American Journal of Physics, November 2023November 2023 Issue,

Volume 91, No. 11

Computational projects with the Landau–Zener problem in the quantum mechanics classroom

The Landau–Zener problem, where a minimum energy separation is passed with constant rate in a two-state quantum-mechanical system, is an excellent model quantum system for a computational project. It requires a low-level computational effort, but has a number of complex numerical and algorithmic issues that can be resolved through dedicated work. It can be used to teach computational concepts, such as accuracy, discretization, and extrapolation, and it reinforces quantum concepts of time-evolution via a time-ordered product and of extrapolation to infinite time via time-dependent perturbation theory. In addition, we discuss the concept of compression algorithms, which are employed in many advanced quantum computing strategies, and easy to illustrate with the Landau–Zener problem.



In this issue: November 2023 by John Essick; Jesse Kinder; Beth Parks; Donald Salisbury; Todd Springer; Keith Zengel. DOI: 10.1119/5.0177701


The first Global e-Competition on Astronomy and Astrophysics by Ioana A. Zelko; Charles Barclay; Tõnis Eenmäe; Taavet Kalda; Hara Papathanassiou; Nikita Poljakov; Gustavo A. Rojas; Tiit Sepp; Greg Stachowski; Aniket Sule. DOI: 10.1119/5.0121242
The 2020 Global e-Competition in Astronomy and Astrophysics was an international Olympiad that successfully moved to an online format with little time to prepare. In this article, the organizers share their experience, which illustrates successful strategies — and pitfalls to avoid — for common challenges of hybrid and online teaching: administering and grading online exams (in multiple languages across many time zones), developing engaging group projects for remote students, and cultivating student interest through challenging assignments. The resources developed for the competition and described here are freely available for use in your own classroom and should also help other faculty who run physics or astronomy competitions.

Coupled oscillations of the Wilberforce pendulum unveiled by smartphones by Thomas Gallot; Daniel Gau; Rodrigo García-Tejera. DOI: 10.1119/5.0138680
The Wilberforce pendulum is a classic physics demonstration involving a spring-mass system with coupled torsional and longitudinal oscillations. Here, the authors discuss the appropriate theoretical and experimental modifications necessary for exploring the system with smartphone sensors. This approach eliminates the traditional need for multiple sensors and results in acceleration-based measurements that differ from the typical displacement-based measurements.

Oblique angle collisions between three or more billiard balls by Rod Cross. DOI: 10.1119/5.0119656
A classic introductory physics problem involves the study of a one-dimensional elastic collision between a moving object and another, initially stationary, object. The final state can be completely determined because the two unknown final velocities can be found from two conservation equations (energy and momentum). Now, imagine the more complicated problem where a moving ball collides off center with two stationary balls (initially touching). In this situation, the number of conservation equations is not sufficient to completely determine the outcome. This article addresses these complications in a way that can be shared with undergraduate students, and collisions involving three and four balls are studied experimentally. Students and instructors will delight in seeing how a slight modification to an elementary problem results in rich physics.

Computational projects with the Landau–Zener problem in the quantum mechanics classroom by Livia A. J. Guttieres; Marko D. Petrović; James K. Freericks. DOI: 10.1119/5.0139717
The authors present a highly adaptable computational project suitable for quantum mechanics courses. Teachers who want to reinforce concepts like time evolution and time-independent perturbation theory, who want their students to engage in computational research style thinking, or who want to add cutting edge concepts like compression algorithms to their curricula will all find useful suggestions in this paper.

Galilean relativity and the path integral formalism in quantum mechanics by Charles Torre. DOI: 10.1119/5.0140018
Under a fixed boost to a new inertial frame of reference, the usual non-relativistic quantum mechanical wave function undergoes a change of phase. The manner in which this occurs can provide additional insight, in advanced undergraduate quantum mechanics and mathematical methods courses, on the nature of symmetries. The author shows that this transformation property is related to a time derivative term that arises when the corresponding classical Lagrangian is transformed. It turns out that one can indeed derive the change of phase by employing this alteration of the Lagrangian in a path integral approach to quantum mechanics.

