“H-Space Diagrams”:  A powerful visualization tool for two-state quantum systems

Quantum Mechanics is often considered a difficult subject, in large part due to a reliance on abstract mathematical formalisms.  While multiple symbolic representations (Dirac notation, matrices, wave functions, etc.) are commonly deployed in quantum courses, visual representations beyond graphs of functions are less common.  The Bloch sphere offers one visualization for two-state systems, used particularly in information science, though research has shown students often struggle to interpret the sphere and its connection to state vectors.  Here I present an alternative visualization tool – Hilbert Space Diagrams – which extends and leverages the analogy between quantum state spaces and traditional coordinate systems.  These diagrams offer a robust and mathematically accurate visualization for any two-state system that can support students in addressing many known difficulties including: orthogonality of quantum states, superpositions, measurement probabilities, overall and relative phase, change of basis, acting operators, and time dependence.  In this talk I’ll present a sequence of activities developed to introduce and leverage H-Space diagrams in the context of a spins-first undergraduate quantum mechanics course and highlight their utility in addressing complicated situations with limited mathematical manipulations.  A preliminary analysis of student engagement with these diagrams will also be discussed.