Student difficulties with complex numbers

Smith, Emily M.; Zwolak, Justyna P.; Manogue, Corinne A.

doi:10.1119/perc.2015.pr.073

Cited by 4 publications

(7 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…About 35% (N ¼ 83) gave correct numerical results for both modulus squared, and 17% made errors in computing j 3 5 þ i 4 5 j 2 . The most salient error appears to be treating modulus squared (e.g., j 3 5 þ i 4 5 j 2 ) as squared [e.g., ð 3 5 þ i 4 5 Þ 2 ], which has been documented in prior research [35][36][37].…”

Section: Resultsmentioning

confidence: 99%

Probing student reasoning in relating relative phase and quantum phenomena

Wan

Emigh

Shaffer

2019

Phys. Rev. Phys. Educ. Res.

View full text Add to dashboard Cite

In quantum mechanics, probability amplitudes are complex numbers and the relative phases between the terms in superposition states have measurable effects. This article describes an investigation into sophomore-and junior-level students' reasoning patterns in relating relative phases and real-world quantum phenomena. The investigation involved one observational experiment and three testing experiments, during which we formulated and tested three hypotheses that allow us to gain insights into why students have difficulty recognizing the measurable effects of relative phases. We found that, in both spin-1=2 and infinite square well contexts, many students do not recognize that quantum states differing only by a relative phase are experimentally distinguishable. Moreover, student ability to recognize the measurable effects of relative phase does not improve when given (i) a task that specifically prompts students to compare the probabilities for a particular observable and (ii) a task that does not require taking inner products or changing basis. We also examined the extent to which lacking proficiency with complex numbers may have hindered student understanding of relative phase. The data indicate that most students are proficient with complex numbers. These findings suggest that many students do not, in fact, recognize the purpose of using complex numbers in superposition states. We discuss possible explanations for why students do not seem to recognize this purpose, and we also provide suggestions for future avenues of research.

show abstract

Section: Resultsmentioning

confidence: 99%

Probing student reasoning in relating relative phase and quantum phenomena

Wan

Emigh

Shaffer

2019

Phys. Rev. Phys. Educ. Res.

View full text Add to dashboard Cite

show abstract

“…Even simple arithmetic involving complex numbers has been found to be difficult for undergraduates. 48,49 On the other hand, Wawro et al 50 have published promising research showing student abilities to critique and understand the purposes of linear algebra and Dirac notations in quantum mechanics and demonstrated students flexibility in reasoning about linear algebra and quantum concepts; this indicates that students should be cognitively able to work with abstract operator algebra.…”

Section: Physics Education Knowledge Base For Teaching Quantum Mechanicsmentioning

confidence: 99%

Should we trade off higher-level mathematics for abstraction to improve student understanding of quantum mechanics?

Freericks

Doughty

2023

Seventeenth Conference on Education and Training in Optics and Photonics: ETOP 2023

View full text Add to dashboard Cite

Undergraduate quantum mechanics focuses on teaching through a wavefunction approach in the position-space representation. This leads to a differential equation perspective for teaching the material. However, we know that abstract representation-independent approaches often work better with students, by comparing student reactions to learning the series solution of the harmonic oscillator versus the abstract operator method. Because one can teach all of the solvable quantum problems using a similar abstract method, it brings up the question, which is likely to lead to a better student understanding? In work at Georgetown University and with edX, we have been teaching a class focused on an operator-forward viewpoint, which we like to call operator mechanics. It teaches quantum mechanics in a representation-independent fashion and allows for most of the math to be algebraic, rather than based on differential equations. It relies on four fundamental operator identities-(i) the Leibniz rule for commutators; (ii) the Hadamard lemma; (iii) the Baker-Campbell-Hausdorff formula; and (iv) the exponential disentangling identity. These identities allow one to solve eigenvalues, eigenstates and wavefunctions for all analytically solvable problems (including some not often included in undergraduate curricula, such as the Morse potential or the Pöschl-Teller potential). It also allows for more advanced concepts relevant for quantum sensing, such as squeezed states, to be introduced in a simpler format than is conventionally done. In this paper, we illustrate the three approaches of matrix mechanics, wave mechanics, and operator mechanics, we show how one organizes a class in this new format, we summarize the experiences we have had with teaching quantum mechanics in this fashion and we describe how it allows us to focus the quantum curriculum on more modern 21st century topics appropriate for the second quantum revolution.

show abstract

“…Many undergraduate engineering students' mathematical content knowledge and arithmetic skills of complex number-related topics often fall short. Furthermore, many studies at various educational levels reveal that students need help with conceptual and procedural knowledge of complex numbers (Ahmad & Shahrill, 2012;Conner et al, 2007;Smith et al, 2015). A study by Hui and Lam (2013) revealed that many students need clarification on geometrical and algebraic representations of complex numbers.…”

Section: Introductionmentioning

confidence: 99%

Developing undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers using GeoGebra

Seloane,

Ramaila,

Ndlovu

2023

PYTHAGORAS

View full text Add to dashboard Cite

This study explored the utilisation of GeoGebra as a modelling tool to develop undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers. This mission was accomplished by implementing GeoGebra-enriched activities, which provided carefully designed representational support to mediate between students’ initially developed conceptual and procedural knowledge gains. The rectangular and polar forms of the complex number were connected and merged using GeoGebra’s computer algebra systems and dynamic geometric systems platforms. Despite the centrality of complex numbers to the undergraduate mathematics curriculum, students tend to experience conceptual and procedural obstacles in mathematics-dependent physics engineering topics such as mechanical vector analysis and electric-circuit theory. The study adopted an exploratory sequential mixed methods design and involved purposively selected first-year engineering mathematics students at a South African university. The constructivist approach and Realistic Mathematical Education underpinned the empirical investigation. Data were collected from students’ scripts. Implementing GeoGebra-enriched activities and providing carefully designed representational support sought to enhance students’ conceptual and procedural knowledge of complex numbers and problem representational competence. The intervention additionally helped students to conceptualise and visualise a complex rectangular number. Implications for technology-enhanced pedagogy are discussed.Contribution: The article provides exploratory insights into the development of undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers using GeoGebra as a dynamic digital tool. Key findings from the study demonstrated that GeoGebra appears to be an effective modelling tool that can be harnessed to demystify the complexity of mathematics students’ conceptual and procedural knowledge of complex numbers.

show abstract

Student difficulties with complex numbers

Cited by 4 publications

References 7 publications

Probing student reasoning in relating relative phase and quantum phenomena

Probing student reasoning in relating relative phase and quantum phenomena

Should we trade off higher-level mathematics for abstraction to improve student understanding of quantum mechanics?

Developing undergraduate engineering mathematics students’ conceptual and procedural knowledge of complex numbers using GeoGebra

Contact Info

Product

Resources

About