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Journal articles on the topic 'Dirac Notation'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

Xu, Yingte, Gilles Barthe, and Li Zhou. "Automating Equational Proofs in Dirac Notation." Proceedings of the ACM on Programming Languages 9, POPL (2025): 1227–59. https://doi.org/10.1145/3704878.

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Dirac notation is widely used in quantum physics and quantum programming languages to define, compute and reason about quantum states. This paper considers Dirac notation from the perspective of automated reasoning. We prove two main results: first, the first-order theory of Dirac notation is decidable, by a reduction to the theory of real closed fields and Tarski's theorem. Then, we prove that validity of equations can be decided efficiently, using term-rewriting techniques. We implement our equivalence checking algorithm in Mathematica, and showcase its efficiency across more than 100 exampl

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Stafford, Randall B., M. Louis Lauzon, Mohammad Sabati, Richard Frayne, and Robert I. Thompson. "A tutorial on the precessional behaviour of hydrogen nuclei in external magnetic fields." Canadian Journal of Physics 88, no. 7 (2010): 465–77. http://dx.doi.org/10.1139/p10-033.

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The purpose of this tutorial is to derive the precessional characteristics of the magnetic moments of hydrogen nuclei in the presence of a constant external magnetic field using the Dirac bra-ket formulation of quantum mechanics (QM). This behaviour has many applications, most notably in nuclear magnetic resonance (NMR) and magnetic resonance (MR) imaging. Many NMR and MR imaging textbooks claim that the QM expectation value of the magnetic moment of a proton in a magnetic field reduces to the classical picture of a precessing magnetic dipole. This paper validates this conclusion by reducing t

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Gao, Yipeng. "A Revisit to the Notation of Martensitic Crystallography." Crystals 8, no. 9 (2018): 349. http://dx.doi.org/10.3390/cryst8090349.

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As one of the most successful crystallographic theories for phase transformations, martensitic crystallography has been widely applied in understanding and predicting the microstructural features associated with structural phase transformations. In a narrow sense, it was initially developed based on the concepts of lattice correspondence and invariant plane strain condition, which is formulated in a continuum form through linear algebra. However, the scope of martensitic crystallography has since been extended; for example, group theory and graph theory have been introduced to capture the crys

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Dr., Kyongyob Min. "Dirac Bra-ket Notation for Interpreting Regional Distribution of Pulmonary Ventilation-Perfusion." International Journal of Case Studies 4, no. 1 (2015): 18–34. https://doi.org/10.5281/zenodo.3525611.

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Newly developing technologies of imaging have shown that the regional distribution of pulmonary ventilation and perfusion is formed from lobules of Miller in varying degrees of ventilation and perfusion. The theoretical study in this paper proposes using Dirac's bra-ket notation for describing the pulmonary ventilation-perfusion relations of the multiple inert-gas elimination technique (MIGET) based on the lobular lung model instead of the classical alveolar model. Bra-ket notation and the corresponding rules of calculation would provide a useful tool to solve the difficulties between pulm

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Dūdėnas, Vytautas, and Thomas Gajdosik. "Feynman rules for Weyl spinors with mixed Dirac and Majorana mass terms." Lithuanian Journal of Physics 56, no. 3 (2016): 149–63. http://dx.doi.org/10.3952/physics.v56i3.3364.

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We present a basic formalism for using the Weyl spinor notation in Feynman rules. We focus on Weyl spinors with mixed Dirac and Majorana mass terms. To clarify the definitions we derive the Feynman rules from the path integral and present two examples: loop corrections for a fermion propagator and a tree level analysis of a seesaw toy model.

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Rao, Allam Srinivasa. "An Intriguing Interpretation of 1D and 2D Non-Diffracting Modes in Cosine Profile." Photonics 10, no. 12 (2023): 1358. http://dx.doi.org/10.3390/photonics10121358.

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We provide a simple analysis based on ray optics and Dirac notation for 1D (one-dimensional) and 2D (two-dimensional) non-diffracting modes in the cosine profile, which are often called Cosine beams. We explore various kinds of structured modes formed by the superposition of two 1D Cosine beams. We then went on to understand the properties of the Bessel beams in terms of Cosine beams. For the first time, we report on the generation of three-dimensional tunable needle structures based on the interference of 1D Cosine beams. These size-tunable optical needles can have multiple advantages in mate

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Gunduz, Orhan, та Mustafa M. Aral. "A Dirac-δ Function Notation for Source/Sink Terms in Groundwater Flow". Journal of Hydrologic Engineering 10, № 5 (2005): 420–27. http://dx.doi.org/10.1061/(asce)1084-0699(2005)10:5(420).

