Review of student difficulties in upper-level quantum mechanics

Abstract

Learning advanced physics, in general, is challenging not only due to the increased mathematical sophistication but also because one must continue to build on all of the prior knowledge acquired at the introductory and intermediate levels.In addition, learning quantum mechanics can be especially challenging because the paradigms of classical mechanics and quantum mechanics are very different. Here, we review research on student reasoning difficulties in learning upper-level quantum mechanics and research on students' problem-solving and metacognitive skills in these courses. Some of these studies were multi-university investigations. The investigations suggest that there is large diversity in student performance in upper-level quantum mechanics regardless of the university, textbook, or instructor, and many students in these courses have not acquired a functional understanding of the fundamental concepts. The nature of reasoning difficulties in learning quantum mechanics is analogous to reasoning difficulties found via research in introductory physics courses. The reasoning difficulties were often due to over-generalizations of concepts learned in one context to another context where they are not directly applicable. Reasoning difficulties in distinguishing between closely related concepts and in making sense of the formalism of quantum mechanics were common. We conclude with a brief summary of the research-based approaches that take advantage of research on student difficulties in order to improve teaching and learning of quantum mechanics.Students taking upper-level quantum mechanics often develop survival strategies for performing reasonably well in their course work. For example, they become proficient at solving algorithmic problems such as the time-independent Schrödinger equation with a complicated potential energy and boundary conditions. However, research suggests that they often struggle to make sense of the material and build a robust knowledge structure. They have difficulty mastering concepts and applying the formalism to answer qualitative questions, e.g., questions related to the properties of Review of Student Difficulties in Upper-Level Quantum Mechanics

Investigating and improving student understanding of the probability distributions for measuring physical observables in quantum mechanics

“…While this study focused on student understanding of probability distribution of measurement outcomes, future investigations will focus on other equally challenging topics in quantum mechanics and strategies to help students learn those concepts better. For example, students also have difficulties with topics such as possible outcomes of measurements, expectation values, and time development of quantum states and time dependence of expectation values [7]. Furthermore, in the study described here, we found that students had difficulties both with Dirac notation and position and momentum representations in the context of probability distributions for measurement outcomes.…”

“…research has shown that students often think that the operator should appear in the expression for calculating probability of measuring a particular eigenvalue of the operator [6,7]. Some interviewed students who incorrectly claimed that the operator should appear in their expression for calculating probability made conceptual arguments claiming that, in quantum mechanics, an operator acting on a generic state must yield an eigenvalue of the corresponding operator.…”

“…) [6,7]. Interviews suggest that the fact that few students were able to recognize the correct expression for the probability of measuring energy E 1 in questions Q1 and Q2 may partly be due to the fact that they thought that the expression for probability of measuring energy E 1 should include the Hamiltonian operator Ĥ .…”

“…They recognized that the probability and expectation value were different, but they still struggled to distinguish between these concepts. They could not write an expression for the probability of measuring E n either using the Dirac notation or in the position representation [6,7]. Difficulties of this type indicate that students often struggle to differentiate between the concepts of probability and expectation value which can lead to attempts to apply concepts in inappropriate situations.…”

“…Therefore, it is important for upper-level undergraduate and PhD students to develop proficiency with the probability distributions of measurement outcomes when different physical observables are measured in a given quantum state of the system. Indeed, several prior studies have found that many advanced students struggle with the foundational issues in quantum mechanics including measurement (e.g., see [1][2][3][4][5][6][7][8][9][10][11][12]). Studies have focused on diverse pedagogical approaches for helping students learn quantum mechanics better (e.g., see [13][14][15][16][17][18][19][20][21][22][23][24][25][26]), and visualization tools such as QuVIS and QuTIP have been developed to help students develop intuition about quantum mechanical phenomena [23,24].…”