Rostock 2019 – scientific programme

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Q: Fachverband Quantenoptik und Photonik

Q 41: Poster: Quantum Optics and Photonics II

Q 41.9: Poster

Wednesday, March 13, 2019, 16:15–18:15, S Fobau Physik

Schrödinger equation for quaternionic quantum mechanics — •Jonathan Steinberg and Matthias Kleinmann — Universität Siegen, Siegen, Deutschland

Standard quantum mechanics is formulated over complex Hilbert spaces with normalized vectors representing states and observables corresponding to hermitian operators. The time evolution is determined by the Schrödinger equation, where −iH/ℏ takes the role of a generator for time shifts. Even in this well known setting the origin of the correspondence between the energy observable H and the generator −iH/ℏ needs to be discussed. We provide arguments showing that this is the only sensible choice, with the only remaining freedom being the numeric value of ℏ. If one now replaces complex Hilbert spaces by quaternionic Hilbert modules, an analogous analysis becomes crucial to identify the correct Schrödinger equation for quaternionic quantum mechanics. By using a quaternionic version of Stone’s theorem for strongly continuous one-parameter groups, we show that there cannot exist a global correspondence between the energy observable and the generator of time shifts for dimensions larger than two. However, for the evolution for two-dimensional quaternionic systems there exist a variety candidates for a Schrödinger equation and we discuss their relations and properties.