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Cai, Ying, Cuihong Lv, Nan Huang, and Nan Jin. "Studying the Two New Convolutions of Fractional Fourier Transform by Using Dirac Notation." Journal of Contemporary Educational Research 6, no. 5 (2022): 38–47. http://dx.doi.org/10.26689/jcer.v6i5.3891.

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Based on quantum mechanical representation and operator theory, this paper restates the two new convolutions of fractional Fourier transform (FrFT) by making full use of the conversion relationship between two mutual conjugates: coordinate representation and momentum representation. This paper gives full play to the efficiency of Dirac notation and proves the convolutions of fractional Fourier transform from the perspective of quantum optics, a field that has been developing rapidly. These two new convolution methods have potential value in signal processing.

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LUNDBERG, WAYNE R. "SUFFICIENT EXTRA DIMENSIONS." International Journal of Modern Physics A 16, supp01c (2001): 998–1000. http://dx.doi.org/10.1142/s0217751x01008709.

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Matrix algebra notation in introduced which extends Dirac abstract vector notation to explicitly describe standard hadrons, leptons and a specific type of 1-brane. The 3×3×3 matrix notation is easily used with the path integral approach to model both strong and weak interactions. The specific 1-brane, called a tripartite string (a type of orientifold), has three extra spatial dimensions and three extra curvature dimensions which are necessary to describe standard particles. The extra spatial dimensions compactify to one area-like metric which, when scaled according to the Randall-Sundrum appro

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10

Khan, Farooq A., and John E. Hansen. "The Dirac (Bracket) Notation in the Undergraduate Physical Chemistry Curriculum: A Pictorial Introduction." Chemical Educator 5, no. 3 (2000): 113–19. http://dx.doi.org/10.1007/s00897000379a.

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11

Tamturk, Muhsin. "Quantum Computing in Insurance Capital Modelling." Mathematics 11, no. 3 (2023): 658. http://dx.doi.org/10.3390/math11030658.

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This paper proposes a quantum computing approach for insurance capital modelling. Using an open-source software development kit, Qiskit, an algorithm for working on a superconducting type IBM quantum computer is developed and implemented to predict the capital of insurance companies in the classical surplus process. With the fundamental properties of quantum mechanics, Dirac notation and Feynman’s path calculation are shown. Furthermore, custom quantum insurance premium and claim gates are investigated in order to build a quantum circuit with respect to initial reserve, premium and claim amoun

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Levy, Jeremy, and Chandralekha Singh. "Teaching quantum formalism and postulates to first-year undergraduates." American Journal of Physics 93, no. 1 (2025): 46–51. https://doi.org/10.1119/5.0209945.

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Traditional approaches to undergraduate-level quantum mechanics require extensive mathematical preparation, preventing most students from enrolling in a quantum mechanics course until the third year of a physics major. Here we describe an approach to teaching quantum formalism and postulates that can be used with first-year undergraduate students and even high school students. The only pre-requisite is a familiarity with vector dot products. This approach enables students to learn Dirac notation and core postulates of quantum mechanics at a much earlier stage in their academic career, which ca

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Marshman, Emily, and Chandralekha Singh. "Investigating and improving student understanding of quantum mechanical observables and their corresponding operators in Dirac notation." European Journal of Physics 39, no. 1 (2017): 015707. http://dx.doi.org/10.1088/1361-6404/aa8e73.

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Hennig, Fabian, Kristóf Tóth, Moritz Förster, and Philipp Bitzenbauer. "A new teaching-learning sequence to promote secondary school students’ learning of quantum physics using Dirac notation." Physics Education 59, no. 4 (2024): 045007. http://dx.doi.org/10.1088/1361-6552/ad353d.

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Abstract This paper describes the design of a new teaching-learning sequence on quantum physics aimed at upper secondary school students. In this teaching-learning sequence, GeoGebra simulations and interactive screen experiments are used to investigate the behaviour of a single photon at beam splitter and single photon interference in a Michelson interferometer. We propose a minimal formalism using Dirac notation, which avoids complex numbers and elaborate vector calculus, to make a quantitative description of the quantum optics experiments accessible to secondary school students. With this n

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Muñoz, J. L. Gómez, and F. Delgado. "QUANTUM: A WolframMathematicaadd-on for Dirac Bra-Ket Notation, Non-Commutative Algebra, and Simulation of Quantum Computing Circuits." Journal of Physics: Conference Series 698 (March 2016): 012019. http://dx.doi.org/10.1088/1742-6596/698/1/012019.

