theorems for which the students will have frequent use in their quantum-mechanics course.
For pedagogical purposes it is demonstrated that with little effort the customary discussion of rotations in introductory courses can be extended to naturally produce a number of angular momentum operator properties, if coordinate-free rotation operators are used in their exponential form. This form gives rise to a particularly useful coordinate-free representation of the generators of rotations in three-space, from which the student can derive the usual angular momentum formalism for the case j=1 by elementary vector algebra alone. Since in this approach the j=1 formalism appears as a set of relations among simple geometric objects in ordinary space, it admits of a geometric interpretation and will leave the student better prepared for a more profound appreciation of the abstract concepts of the full theory.
This concise textbook introduces an innovative computational approach to quantum mechanics. Over the course of this engaging and informal book, students are encouraged to take an active role in learning key concepts by working through practical exercises. The book equips readers with some basic methodology and a toolbox of scientific computing methods, so they can use code to simulate and directly visualize how quantum particles behave. The important foundational elements of the wave function and the Schrödinger equation are first introduced, then the text gradually builds up to advanced topics including relativistic, open, and non-Hermitian quantum physics. This book assumes familiarity with basic mathematics and numerical methods, and can be used to support a two-semester advanced undergraduate course. Source code and solutions for every book exercise involving numerical implementation are provided in Python and MATLAB®, along with supplementary data. Additional problems are provided online for instructor use with locked solutions.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
This concise textbook introduces an innovative computational approach to quantum mechanics. Over the course of this engaging and informal book, students are encouraged to take an active role in learning key concepts by working through practical exercises. The book equips readers with some basic methodology and a toolbox of scientific computing methods, so they can use code to simulate and directly visualize how quantum particles behave. The important foundational elements of the wave function and the Schrödinger equation are first introduced, then the text gradually builds up to advanced topics including relativistic, open, and non-Hermitian quantum physics. This book assumes familiarity with basic mathematics and numerical methods, and can be used to support a two-semester advanced undergraduate course. Source code and solutions for every book exercise involving numerical implementation are provided in Python and MATLAB®, along with supplementary data. Additional problems are provided online for instructor use with locked solutions.