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Canadian Journal of Physics

Quantum mechanics in rotating frames. II. The lattice structure of current circulations for a rotating single-particle fluid

P. Gulshani, DAVID J. ROWE

Published Apr 1, 1978• 19 citations
Publisher

Abstract

It is shown that, in general, the current flow for a single particle in quantum mechanics exhibits circulations. These circulations are studied in detail for the anisotropic oscillator system in a slowly rotating frame.
DOI
10.1139/p78-061

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The Hamiltonian describing a system of particles in a rotating coordinate system is derived and it is shown that the simple classical solution of rigid flow is forbidden in quantum mechanics, even at very low angular velocities. This effect is closely parallel to the Aharonov–Bohm effect, which likewise has its origin in the single-valuedness requirement of the wave function. An analytical approach to perturbation theory is used to include the effects of the Coriolis and centrifugal forces and to derive the current flows for some independent-particle systems, that is, for the Inglis cranking model. It is shown, by explicit construction, that the currents are not rigid even when the moment of inertia assumes the rigid-flow value, as it does for the harmonic oscillator single-particle potential under conditions of self-consistency. Furthermore, it is shown that, for a more general potential, even the moment of inertia is not rigid.
It is shown that a natural extension of the hydrodynamical formalism of quantum mechanics for a Schrödinger particle to include vortical flows leads to the hydrodynamical formalism of quantum mechanics for a spinning particle. This latter formalism is then analysed in regard to its characteristic features, especially the subsidiary condition connecting the vorticity of flow with the inhomogeneity of spin field and the existence of spin stress. Also the formalism is brought to completion by establishing the global condition that quantizes circulation around a singular vortex line. The geometro-hydrodynamical formalism which is equivalent to the above hydrodynamical formalism but introduces a triad structure underlying the classical spin is reconstructed on its own footing. The geometrical property of the triad implies the invariance of theory with respect to the rotation of each triad around its symmetry axis by an arbitrary angle, and this necessitates the introduction of the electromagnetic potential, providing the geometrical interpretation of local gauge invariance. Applications of theory to various special cases and typical examples are deferred to Part II.