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Gunduz, Orhan, та Mustafa M. Aral. "Errata for “A Dirac-δ Function Notation for Source/Sink Terms in Groundwater Flow” by Orhan Gunduz and Mustafa M. Aral". Journal of Hydrologic Engineering 11, № 1 (2006): 89. http://dx.doi.org/10.1061/(asce)1084-0699(2006)11:1(89).

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Yang, Yingrui. "Principles of Economic Externality Dynamics Economic Dynamics and the Standard Model (V)." Journal of Electrical Electronics Engineering 3, no. 4 (2024): 01–13. http://dx.doi.org/10.33140/jeee.03.04.06.

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The present paper discusses the importance of economic externality and the economic phenomena involved. Section 1 draws on the framework of weak force theory in physics, we propose a new approach of dynamic analysis for economic externality. From a mesoscopic (intermediate scale) perspective, no distinction is made between the macro-demand and individual micro-buyers nor between the macro-supply and individual micro-sellers. Section 2 applies the isospin modeling method to construct two economic isospin models. One is between the achieving impulse and the fear-of-failure impulse, under the eff

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Minh Do, Canh, and Kazuhiro Ogata. "Symbolic model checking quantum circuits in Maude." PeerJ Computer Science 10 (June 20, 2024): e2098. http://dx.doi.org/10.7717/peerj-cs.2098.

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This article presents a symbolic approach to model checking quantum circuits using a set of laws from quantum mechanics and basic matrix operations with Dirac notation. We use Maude, a high-level specification/programming language based on rewriting logic, to implement our symbolic approach. As case studies, we use the approach to formally specify several quantum communication protocols in the early work of quantum communication and formally verify their correctness: Superdense Coding, Quantum Teleportation, Quantum Secret Sharing, Entanglement Swapping, Quantum Gate Teleportation, Two Mirror-

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Ait, Khalid, Saddik Dachraoui, Taoufik Hassouni, El mehdi, Elmahjoub Chakir, and Aimad Belboukhari. "Development of a Survey to Assess Conceptual Understanding of Quantum Mechanics among Moroccan Undergraduates." European Journal of Educational Research 11, no. 4 (2022): 2219–43. http://dx.doi.org/10.12973/eu-jer.11.4.2219.

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<p style="text-align: justify;">We developed a Quantum Mechanics Conceptual Understanding Survey (QMCUS) in this study. The survey was conducted using a quantitative methodology. A multiple-choice survey of 35 questions was administered to 338 undergraduate students. Three experienced quantum mechanics instructors examined the validity of the survey. The reliability of our survey was measured using Cronbach's alpha, the Fergusson delta index, the discrimination index, and the point biserial correlation coefficient. These indices showed that the developed survey is reliable. The statistic

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Xu, Qining, Jun Zhu, and Liuli Qin. "Tunable and super-wide angle defocusing lens large phase modulator." Physica Scripta 99, no. 2 (2024): 025510. http://dx.doi.org/10.1088/1402-4896/ad19ba.

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Abstract Traditional terahertz wavefront modulators exhibit significant limitations in controlling optical wavefronts. In order to address the adjustability issues of terahertz wavefront modulators, we have developed a novel optical device based on the tunability of Fermi energy levels. Structurally, we find the ability to modulate the Fermi energy levels of graphene by modulating the carrier concentration. We can also change the phase of the light, causing the transmission path of the light as it passes through the graphene array to change, affecting the position of the backfocusing focal poi

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ZHANG, Yuyu, Zixiang HU, Jianjun DONG, and Keling WANG. "THE DIMENSION OF THE DIRAC NOTATIONIN QUANTUM MECHANICS." Physics and Engineering 34, no. 4 (2024): 110–13. https://doi.org/10.26599/phys.2024.9320417.

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Bulathsinghala, D. L., and K. A. I. L. Wijewardena Gamalath. "Implementation of a Quantized Line Element in Klein-Gordon and Dirac Fields." International Letters of Chemistry, Physics and Astronomy 48 (March 2015): 68–86. http://dx.doi.org/10.18052/www.scipress.com/ilcpa.48.68.

