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Physical Review Physics Education Research

Investigating how students relate inner products and quantum probabilities

Tong Wan, Paul J. Emigh, Peter S. Shaffer

Published Mar 12, 2019• 26 citations
Publisher

Abstract

The Born rule, which describes the formalism for determining probabilities, is one of the most fundamental postulates in quantum mechanics.This paper presents results from an investigation into how students apply the Born rule to determine probabilities for energy and position measurements.The investigation includes two stages with independent methods: a quantitative analysis of student written work and a qualitative analysis of student individual interviews.The data from written tasks suggest that after instruction many students have not developed a coherent model for determining probabilities that they can apply to observables regardless of whether the eigenvalues are discrete or continuous.Moreover, many students seem to lack a functional understanding of quantum states and inner products that allows them to translate between Dirac notation and wave function representation.These results motivate student interviews, which allow us to probe student reasoning in depth.Prior research suggests that various features of each notation used in quantum mechanics may have an impact on how students perform computations.We postulate that the features of quantum notations may also interact with student sensemaking.Therefore, we analyze student interviews through the lens of the structural features of quantum notations framework.In particular, we discuss how different structural features may facilitate or hinder student sensemaking about concepts relevant to determining probabilities.The results from both quantitative and qualitative data suggest that unsuccessfully differentiating between a wave function and its associated state vector in Dirac notation may be a primary barrier for students to develop a model for determining probabilities for discrete and continuous cases.

DOI
10.1103/physrevphyseducres.15.010117

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The Born rule, which describes the formalism for determining probabilities, is one of the most fundamental postulates in quantum mechanics.This paper presents results from an investigation into how students apply the Born rule to determine probabilities for energy and position measurements.The investigation includes two stages with independent methods: a quantitative analysis of student written work and a qualitative analysis of student individual interviews.The data from written tasks suggest that after instruction many students have not developed a coherent model for determining probabilities that they can apply to observables regardless of whether the eigenvalues are discrete or continuous.Moreover, many students seem to lack a functional understanding of quantum states and inner products that allows them to translate between Dirac notation and wave function representation.These results motivate student interviews, which allow us to probe student reasoning in depth.Prior research suggests that various features of each notation used in quantum mechanics may have an impact on how students perform computations.We postulate that the features of quantum notations may also interact with student sensemaking.Therefore, we analyze student interviews through the lens of the structural features of quantum notations framework.In particular, we discuss how different structural features may facilitate or hinder student sensemaking about concepts relevant to determining probabilities.The results from both quantitative and qualitative data suggest that unsuccessfully differentiating between a wave function and its associated state vector in Dirac notation may be a primary barrier for students to develop a model for determining probabilities for discrete and continuous cases.
A solid grasp of the probability distributions for measuring physical observables is central to connecting the quantum formalism to measurements. However, students often struggle with the probability distributions of measurement outcomes for an observable and have difficulty expressing this concept in different representations. Here we first describe the difficulties that upper-level undergraduate and PhD students have with the probability distributions for measuring physical observables in quantum mechanics. We then discuss how student difficulties found in written surveys and individual interviews were used as a guide in the development of a quantum interactive learning tutorial (QuILT) to help students develop a good grasp of the probability distributions of measurement outcomes for physical observables. The QuILT strives to help students become proficient in expressing the probability distributions for the measurement of physical observables in Dirac notation and in the position representation and be able to convert from Dirac notation to position representation and vice versa. We describe the development and evaluation of the QuILT and findings about the effectiveness of the QuILT from in-class evaluations.
Dirac notation is often used in upper level quantum mechanics courses, but students struggle with this representation.To investigate the difficulties that advanced students (i.e., upper-level undergraduate and graduate students) have while translating state vectors in Dirac notation to wave functions in position and momentum representations, we administered free-response and multiple-choice questions and conducted individual interviews with students.We find that students display common difficulties with these topics.
This paper is part of the Focused Collection on Upper Division Physics Courses.]The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation.The varied features of these representations affect how computations are performed.For example, identifying probabilities of measurement outcomes for a state described in Dirac notation may involve identifying expansion coefficients by inspection, but if the state is described as a function, identifying those expansion coefficients often involves performing integrals.In this study, we focus on three notational systems: Dirac notation, algebraic wavefunction notation, and matrix notation.These quantum notations must include information about basis states and their associated complex probability amplitudes.In this theory paper, we identify four structural features of quantum notations, which we term individuation, degree of externalization, compactness, and symbolic support for computational rules.We illustrate how student reasoning interacts with these structural features with episodes from interviews with advanced undergraduate physics majors reasoning about a superposition state of an infinite square well system.We find evidence of the students coordinating different notations through the use of Dirac notation, using an expression in Dirac notation to guide their work in another notation.These uses are supported by the high degree of individuation, compactness, and symbolic support for computation and the moderate degree of externalization provided by Dirac notation.
Abstract Learning quantum mechanics (QM) is widely recognized as a challenging task, as it involves highly abstract concepts and requires a significant shift from classical physics reasoning. Developing expertise in QM requires students to represent physical concepts using multiple formalisms, such as Dirac notation and algebraic representations, and to move flexibly between them. This study investigates two main categories of student difficulties related to Dirac notation. The first concerns difficulties in expressing a generic state vector in the position and momentum representations. The second relates to challenges in writing the eigenvectors of the position and momentum operators in Dirac notation. To explore these issues, we administered questionnaires to third-year undergraduate physics students enrolled in a QM course at the Lyon 1 University. We present a detailed analysis of student responses, identify the principal difficulties encountered, and compare our findings with results reported in prior American studies. Our results show that many students struggle to establish a clear connection between Dirac notation and its functional algebraic representation. In particular, students find it easier to recognize state vectors and eigenvectors than to construct them explicitly. Additionally, common difficulties include confusion between bras (e.g. x | and p |) and the corresponding operators, as well as confusion between the Dirac delta distribution and the Kronecker delta. Finally, we propose some interpretations of our results in light of those reported in previous studies.
Quantum uncertainty is a fundamental concept in quantum mechanics, but challenging for students to master.In this article, we describe student difficulties with visual and conceptual understanding of quantum uncertainty in the context of discrete probability distributions such as those for a spin 1/2 particle.We collected written responses from students at two institutions to a homework activity focusing on uncertainty of spin measurement outcomes, as well as written responses to a test question from one of the institutions.We also conducted interviews with six students to gain further insight into difficulties found.Common incorrect ideas found included a depiction of uncertainty as the error around each of the individual measurement outcomes, not depicting the uncertainty region from the expectation value outwards, and the idea that quantum uncertainty of an observable can never be zero.These ideas may indicate a confusion between quantum uncertainty and errors due to instrumental imperfections of the measurement apparatus, a lack of conceptual understanding of quantum uncertainty as the standard deviation of the probability distribution with respect to its mean, and an incorrect interpretation of the uncertainty relation between two incompatible observables to deduce that quantum uncertainty can never be zero.The results of this study show the importance of supporting students in visual and conceptual understanding of quantum uncertainty.
As part of an effort to examine student understanding of expressions for probability in an upper-division spins-first quantum mechanics (QM) context, clinical think-aloud interviews were conducted with students following relevant instruction.Students were given various tasks to showcase their conceptual understanding of the mathematics and physics underpinning these expressions.The symbolic forms framework was used as a lens through which to analyze their understanding.Various symbol templates and conceptual schemata were identified, in Dirac and function notations, with multiple schemata paired with different templates.The overlapping linking suggests that defining strict template-schema pairs may not be feasible or productive for studying student interpretations of expressions for probability in upper-division QM courses.