Conceptual Difficulties and a Flipped–Interactive Intervention on the Schrödinger Equation: Evidence from Vietnamese Undergraduates

This study investigates Vietnamese undergraduate students’ conceptual difficulties in learning the Schrödinger equation and the wave function [Formula: see text], and examines whether a flipped–interactive teaching model can help reduce such difficulties. A mixed-method design was used with a total of 128 students enrolled in an introductory quantum physics course. Two comparable classes were formed: Class A followed a traditional lecture-based approach, while Class B received a flipped–interactive intervention that combined pre-class materials, in-class conceptual discussions, simulations, and guided reflection. Conceptual understanding was measured across four core topics: (i) physical meaning of the wave function [Formula: see text], (ii) distinction between eigenstate and eigenvalue, (iii) superposition and interference, and (iv) measurement and probability. Quantitative results show a moderate normalized gain for the traditional group (Class A, [Formula: see text]) and a substantial gain for the flipped group (Class B, [Formula: see text], [Formula: see text]). An additional extended survey ([Formula: see text]) confirmed that the most persistent conceptual difficulties included treating [Formula: see text] as a classical mechanical oscillation, interpreting [Formula: see text] as a mere algebraic square rather than a probability density, and misreading superposition as “being in two places at once.” Based on these findings, the paper proposes the notion of a Cognitive Quantum Condition (CQC) to provide a descriptive framework for interpreting the observed shift in reasoning from classical–deterministic reasoning toward quantum–probabilistic reasoning. This transition is illustrated through everyday metaphors previously observed in classroom interaction (e.g. six-faced dice for classical probability, the vector-sum view of “[Formula: see text]” or “[Formula: see text],” and the idea that quantum reasoning becomes more salient when the learner’s “cognitive distance” is compressed, used here as a pedagogical analogy rather than a literal physical claim). The study contributes an adaptation of the international Physics Education Research (PER) framework to the Vietnamese context and offers a model to design future quantum mechanics instruction that targets conceptual change rather than procedural problem solving.