Abstract Moving from the research tradition on the use of Feynman’s sum over paths approach in education, we have developed an educational reconstruction of elementary quantum physics which uses Feynman paths at fixed energy and is capable of a simple conceptual explanation of discrete energy levels in bound systems. The reconstruction has been found to offer several educational advantages, may help students think of quantum physics in a way more akin to how the subject is used by scientists, and allows to shed light on the relationship between energy and time.
Abstract Moving from the research tradition on the use of Feynman’s sum over paths approach in education, we have developed an educational reconstruction of elementary quantum physics which uses Feynman paths at fixed energy and is capable of a simple conceptual explanation of discrete energy levels in bound systems. The reconstruction has been found to offer several educational advantages, may help students think of quantum physics in a way more akin to how the subject is used by scientists, and allows to shed light on the relationship between energy and time.
In this paper, we present an overview of recent developments in the Feynman sum over paths approach for teaching introductory quantum mechanics to high school students and university undergraduates. A turning point in recent research is identified in the clarification of the distinction between the time-dependent and time-independent approaches, and it is shown how the adoption of the latter has allowed new educational reconstructions to proceed much farther beyond what had previously been achieved. It is argued that sum over paths has now reached full maturity as an educational reconstruction of quantum physics and offers several advantages with respect to other approaches in terms of leading students to develop consistent mental models of quantum phenomena, achieving better conceptual understanding and a higher degree of longitudinal integration of knowledge.
We present a research-based teaching sequence in introductory quantum physics using the Feynman sum over paths approach. Our reconstruction avoids the historical pathway, and starts by reconsidering optics from the standpoint of the quantum nature of light, analysing both traditional and modern experiments. The core of our educational path lies in the treatment of conceptual and epistemological themes, peculiar of quantum theory, based on evidence from quantum optics, such as the single photon Mach–Zehnder and Zhou–Wang–Mandel experiments. The sequence is supported by a collection of interactive simulations, realized in the open source GeoGebra environment, which we used to assist students in learning the basics of the method, and help them explore the proposed experimental situations as modeled in the sum over paths perspective. We tested our approach in the context of a post-graduate training course for pre-service physics teachers; according to the data we collected, student teachers displayed a greatly improved understanding of conceptual issues, and acquired significant abilities in using the sum over path method for problem solving.
We present an alternative treatment for simple time-independent quantum systems in one dimension, which can be used in the context of an elementary introduction to quantum physics using the Feynman approach. The method is based on representation of the energy-dependent propagator (or Green function) as a sum of complex amplitudes over all possible paths, classical and non-classical, at fixed energy. We treat both confined and open systems with piecewise-constant potentials, obtaining exact results. We introduce an approximation scheme to extend the method to smooth potentials, recovering the Van Vleck-Gutzwiller propagator. Finally, we discuss the educational application of the method.
Quantum mechanics (QM) has taken on a paradigmatic role in how physics describes and builds new knowledge about the world. It also has a growing role in our society thanks to recent and rapid developments in the field of new quantum technologies. These are becoming increasingly important in our society, in the world of work and which will also decisively influence future socio-economic choices. Also for this reason QM has an increasing role in the education of young people and has been introduced in many different ways also in secondary school in all continents. The change in the theoretical and cultural frame has profound implications that are often not intuitive, which involve the appropriation of a specific way of thinking. Neither macroscopic phenomenology, nor intuition help, because there is no simple evidence of how quantum objects must be thought to interpret that unobservable microscopic world and for which we must enter another interpretative paradigm even if we want to limit ourselves to describe it. PER has carried out several studies on the learning of key concepts, on the learning nodes, the ways of reasoning, the difficulties of the students, their spontaneous ideas, according to two main strands: above all at university level, conceptual difficulties and misunderstandings often linked to a formal approach, which, moreover, underpins the very nature of the entities involved; at secondary school level in relation to how students acquire a quantum view of phenomena on the basis of specific approaches adopted to address QM. In particular, a vast literature, which has increased considerably and especially in relatively recent times, has studied how university students deal with certain concepts such as that of state, superposition, measurement in quantum mechanics rather than how quantum quantities evolve over time. These studies are based on the use of questionnaires that allow to explore different levels: the verbal, formal and representative one, with different representations (eg. wave function, Dirac notation), rather than different representative modes (eg. Graphs of probability amplitudes, or probability distributions). At the secondary school level, the most important studies on student learning were instead conducted evaluating the educational impact and outlining the learning paths of students who had followed formative activities based on differentiated innovative research-based educational paths. In particular, the learning outcomes were compared to those activated by proposals based on the historical reconstruction approach of quantum physics. This approach is privileged in traditional curricula and school texts and consequently in the ways of teachers approaching the topic. It was developed in schools with a narrative or descriptive dimension of solutions based on old physics of quanta for the interpretation of phenomena not explained by classical physics. The innovative approaches instead make use of contexts whose formal treatment is simple (quantum properties described in a two-state space such as polarization or spin) with the aid of simulations that compensate for the problem of students” limited mathematical-formal skills, avoiding any historical reference. In particular, there is a significant difference between the survey tools and the survey contexts in the high school environment compared to surveys conducted with university students. This paper presents a review of the research literature, focusing in particular on the learning processes of secondary school students and outlining how research perspectives have evolved.
We analyse part of the implementation of a didactic sequence to teach different aspects of light in a unified non traditional framework. The goal was to propose the quantum theory of light as a universal framework to describe different phenomena observed. The laws of quantum mechanics for light using Feynman´s “Sum of all Paths” approach adapted to the mathematical level of the students was proposed as a model to explain the experiences. This particular denomination of Feynman´s approach is an intentional choice to avoid the language of "integral" because the students haven't had calculus. Graphic representations and basic operations with vectors capturing the essential aspects of the theory, were used. Simulations made with the software GeoGebra(R) and Modellus were created to help students visualize the formulation. The sequence was carried out in four courses (aged 15-16). For the data analysis, an answer categorization was formulated, considering among other aspects the quantum reformulation of experiment shown herein. This analysis seeks to understand the student´s conceptualization process about quantum interpretation. The results support the conclusion that the conceptualization is complex, and slow, due to both the concepts involved and the representation systems demanded by the situations.