Introducing the concepts of energy and dynamics in quantum mechanics by engaging students in theoretical modeling

In the last decades, empirical investigations have shown the existence of deep and persistent difficulties with teaching/learning the time independent Schrödinger equation and time evolution. At the same time, research has provided evidence in favor of engaging students in modeling activities to help them improve their understanding. However, there is a lack of model-based curriculum materials for teaching the concepts of energy and dynamics in quantum mechanics. To illustrate how modeling could be included in the learning of these topics, we outline an educational proposal in the context of a spins-first approach. As a starting point to introduce the energy concept for atoms and particles, the learning path leverages student knowledge of the Bohr model. The energy level structure of a hydrogen atom with transitions accompanied by the absorption or emission of a photon provides an intuitive basis for activating the modeling of energy, that is organized around the analysis of contrasting cases on the possible results of spin flip transitions in a magnetic field. In this process, students are guided to build a mathematical representation of energy and to investigate its nature up to a derivation of the energy eigenvalue equation. The learning path allows students to compare and contrast the semi-classical Bohr model with the quantum model, as a support to help them build a consistent understanding of the latter. A different set of contrasting cases on the possible results of spin measurements after different amounts of time in the field is used to activate a qualitative modeling process on time evolution, which is then quantitatively examined by analyzing graphs of the state tomography, guiding students to explore the basic features of quantum dynamics up to a derivation of the Schrödinger equation.