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In this paper an ansatz that the anti-commutation rules hold only as integrated average over time intervals and not at every instant giving rise to a time-discrete form of Klein-Gordon equation is examined. This coarse-grained validation of the anti-commutation rules enables us to show that the relativistic energy-momentum relation holds only over discrete time intervals, fitting well with the time-energy uncertainty relation. When this time-discrete scheme is applied to four vector notations in relativity, the line-element can be quantized and thereby how the physical attributes associated wi

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Bulathsinghala, D. L., and K. A. I. L. Wijewardena Gamalath. "Implementation of a Quantized Line Element in Klein-Gordon and Dirac Fields." International Letters of Chemistry, Physics and Astronomy 48 (March 25, 2015): 68–86. http://dx.doi.org/10.56431/p-36k0sm.

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In this paper an ansatz that the anti-commutation rules hold only as integrated average over time intervals and not at every instant giving rise to a time-discrete form of Klein-Gordon equation is examined. This coarse-grained validation of the anti-commutation rules enables us to show that the relativistic energy-momentum relation holds only over discrete time intervals, fitting well with the time-energy uncertainty relation. When this time-discrete scheme is applied to four vector notations in relativity, the line-element can be quantized and thereby how the physical attributes associated wi

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Tamturk, Muhsin, Dominic Cortis, and Mark Farrell. "Examining the Effects of Gradual Catastrophes on Capital Modelling and the Solvency of Insurers: The Case of COVID-19." Risks 8, no. 4 (2020): 132. http://dx.doi.org/10.3390/risks8040132.

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This paper models the gradual elements of catastrophic events on non-life insurance capital with a particular focus on the impact of pandemics, such as COVID-19. A combination of actuarial and epidemiological models are handled by the Markovian probabilistic approach, with Feynman’s path calculation and Dirac notations, in order to observe how a pandemic risk may affect an insurer via reduced business. We also examine how the effects of a pandemic can be taken into account both during and at the end of the process. Examples are also provided showing the potential effects of a pandemic on diffe

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Cortis, Dominic, and Muhsin Tamturk. "Accumulators and Bookmaker’s Capital with Perturbed Stochastic Processes." AppliedMath 2, no. 2 (2022): 170–84. http://dx.doi.org/10.3390/appliedmath2020010.

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The sports betting industry has been growing at a phenomenal rate and has many similarities to the financial market in that a payout is made contingent on an outcome of an event. Despite this, there has been little to no mathematical focus on the potential ruin of bookmakers. In this paper, the expected profit of a bookmaker and probability of multiple soccer matches are observed via Dirac notations and Feynman’s path calculations. Furthermore, we take the unforeseen circumstances into account by subjecting the betting process to more uncertainty. A perturbed betting process, set by modifying

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Boyd, Jeffrey. "Paul Diracâ€™s view of the Theory of Elementary Waves." JOURNAL OF ADVANCES IN PHYSICS 13, no. 3 (2017): 4731–34. http://dx.doi.org/10.24297/jap.v13i3.5921.

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Is science open to a new idea? Thomas Kuhn says paradigm shifts sound like gibberish to scientificleaders, and are rejected for that reason. The Theory of Elementary Waves (TEW) is such an idea:quantum particles follow waves moving in the opposite direction. Time always goes forwards. Wefocus on Paul Diracâ€™s 1930 book The Principles of Quantum Mechanics, applied to TEW. We keepDirac notation and quantum math but replace the picture of how nature is organized. Waveinterference and probabilistic effects occur prior to particle emission. Wave function collapse occursat emission & there is n

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Cotăescu, Ion I. "Relativistic Persistent Currents in Ideal Aharonov-Bohm Rings and Cylinders." Annals of West University of Timisoara - Physics 59, no. 1 (2016): 49–67. http://dx.doi.org/10.1515/awutp-2016-0006.

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AbstractIn this report we revisit the results obtained in [1, 2] where the relativistic Aharonov-Bohm was studied for the first time. The method is based on the exact solutions of the complete (1+3)-dimensional Dirac equation of fermions moving in ideal Aharonov-Bohm (AB) rings and cylinders which are used for deriving the exact expressions of the relativistic partial currents. It is shown that these currents can be related to the derivative of the fermion energy with respect to the flux parameter, just as in the non-relativistic case. However, a new and remarkable relativistic effect is the s

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Chen, Zhenyu, Xiangyu Wang, C. W. Lim, and Fan Shi. "Robust large-area elastic transverse wave transport in active acoustic metamaterials." Journal of Applied Physics 131, no. 18 (2022): 185112. http://dx.doi.org/10.1063/5.0087988.

