Abstract
After instruction, many students lack a functional understanding of quantum states and inner products that would allow them to translate between Dirac notation and wave function representation.
Highlights
The Born rule describes how the probabilities for any observable can be determined in a unified way whether the eigenvalues are discrete, continuous, or a mixture of both
This article describes an investigation into how students relate inner products and probabilities for discrete and continuous cases
The results motivated our qualitative analysis of interview data through the lens of the structural features of quantum notations framework [25]
Summary
The Born rule describes how the probabilities for any observable can be determined in a unified way whether the eigenvalues are discrete, continuous, or a mixture of both It is introduced at the beginning of many textbooks, typically using either a position or a spin context. In order for students to construct a coherent model for determining probabilities, they must develop a functional understanding of quantum states and inner products. Students typically encounter both Dirac notation and wave functions, and are expected to be able to solve problems with both. The framework was initially developed in the context of students’ performing computations, we postulated that the structural features of those notations may impact student sensemaking about abstract concepts, such as quantum states and inner products.
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