Investigating how students relate inner products and quantum probabilities

The Born rule, which describes the formalism for determining probabilities, is one of the most fundamental postulates in quantum mechanics. This paper presents results from an investigation into how students apply the Born rule to determine probabilities for energy and position measurements. The investigation includes two stages with independent methods: a quantitative analysis of student written work and a qualitative analysis of student individual interviews. The data from written tasks suggest that after instruction many students have not developed a coherent model for determining probabilities that they can apply to observables regardless of whether the eigenvalues are discrete or continuous. Moreover, many students seem to lack a functional understanding of quantum states and inner products that allows them to translate between Dirac notation and wave function representation. These results motivate student interviews, which allow us to probe student reasoning in depth. Prior research suggests that various features of each notation used in quantum mechanics may have an impact on how students perform computations. We postulate that the features of quantum notations may also interact with student sensemaking. Therefore, we analyze student interviews through the lens of the structural features of quantum notations framework. In particular, we discuss how different structural features may facilitate or hinder student sensemaking about concepts relevant to determining probabilities. The results from both quantitative and qualitative data suggest that unsuccessfully differentiating between a wave function and its associated state vector in Dirac notation may be a primary barrier for students to develop a model for determining probabilities for discrete and continuous cases.