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We investigate robust large-area elastic transverse wave propagation in an actively tunable membrane-type acoustic metamaterial. The waveguide with multiple degrees of freedom to control the width of the interface mode is realized by designing a heterostructure including three domains. One central domain is constructed by periodic unit cells in an ordinary state, where a Dirac cone can be observed in the band structure. The other two domains consist of periodic unit cells possessing opposite valley Chern numbers, respectively. By employing a finite element model, the topologically protected in

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Riihiluoma, William D., Zeynep Topdemir, and John R. Thompson. "Network analysis of students’ conceptual understanding of mathematical expressions for probability in upper-division quantum mechanics." Physical Review Physics Education Research 20, no. 2 (2024). http://dx.doi.org/10.1103/physrevphyseducres.20.020102.

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One expected outcome of physics instruction is for students to be capable of relating physical concepts to multiple mathematical representations. In quantum mechanics (QM), students are asked to work across multiple symbolic notations, including some they have not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, a survey was developed and distributed to students at six different institutions. All of the courses studied were structured as “spins-first,” in which the course begins with spin-1/2 systems and Dira

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Walters, Ian. "Dirac Notation for Evolution Equations: A Note." SSRN Electronic Journal, 2024. http://dx.doi.org/10.2139/ssrn.4790698.

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Wawro, Megan, Kevin Watson, and Warren Christensen. "Students’ metarepresentational competence with matrix notation and Dirac notation in quantum mechanics." Physical Review Physics Education Research 16, no. 2 (2020). http://dx.doi.org/10.1103/physrevphyseducres.16.020112.

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Hennig, Fabian, Kristóf Tóth, Joaquin Veith, and Philipp Bitzenbauer. "Introducing quantum physics concepts and Dirac notation at the secondary school level: Insights into student reasoning from an acceptance survey." Physical Review Physics Education Research 20, no. 2 (2024). http://dx.doi.org/10.1103/physrevphyseducres.20.020147.

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Evidence from physics education research suggests that the introduction of a formalistic description of quantum phenomena can be beneficial to student learning, particularly in terms of helping students move away from naive realist views of quanta or space-time descriptions of quantum phenomena. However, the mathematical requirements for mastering quantum formalism are typically far beyond the scope of secondary school education. Therefore, in previous research, we have proposed a reduced Dirac notation tailored to the capabilities of secondary school students, which is included in a novel tea

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Marshman, Emily, Alexandru Maries, and Chandralekha Singh. "Using multiple representations to improve student understanding of quantum states." Physical Review Physics Education Research 20, no. 2 (2024). https://doi.org/10.1103/physrevphyseducres.20.020152.

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One hallmark of expertise in physics is the ability to translate between different representations of knowledge and use the representations that make the problem-solving process easier. In quantum mechanics, students learn about several ways to represent quantum states, e.g., as state vectors in Dirac notation and as wave functions in position and momentum representation. Many advanced students in upper-level undergraduate and graduate quantum mechanics courses have difficulty translating state vectors in Dirac notation to wave functions in the position or momentum representation and vice vers

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Riihiluoma, William D., Zeynep Topdemir, and John R. Thompson. "Comparative analysis of spins-first and wave functions-first students’ understanding of expressions in quantum mechanics." Physical Review Physics Education Research 21, no. 1 (2025). https://doi.org/10.1103/physrevphyseducres.21.010113.

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[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] Instructors teaching upper-division quantum mechanics have had two primary options when it comes to textbook choice and thus curriculum sequence: starting with wave functions and the Schrödinger equation, referred to as “wave functions-first;” and starting with discrete spin-1/2 systems and Dirac notation, known as “spins-first” courses. Given the very different structures of these courses, particularly as it pertains to the notations and formalisms both emphasized and used, it beg

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Redo-Sanchez, Albert, Pablo Luesia Lahoz, Diego Gutierrez, and Adolfo Muñoz. "A Cohesive Framework for Non-Line-of-Sight Imaging based on Dirac Notation." Optics Express, February 8, 2024. http://dx.doi.org/10.1364/oe.518466.

