February 2023 Issue,

Volume 91, No. 2

 

A wave packet approach to resonant scattering

Resonant transmission occurs when constructive interference results in the complete passage of an incoming wave through an array of barriers. In this paper, we explore such a scenario with one-dimensional models. We adopt wave packets with finite width to illustrate the deterioration of resonance with decreasing wave packet width and suggest an approximate wave function for the transmitted and reflected components, derived from aspects of both the wave packet and plane wave approaches. A comparison with exact numerical calculations shows excellent agreement and provides insight into the scattering process.

 

EDITORIAL

In this issue: February 2023 by John Essick, Adam Fritsch, Claire A. Marrache-Kikuchi, Beth Parks, B. Cameron Reed, Todd Springer and Donald Salisbury. DOI: American Journal of Physics 91, 85 (2023); https://doi.org/10.1119/5.0139314

BACK OF THE ENVELOPE

Invariants: Finding constancy in a sea of change by Sanjoy Mahajan. DOI: American Journal of Physics 91, 87 (2023); https://doi.org/10.1119/5.0139569
This Back-of-the-Envelope paper shows how invariants can be useful in estimations. Sometimes they even yield exact solutions!

PAPERS

Low-energy scattering parameters: A theoretical derivation of the effective range and scattering length for arbitrary angular momentum by Jordi Pera and Jordi Boronat. DOI: American Journal of Physics 91, 90 (2023); https://doi.org/10.1119/5.0079744
While scattering theory is commonly studied by upper-level physics students, how it is presented and studied can vary widely. This paper presents derivations for the angular-momentum scattering parameters of four different model potentials: hard-sphere, soft-sphere, spherical well, and well-barrier potentials. Working through these derivations could be beneficial for students as they learn scattering theory basics and apply their knowledge to various cases such as the study of ultracold quantum gases.

A wave packet approach to resonant scattering by A. M. Michalik and F. Marsiglio. DOI: American Journal of Physics 91, 102 (2023); https://doi.org/10.1119/5.0106701
Most treatments of scattering theory in upper-level quantum mechanics classes use time-independent plane-wave formulations, but this limits qualitative understanding of the details of the scattering process, particularly in cases of resonant scattering where constructive interference leads to the complete passage of an incoming wave through an array of barriers. This paper offers deeper insight into the scattering process by modeling incoming particles as time-dependent Gaussian wave packets which retain their shape while propagating, an approach which reveals that the reflected wave packet at resonance is in fact non-zero and is no longer Gaussian. Rather, the reflected wave packet, while being of very small amplitude, has a surprising double-Gaussian shape. Appropriate for upper-level and graduate quantum mechanics classes.

“A call to action”: Schrödinger's representation of quantum mechanics via Hamilton's principle by Michele Marrocco. DOI: American Journal of Physics 91, 110 (2023); https://doi.org/10.1119/5.0083015
Quantum mechanics courses often present the Schrödinger equation as a fundamental postulate. This approach can leave students disoriented without support from their knowledge gained in earlier courses. This article reviews some other common ways to introduce the Schrödinger equation and develops its own pedagogical approach (which has its roots in the foundations of quantum mechanics) based on classical Hamilton-Jacobi theory. This way of motivating the Schrödinger equation may be less jarring to students who have knowledge of the action principle in classical mechanics, and anyone interested in the relationship between classical and quantum physics will find this article to be an interesting read.

Science on a stick: An experimental and demonstration platform for learning several physical principles by Dhananjay V. Gadre, Harsh Sharma, Sangeeta D. Gadre and Smriti Srivastava. DOI: American Journal of Physics 91, 116 (2023); https://doi.org/10.1119/5.0080881
The authors give detailed instructions for building a simple apparatus that can be used in the laboratory or at home to study both mechanics (gravitational acceleration) and electromagnetism (induced currents).

Thermal infrared astronomy for the introductory laboratory by Clifford W. Padgett, William H. Baird, J. Spencer Coile, Wayne M. Johnson, Erin N. Groneck and Robert A. Rose. DOI: American Journal of Physics 91, 122 (2023); https://doi.org/10.1119/5.0081072
Although the electromagnetic spectrum is wide, labs often restrict themselves to the study of visible light or, in more advanced courses, microwaves. But thermal infrared cameras can nowadays be purchased at a minimal cost to investigate thermal leaks in houses. So why not place them at a telescope’s eyepiece, and observe the Moon and other celestial bodies in the infrared range? Students can verify that the laws of refraction are still valid, while realizing that the images that they acquire can be drastically different from what they obtain in the visible range. From there, thermal properties of various bodies can be investigated.

INSTRUCTIONAL LABORATORIES AND DEMONSTRATIONS

Low-cost quadrature optical interferometer by Tanner M. Melody, Krishna H. Patel, Peter K. Nguyen and Christopher L. Smallwood. DOI: American Journal of Physics 91, 132 (2023); https://doi.org/10.1119/5.0110405
This paper presents a Mach-Zehnder interferometer designed to illustrate the qualitative functionality of polarization-based quadrature interferometry at a fraction of the cost of commercially available alternatives. Background theory on quadrature-detected interferometry is given, followed by details on the construction of the low-cost instrument comprised of a laser pointer, affordable polarization optics, 3-D-printed optics mounts, homebuilt photodetectors, and low-cost microcontroller-based analog-to-digital signal conversion. Quality of performance of the system is exhibited by plots of detector outputs and a thermal expansion coefficient measurement. This work provides an accessible and affordable introduction to the topic of quadrature interferometry for optics instructional laboratory instructors and students.

Picometer measurements of strain coefficients by quadrature interferometry and lock-in amplification by Alec Nilson and Kurt Wick. DOI: American Journal of Physics 91, 142 (2023); https://doi.org/10.1119/5.0102091
The authors were motivated by LIGO to ask how accurately they could measure distances in an advanced undergraduate lab. The answer? Using quadrature interferometry, they were able to measure displacements with an uncertainty of λ/150,000. This manuscript explains the technique, starting with a simple model and gradually adding more realistic complications, and also provides details for how to implement a similar laboratory project.

ADVANCED TOPICS

Still learning about space dimensionality: From the description of hydrogen atom by a generalized wave equation for dimensions D ≥ 3 by Francisco Caruso, Vitor Oguri and Felipe Silveira. DOI: American Journal of Physics 91, 153 (2023); https://doi.org/10.1119/5.0058930
A glimpse into the history and current status of explorations of extra spatial dimensions helps to place in context this investigation of a Schrödinger-like equation for a hydrogen atom in higher dimensions. One significant alteration is the appearance of the Laplacian operator at higher integer powers, with surprising restrictions that lead to bound states.

BOOK REVIEWS

On the Trail of Blackbody Radiation: Max Planck and the Physics of his Era by Don S. Lemons, William R. Shanahan, and Louis J. Buchholtz Peter W. Milonni, Reviewer American Journal of Physics 91, 159 (2023); https://doi.org/10.1119/5.0137737

 

Additional Resources