Degenerate perturbation theory (DPT) is a powerful approximation method for finding the energies and the energy eigenstates for a system for which the time-independent Schrödinger equation is not exactly solvable and there is degeneracy in the unperturbed energy spectrum. However, many students struggle with DPT because they have difficulty identifying whether a given basis is a good basis for finding perturbative corrections and determining a good basis for a given system. Here, we first discuss an investigation of student difficulties with determining a good basis and finding the corrections to the energies and energy eigenstates in the context of DPT 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 this topic. We then describe how the research on student difficulties was used as a guide to develop and validate a quantum interactive learning tutorial (QuILT), which strives to help students develop a functional understanding of a good basis for finding the perturbative corrections in the context of DPT. We discuss the development and validation of the QuILT on DPT and its evaluation in undergraduate and graduate courses.
We discuss an investigation of student difficulties with degenerate perturbation theory (DPT) 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 this topic. We used the difficulties found via research as resources to develop and evaluate a Quantum Interactive Learning Tutorial (QuILT) which strives to help students develop a functional understanding of DPT. We discuss the development of the DPT QuILT and its preliminary evaluation in the undergraduate and graduate courses.
Understanding when and how to make limiting case approximations and why they are valid in a particular situation is a hallmark of expertise in physics. Using limiting cases can simplify the problem-solving process significantly and they often provide a means to check that the results obtained are reasonable. We discuss an investigation of student difficulties with the corrections to the energy spectrum of the hydrogen atom for the limiting cases of the strong and weak field Zeeman effects using degenerate perturbation theory. This investigation was carried out in advanced quantum mechanics courses by administering written free-response and multiple-choice questions and conducting individual interviews with students. Here we first discuss the common student difficulties related to these concepts. We then 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 strong field and weak field Zeeman effects. The development of the QuILT and its evaluation in the undergraduate and PhD level courses are presented.
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.
We discuss an investigation of student difficulties with whether a given total energy is possible for the system under given constraints, the possible outcomes of an energy measurement, and the probability of obtaining a particular energy if we randomly measured the energy of one particle and concepts involved in determining the number of distinct many-particle states in a system of noninteracting identical particles with a fixed total energy. 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 also discuss how the research was used as a guide to develop, validate, and evaluate a Quantum Interactive Learning Tutorial (QuILT), which strives to help students develop a functional understanding of the relevant concepts for a system of non-interacting identical particles with a fixed total energy. We discuss the development and validation of the identical particles QuILT and its evaluation in the undergraduate and graduate courses that focused on these issues.
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