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Mario, Mastriani, and Naiouf Marcelo. "Quantum Enhanced Correlation Matrix Memories via States Orthogonalisation." November 4, 2013. https://doi.org/10.5281/zenodo.1089166.

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This paper introduces a Quantum Correlation Matrix Memory (QCMM) and Enhanced QCMM (EQCMM), which are useful to work with quantum memories. A version of classical Gram-Schmidt orthogonalisation process in Dirac notation (called Quantum Orthogonalisation Process: QOP) is presented to convert a non-orthonormal quantum basis, i.e., a set of non-orthonormal quantum vectors (called qudits) to an orthonormal quantum basis, i.e., a set of orthonormal quantum qudits. This work shows that it is possible to improve the performance of QCMM thanks QOP algorithm. Besides, the EQCMM algorithm has a lot of a

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Malcher, Viliam. "Operators and meaning of wave function." JOURNAL OF ADVANCES IN PHYSICS, July 30, 2016, 4039–42. http://dx.doi.org/10.24297/jap.v11i8.216.

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The interpretation problems of quantum theory are considered. In the formalism of quantum theory the possible states of a system are described by a state vector. The state vector, which will be represented as |Ïˆ> in Dirac notation, is the most general form of the quantum mechanical description. The central problem of the interpretation of quantum theory is to explain the physical significance of the |Ïˆ>. In this paper we have shown that one of the best way to make of interpretation of wave function is to take the wave function as an operator.

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Hu, Peter, Yangqiuting Li, and Chandralekha Singh. "Challenges in addressing student difficulties with basics and change of basis for two-state quantum systems using a multiple-choice question sequence in online and in-person classes." European Journal of Physics, August 31, 2023. http://dx.doi.org/10.1088/1361-6404/acf5b3.

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Abstract Research-validated multiple-choice questions comprise an easy-to-implement instructional tool for scaffolding student learning and providing formative assessment of students’ knowledge. We present findings from the implementation of a research-validated multiple-choice question sequence on the basics of two-state quantum systems, including inner products, outer products, translation between Dirac notation and matrix representation in a particular basis, and change of basis. This study was conducted in an advanced undergraduate quantum mechanics course, in both online and in-person lea

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Evans, Daniel. "Quick Quantum Circuit Simulation." Journal of Computer Science Research 3, no. 4 (2021). http://dx.doi.org/10.30564/jcsr.v3i4.3567.

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Quick Quantum Circuit Simulation (QQCS) is a software system for computing the result of a quantum circuit using a notation that derives directly from the circuit, expressed in a single input line. Quantum circuits begin with an initial quantum state of one or more qubits, which are the quantum analog to classical bits. The initial state is modified by a sequence of quantum gates, quantum machine language instructions, to get the final state. Measurements are made of the final state and displayed as a classical binary result. Measurements are postponed to the end of the circuit because a quant

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Hennig, Fabian, Kristóf Tóth, Joaquin Veith, and Philipp Bitzenbauer. "Mathematical sense making of quantum phenomena using Dirac notation: its effect on secondary school students’ functional thinking about photons." EPJ Quantum Technology 11, no. 1 (2024). http://dx.doi.org/10.1140/epjqt/s40507-024-00274-6.

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AbstractPrevious research has consistently demonstrated that students often possess an inadequate understanding of fundamental quantum optics concepts, even after formal instruction. Findings from physics education research suggest that introducing a mathematical formalism to describe quantum optical phenomena may enhance students’ conceptual understanding of quantum optics. This paper investigates whether using formal descriptions of quantum optics phenomena – such as photon anticorrelation at a beamsplitter or single-photon interference in a Michelson interferometer – expressed in Dirac nota

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Piña, A. R., Zeynep Topdemir, and John R. Thompson. "Student understanding of eigenvalue equations in quantum mechanics: Symbolic blending and sensemaking analysis." Physical Review Physics Education Research 20, no. 1 (2024). http://dx.doi.org/10.1103/physrevphyseducres.20.010153.

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As part of an effort to examine students’ mathematical sensemaking (MSM) in a spins-first quantum mechanics course during the transition from discrete (spin) to continuous (position) systems, students were asked to construct an eigenvalue equation for a one-dimensional position operator. A subset of responses took the general form of an eigenvalue equation written in Dirac notation. Symbolic blending, a combination of symbolic forms and conceptual blending, as well as a categorical framework for MSM, were used in the analysis. The data suggest two different symbolic forms for an eigenvalue equ

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Merzel, Avraham, Efraim Yehuda Weissman, Nadav Katz, and Igal Galili. "Mathematical structures in quantum physics education for high school students: Unveiling the power of Dirac notation for conceptual and problem-solving proficiency." Physical Review Physics Education Research 20, no. 2 (2024). http://dx.doi.org/10.1103/physrevphyseducres.20.020134.

