Student Understanding of Superposition: Vectors and Wave Functions

The ability to switch between various representations is an invaluable problem-solving skill in physics. In addition, research has shown that using multiple representations can greatly enhance a person's understanding of mathematical and physical concepts. This paper describes a study of student difficulties regarding interpreting, constructing, and switching between representations of vector fields, using both qualitative and quantitative methods. We first identified to what extent students are fluent with the use of field vector plots, field line diagrams, and symbolic expressions of vector fields by conducting individual student interviews and analyzing in-class student activities. Based on those findings, we designed the Vector Field Representations test, a free response assessment tool that has been given to 196 second-and third-year physics, mathematics, and engineering students from four different universities. From the obtained results we gained a comprehensive overview of typical errors that students make when switching between vector field representations. In addition, the study allowed us to determine the relative prevalence of the observed difficulties. Although the results varied greatly between institutions, a general trend revealed that many students struggle with vector addition, fail to recognize the field line density as an indication of the magnitude of the field, confuse characteristics of field lines and equipotential lines, and do not choose the appropriate coordinate system when writing out mathematical expressions of vector fields.

Section: Discussion and Implications For Instructionmentioning

confidence: 99%

Student difficulties regarding symbolic and graphical representations of vector fields

Bollen

Kampen

Baily

et al. 2017

“…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

Self 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.

“…So, the nature of the physics context seems to matter. Emigh et al [7], however, found that the types of incorrect reasoning students made were roughly similar for different contexts.…”

Section: Introductionmentioning

confidence: 96%

“…Previous research has shown that many incoming students at the university level lack basic knowledge of vectors that they need to tackle their physics subjects successfully [1]. In recent years, a number of studies explored students' difficulties with a large variety of vector concepts in detail [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Investigating student understanding of cross products in a mathematical and two electromagnetism contexts

Deprez

Gijsen

Deprez

et al. 2019

We investigated the influence of context on students' understanding of cross products of vectors using three isomorphic multiple-choice tests asking for the direction of a cross product in different geometrical settings. One version of the test involved the Lorentz force, the second version involved the torque on an electric dipole, and the third version was without physics context. We administered the tests to 216 first-year pre-med students at a Belgian (Flemish) university. We found that students perform significantly better in the context of the Lorentz force. Students more often chose the incorrect alternative corresponding to the vector sum in the test versions involving an electric dipole or without physics context when the vectors are not orthogonal. For orthogonal vectors, a sign error-i.e., inverting the direction of the resulting vector-was the most common mistake in both tests with physics context, while without physics context selecting the alternative corresponding to the sum remained the most common mistake. Prior familiarity with a right-hand rule in a specific context seems to be able to explain improved scores in the test version concerning the Lorentz force. Instructors and curriculum developers can benefit from adopting an integrated approach in which the mathematical aspects of the cross product are treated together with multiple examples in physics, allowing students to transition from using specific rules to determine a cross product, to a more integrated understanding of it.