S-matrices for simple quantum systems by Leo de Wit. DOI: 10.1119/5.0078607
Feynman diagrams are both attractive visualizations of physical processes and powerful calculational tools used in quantum field theory, a subject traditionally taught at the graduate level. Because they are employed at the forefront of research, Feynman diagrams tend to inspire intense curiosity among undergraduate students. Undergraduates can easily learn recipes to use these diagrams, but a deeper understanding arises by learning where the diagrams come from and understanding their inner workings. In this paper, a few toy models are introduced that illustrate essential features of field theory and the construction of Feynman diagrams. We hope students and instructors will enjoy this pedagogical introduction to field theory scattering in a simplified context.


Electric field lines of an arbitrarily moving charged particle by S. G. Arutunian; M. A. Aginian; A. V. Margaryan; M. Chung; E. G. Lazareva. DOI: 10.1119/5.0124544
In this article, the authors derive a set of linear differential equations for constructing electric field lines of a charged particle in arbitrary motion. In some cases, these equations yield simple analytic solutions and geometric constructions. The authors construct field lines for a linearly oscillating charge, synchrotron motion, and the “figure eight” motion of a charged particle moving under the influence of an electromagnetic plane wave, including highly relativistic motion in strong fields. Readers can generate their own field lines using a script in the supplementary online material.

Leading quantum correction to the classical free energy by Markus Deserno; O. Teoman Turgut. DOI: 10.1119/5.0106687
How different is a thermal quantum system from a classical one? This article delves into the fascinating realm of quantum statistical mechanics and examines the leading-order quantum correction to the classical free energy. A new derivation of a famous formula (due to Wigner) is provided, and results are then illustrated using several familiar systems (the harmonic oscillator and the particle in a box). Students will find this article aids in building intuition about the differences between classical and quantum systems. Instructors may be inspired to assign extensions or generalizations of these results as student projects.

Force on a moving object in an ideal quantum gas by Wittaya Kanchanapusakit; Pattarapon Tanalikhit. DOI: 10.1119/5.0127334
An object moving through fluid experiences drag force, and treating this drag force is a standard part of classical fluid dynamics. But what if the motion must be treated quantum mechanically? This paper derives the temperature dependence of the drag force for classical gases, Fermi gases, and Bose gases.


Multiplicity counting using organic scintillators to distinguish neutron sources: An advanced teaching laboratory by Flynn B. Darby; Michael Y. Hua; Oskari V. Pakari; Shaun D. Clarke; Sara A. Pozzi. DOI: 10.1119/5.0139531
A versatile experimental setup for carrying out a modern nuclear assay technique is presented. The experimental setup is described in detail, along with the nondestructive assay methods it makes possible. In addition, abundant references are given to fill in background knowledge and encourage further study. As demonstrated in the experiment, these assay methods provide the ability to distinguish special nuclear materials (such as plutonium and enriched uranium) from other neutron sources. Such comparisons can help inspectors spot clandestine attempts to “relocate” fission sources by replacing them with sources that undergo (alpha, neutron) reactions. This paper will be of interest to instructional laboratory instructors wishing to adopt an up-to-date nuclear physics experiment as well as to those looking to learn more about nuclear nonproliferation techniques.


An alternative derivation of propagator for a linear potential by Xi-Jun Ren. DOI: 10.1119/5.0103857
In classical mechanics, a particle experiencing a constant force (a linear potential) is both simple to analyze and fundamentally important. The corresponding quantum problem is surprisingly difficult and has been approached in a variety of ways in the literature. This short note employs momentum space to present a clear and straightforward derivation of the propagator and serves as a complement to other approaches. We hope that readers will enjoy a heightened understanding after examining the problem through this alternate lens.


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