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[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] Teaching quantum physics (QP) to high school (HS) students is gaining momentum, necessitating the exploration of various effective methods. Specifically, the research on quantitative teaching methods is still in its early stages. Understanding the power of Dirac notation (DN) in teaching is crucial for grasping the complexities of QP, as it provides a concise and powerful symbolic framework that facilitates both conceptual and computational understanding. This study investigates th

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"A spectral solution of the Boltzmann equation for the infinitely strong shock." Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 330, no. 1611 (1990): 217–52. http://dx.doi.org/10.1098/rsta.1990.0015.

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We formulate and implement a new spectral method for the solution of the Boltzmann equation, making extensive use of the theory of irreducible tensors together with the symbolic notation of Dirac. These tools are shown to provide a transparent organization of the algebra of the method and the efficient automation of the associated calculations. I he power of the proposed method is demonstrated by application to the highly nonlinear problem of the infinitely strong shock. It is shown that the distribution function can in this limit be decomposed into a singular part corresponding to the molecul

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Keçeci, Mehmet. "Understanding Quantum Mechanics through Hilbert Spaces: Applications in Quantum Computing." May 20, 2025. https://doi.org/10.5281/zenodo.15468794.

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Understanding Quantum Mechanics through Hilbert Spaces: Applications in Quantum Computing     <strong>Mehmet Keçeci<sup>1</sup></strong> <strong><sup>1</sup></strong><strong>ORCID </strong>: https://orcid.org/0000-0001-9937-9839, İstanbul, Türkiye   <strong>Received</strong>: 20.05.2025   <strong>Abstract:</strong><strong> </strong> <strong> </strong> Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of qu

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Riihiluoma, William D., Zeynep Topdemir, and John R. Thompson. "Symbolic forms analysis of expressions for probability in Dirac and wave function notations for spins-first students." Physical Review Physics Education Research 21, no. 1 (2025). https://doi.org/10.1103/physrevphyseducres.21.010105.

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The ability to relate physical concepts and phenomena to multiple mathematical representations—and to move fluidly between these representations—is a critical outcome expected of physics instruction. In upper-division quantum mechanics, students must work with multiple symbolic notations, including some that they have not previously encountered. Thus, developing the ability to generate and translate expressions in these notations is of great importance, and the extent to which students can relate these expressions to physical quantities and phenomena is crucial to understand. To investigate st

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Keçeci, Mehmet. "Investigating Layered Structures Containing Weyl and Majorana Fermions via the Stratum Model." May 22, 2025. https://doi.org/10.5281/zenodo.15489075.

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Investigating Layered Structures Containing Weyl and Majorana Fermions via the Stratum Model <strong>Mehmet Keçeci<sup>1</sup></strong> <strong><sup>1</sup></strong><strong>ORCID </strong>: https://orcid.org/0000-0001-9937-9839, İstanbul, Türkiye <strong>Received</strong>: 22.05.2025 <strong>Abstract:</strong><strong> </strong>The development of quantum computers represents one of the most exciting and challenging endeavours in modern science. Progress in this field, particularly with the advent of the second quantum revolution, necessitates a profound interplay between diverse dis

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Chaudhary, Rozina, Asma Afzal, Stefaan Poedts, Shahid Ali та Hassan A. Shah. "Signatures of quantized magnetic ﬁeld on the nonlinear Landau damping of transverse electromagnetic waves". Physica Scripta, 14 липня 2023. http://dx.doi.org/10.1088/1402-4896/ace79e.

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Abstract A quantum kinetic approach along with the Landau theory of quantization (LQ) is utilized to study the impact of the magnetic field on the nonlinear Landau damping (NLD) of transverse electromagnetic (EM) waves in a degenerate electron-ion plasma. The gyratory motion of fermions around the magnetic field (H) lines gets quantized into the Landau levels and consequently the associated Fermi-Dirac distribution function becomes modified with the fermion cyclotron frequency under the limit lℏω_{ce}-ε_{Fe}≫k_{B}T_{e}, where l is the orbital quantum number with all other standard notations. I

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