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Based on a geometric view on the famous Dirac belt trick in terms of a Möbius strip, we propose a simple haptic model for spin states as standing waves in non-trivial geometry. We compare two representations of the model: the spinor representation in SU(2), which geometrically is described by ‘Dirac belt’ states on S3, and the well-known Bloch-sphere representation on S2. We show that after Hopf-mapping , the position and number of nodal points are sufficient to describe the spin state and make the relation between the representations on S2 and on S3 explicit in simple haptic models. Our approach is well-suited for graduate physics courses, and provides a thorough understanding of the complex geometry of spin states beyond the usual Bloch-sphere visualisation. It turns out that an even number 2l of nodal points can be related to nodal lines of spherical harmonics Ylm, an odd number can be traced back to spin states. In this sense, our model provides a unified view for orbital and spin states based on a simple picture of standing waves.

A generalization of the famous Dirac belt trick opens up the way to a haptic model for quantum phases of fermions and bosons in Hilbert space based on knot theory. We introduce a simple paper strip model as an aid for visualization of the quantum phases before and after Hopf-mapping, which can be extended to arbitrary spin states with almost no mathematical formalism. Knot theory arises naturally, leading to the Jones polynomials derived from Artin’s braid group for fermionic knots and for bosonic links. The paper strip model explicitly illuminates the relation between these knots and links within the S U ( 2 ) -representation of spin-jstates in C 2 j + 1 before Hopf-mapping and the number p = 2 j of nodes in the stellar representation in C P 1 after Hopf mapping.

The Heegaard splitting of S U ( 2 ) is a particularly useful representation for quantum phases of spin j-representation arising in the mapping S 1 → S 3, which can be related to ( 2 j , 2 ) torus knots in Hilbert space. We show that transitions to homotopically equivalent knots can be associated with gauge invariance, and that the same mechanism is at the heart of quantum entanglement. In other words, (minimal) interaction causes entanglement. Particle creation is related to cuts in the knot structure. We show that inner twists can be associated with operations with the quaternions ( I , J , K ), which are crucial to understand the Hopf mapping S 3 → S 2. We discuss the relationship between observables on the Bloch sphere S 2, and knots with inner twists in Hilbert space. As applications, we discuss selection rules in atomic physics, and the status of virtual particles arising in Feynman diagrams. Using a simple paper strip model revealing the knot structure of quantum phases in Hilbert space including inner twists, a h a p t i c model of entanglement and gauge symmetries is proposed, which may also be valid for physics education.

A poorly elaborated learner’s understanding of models has been reported to be one of the major sources for learning difficulties in the quantum domain. To be able to provide physics education in schools with evidence as to how this problem can be tackled, a deeper theoretical understanding of the structure of learners’ mental models in quantum physics seems essential. In this respect, previous research has proposed two dimensions in learners’ mental models in the atomic hull context, labelled Fidelity of Gestalt and Functional Fidelity. In this article, we investigate whether this proposed two-factorial structure can be transferred to quantum concepts beyond the atomic hull context. To approach this, we surveyed the structure of students’ mental models in the context of photons’ properties and behavior. We conducted a questionnaire study: 170 secondary school students completed a survey instrument adapted from the literature. Using exploratory factor analysis, the two factors Fidelity of Gestalt and Functional Fidelity to describe the students’ mental models could be replicated for the photon context. We provide a selection of results from physics education literature to reveal that our two-factor framework to describe the students’ mental models seems to be a promising endeavor in the landscape of science education research in general.

An easy circuit for measuring the power of a solar panel in physics classroom by using the microcontroller Arduino will be introduced in this article. The measured data is transferred via Bluetooth to the smartphone app ‘phyphox’ where it is displayed graphically. The circuitry enables measuring the power of a solar panel in different situations of light intensity. Several model experiments for students will be described.

With new quantum technologies and new applications comes a new need for specialists, the new quantum workforce. This brings new challenges for education that the typical quantum mechanics courses for physicists do not address. Requirements for the future quantum workforce need to be collected and training programmes created. In between, there should be a European competence framework on which to build the training programmes. One goal of the European Flagship project QTEdu is to develop this framework for second-generation quantum technologies. The Delphi study presented here serves as a basis for this: The aim is to identify knowledge and competences in the field of quantum information technologies, which are partly already needed in industry today, but especially in the future.

Entropy has played an essential role in the history of physics. Its mathematical definition and applications have changed over time till today. In this paper, we first review the historical evolution of these various points of view, from the thermodynamic definition to information entropy from Shannon in classical physics, up to the modern concept of Neumann’s quantum entropy. As a specific example, we consider entanglement entropy and compare the phase space approach in classical physics to the Hilbert space approach in quantum physics in simple model systems. We derive a general expression for the entanglement entropy of fermions and bosons in arbitrary partitions of Hilbert space, valid beyond the thermodynamic limit. Next, we compare thermodynamic heat engines with quantum heat engines. Finally, we proceed to the more general concept of quantum (computational) complexity and argue, using the concept of entanglement entropy, that the Heisenberg time in classically chaotic systems coincides with the time when maximal complexity is reached in the quantum case for systems with all-all interactions.