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Quantum mechanics of a generalised rigid body. (English) Zbl 1344.81107

Summary: We consider the quantum version of Arnold’s generalisation of a rigid body in classical mechanics. Thus, we quantise the motion on an arbitrary Lie group manifold of a particle whose classical trajectories correspond to the geodesics of any one-sided-invariant metric. We show how the derivation of the spectrum of energy eigenstates can be simplified by making use of automorphisms of the Lie algebra and (for groups of type I) by methods of harmonic analysis. We show how the method can be extended to cosets, generalising the linear rigid rotor. As examples, we consider all connected and simply connected Lie groups up to dimension 3. This includes the universal cover of the archetypical rigid body, along with a number of new exactly solvable models. We also discuss a possible application to the topical problem of quantising a perfect fluid.

MSC:

81R25 Spinor and twistor methods applied to problems in quantum theory
70E05 Motion of the gyroscope
22E05 Local Lie groups
53C22 Geodesics in global differential geometry
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups

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