Improving student understanding of corrections to the energy spectrum of the hydrogen atom for the Zeeman effect

Abstract

We discuss an investigation of student difficulties with the corrections to the energy spectrum of the hydrogen atom for the intermediate field Zeeman effect using degenerate perturbation theory (DPT). The investigation was carried out in advanced quantum mechanics courses by administering free-response and multiple-choice questions and conducting individual interviews with students. We find that students share many common difficulties related to relevant physics concepts. They had difficulty with mathematical sense making in this context of quantum mechanics, which requires the ability to interpret the implications of the degeneracy in the unperturbed energy spectrum and how the Zeeman perturbation will impact the splitting of the energy levels. Many of the common student difficulties arise from challenges in mathematical sense making and applying linear algebra concepts incorrectly in this novel context of quantum mechanics. We describe how the research on student difficulties was used as a guide to develop and evaluate a Quantum Interactive Learning Tutorial (QuILT), which strives to help students develop a functional understanding of the concepts necessary for finding the corrections to the energy spectrum of the hydrogen atom for the intermediate field Zeeman effect using the DPT. We also discuss the development and validation of the DPT QuILT focusing on these issues and its in-class evaluation in the undergraduate and graduate courses.

Challenges in addressing student difficulties with time-development of two-state quantum systems using a multiple-choice question sequence in virtual and in-person classes

“…For such difficulties as those described, researchvalidated learning tools can effectively help students develop a robust knowledge structure [19][20][21][22][23][24]. For example, quantum interactive learning tutorials (QuILTs) have been developed, validated and implemented with encouraging results on many topics in quantum mechanics, including quantum measurement of physical observables [25][26][27], addition of angular momentum [28], perturbation theory and corrections to the energy spectrum of the hydrogen atom [29][30][31][32], systems of identical particles [13,33], quantum key distribution [34], Larmor precession [35], as well as the uncertainty principle and Mach-Zehnder interferometer [36,37]. Similarly, clicker questions, first popularized by Eric Mazur using his Peer Instruction method, are conceptual multiple-choice questions presented to a class for students to answer anonymously, individually first and again after discussion with peers, and with immediate feedback [38].…”

Effectiveness of an online homework tutorial about changing basis in quantum mechanics

“…Quantum mechanics is an essential course for undergraduate physics majors, but the material is known to be conceptually challenging for students [13][14][15][16][17]. Research-based tutorialsguided-inquiry worksheets that assist students in constructing physics knowledge for themselves-have repeatedly proven to be a highly effective complement to lectures in physics courses, and are especially targeted at supporting students' conceptual understanding [10,[18][19][20][21]. Our collaboration has developed a collection of tutorials for upper-division quantum mechanics (QM) that has been adopted by numerous QM instructors nationwide [12].…”

Investigating student understanding of the stationary state wavefunction for a system of identical particles

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. We have been involved in a number of research studies aimed at investigating student reasoning in QM and improving student understanding of QM [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. However, there have been few investigations into student difficulties with fundamental concepts involving a system of identical particles.…”