Modelling spin

摘要

Based on a geometric view on the famous Dirac belt trick in terms of a M?bius strip, we propose a simple haptic model for spin states as standing waves in non-trivial geometry.We compare two representations of the model: the spinor representation in SU(2), which geometrically is described by 'Dirac belt' states on S_3, and the well-known Bloch-sphere representation on S_2. We show that after Hopf-mapping S_3 → S_2, the position and number of nodal points are sufficient to describe the spin state and make the relation between the representations on S_2 and on S_3 explicit in simple haptic models.Our approach is well-suited for graduate physics courses, and provides a thorough understanding of the complex geometry of spin states beyond the usual Bloch-sphere visualisation.It turns out that an even number 2l of nodal points can be related to nodal lines of spherical harmonics Y_(lm), an odd number can be traced back to spin states.In this sense, our model provides a unified view for orbital and spin states based on a simple picture of standing waves.