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Abstract
Based on a geometric view on the famous Dirac belt trick in terms of a Möbius 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 S3, and the well-known Bloch-sphere representation on S2. We show that after Hopf-mapping , the position and number of nodal points are sufficient to describe the spin state and make the relation between the representations on S2 and on S3 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 Ylm, 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.
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