Writing Position Vectors in 3-d Space: A Student Difficulty With Spherical Unit Vectors in Intermediate E&amp;M

Hinrichs, Brant E.; Singh, Chandralekha; Sabella, Mel; Rebello, Sanjay

doi:10.1063/1.3515190

Cited by 14 publications

(24 citation statements)

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“…Nonetheless, here we show that even without such a procedure, but by explicitly including the seldom considered Landau coefficient a 6 into the analysis, one is able to extract critical exponents for the 2D Bose-Hubbard system which agree to better than 1% with those computed for the lambda transition [6], thus providing fair evidence in favor of universality. Our ad hoc procedure still requires formal justification and hence should be regarded as preliminary, but quite similar results are obtained by applying variational perturbation theory [19]. Some conclusions are drawn in the final Sec.…”

Section: Introductionmentioning

confidence: 60%

“…Yet, our findings still have to be regarded as preliminary. Subsequent steps to be taken now should involve a more systematic processing of the perturbative data, combined with an improved fitting procedure and a reliable error estimate, and it will be important to answer the question whether the encouraging first results reported here can be made more precise [19]. [6].…”

Section: Discussion and Outlookmentioning

confidence: 99%

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Perturbative calculation of critical exponents for the Bose–Hubbard model

2013

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We develop a strategy for calculating critical exponents for the Mott insulator-to-superfluid transition shown by the Bose-Hubbard model. Our approach is based on the field-theoretic concept of the effective potential, which provides a natural extension of the Landau theory of phase transitions to quantum critical phenomena. The coefficients of the Landau expansion of that effective potential are obtained by high-order perturbation theory. We counteract the divergency of the weak-coupling perturbation series by including the seldom considered Landau coefficient a6 into our analysis. Our preliminary results indicate that the critical exponents for both the condensate density and the superfluid density, as derived from the two-dimensional Bose-Hubbard model, deviate by less than 1% from the best known estimates computed so far for the three-dimensional XY universality class.

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Section: Introductionmentioning

confidence: 60%

Section: Discussion and Outlookmentioning

confidence: 99%

Perturbative calculation of critical exponents for the Bose–Hubbard model

2013

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“…Previous studies of students' understanding of vector concepts can be clustered into three groups: (1) those that analyze their understanding of vector concepts in problems without a physical context [14][15][16][17][18][19][20][21][22][23][24][25][26][27], (2) studies that investigate their understanding in problems with a physical context [16][17][18][19]24,[27][28][29][30][31][32], and (3) studies that compare students' performance on both types of problems, with and without a physical context [17][18][19]24,27,31]. Note that some of these studies pertain to more than one group.…”

Section: Previous Researchmentioning

confidence: 99%

“…The studies in the first group [14][15][16][17][18][19][20][21][22][23][24][25][26][27] are closely related to our investigation, since they analyze students' understanding of vector concepts in problems without a physical context. From this group, six studies [14][15][16][17][18][19] (by other researchers) identify frequent errors that university students make when they are learning vector concepts.…”

Section: Previous Researchmentioning

confidence: 99%

Test of understanding of vectors: A reliable multiple-choice vector concept test

Barniol

Zavala

2014

Phys. Rev. ST Phys. Educ. Res.

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In this article we discuss the findings of our research on students' understanding of vector concepts in problems without physical context. First, we develop a complete taxonomy of the most frequent errors made by university students when learning vector concepts. This study is based on the results of several test administrations of open-ended problems in which a total of 2067 students participated. Using this taxonomy, we then designed a 20-item multiple-choice test [Test of understanding of vectors (TUV)] and administered it in English to 423 students who were completing the required sequence of introductory physics courses at a large private Mexican university. We evaluated the test's content validity, reliability, and discriminatory power. The results indicate that the TUV is a reliable assessment tool. We also conducted a detailed analysis of the students' understanding of the vector concepts evaluated in the test. The TUV is included in the Supplemental Material as a resource for other researchers studying vector learning, as well as instructors teaching the material.

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“…Whereas the first hopping order of the effective poten- tial Landau theory leads to similar results as mean-field theory [4], higher hopping orders have recently been evaluated via the process-chain approach [47][48][49][50][51], which determines the location of the quantum phase transition for the single component Hubbard model for cubic as well as triangular and hexagonal optical lattices to a similar precision as demanding quantum Monte Carlo simulations [52,53]. Thus, it becomes even possible to calculate the critical exponents of the corresponding quantum phase transition [54,55]. We now present a generalized effective potential Landau theory (GEPLT), which extends those concepts to multi-component systems and to phases with non-trivial crystalline order parameters.…”

Section: Introductionmentioning

confidence: 99%

Generalized effective-potential Landau theory for bosonic quadratic superlattices

et al. 2013

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We study the properties of the Bose-Hubbard model for square and cubic superlattices. To this end we generalize a recently established effective potential Landau theory for a single component to the case of multi components and find not only the characteristic incompressible solid phases with fractional filling, but also obtain the underlying quantum phase diagram in the whole parameter region at zero temperature. Comparing our analytic results with corresponding ones from quantum Monte Carlo simulations demonstrates the high accuracy of the generalized effective potential Landau theory (GEPLT). Finally, we comment on the advantages and disadvantages of the GEPLT in view of a direct comparison with a corresponding decoupled mean-field theory.

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Writing Position Vectors in 3-d Space: A Student Difficulty With Spherical Unit Vectors in Intermediate E&M

Cited by 14 publications

References 0 publications

Perturbative calculation of critical exponents for the Bose–Hubbard model

Perturbative calculation of critical exponents for the Bose–Hubbard model

Test of understanding of vectors: A reliable multiple-choice vector concept test

Generalized effective-potential Landau theory for bosonic quadratic superlattices

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