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Geometric phase; distance function; length of the curves

The concept of a curve traced by a state vector in the Hilbert space is introduced into the general context of quantum evolutions and its length defined. Three important curves are identified and their relation to the dynamical phase, the geometric phase and the total phase are studied. These phases are reformulated in terms of the dynamical curve, the geometric curve and the natural curve. For any arbitrary cyclic evolution of a quantum system, it is shown that the dynamical phase, the geometric phase and their sums and/or differences can be expressed as the integral of the contracted length of some suitably-defined curves. With this, the phases of the quantum mechanical wave function attain new meaning. Also, new inequalities concerning the phases are presented.

Arun Kumar Pati¹

1. Theoretical Physics Division, Bhabha Atomic Research Centre, Bombay - 400 085, India

Published

June 1994

Manuscript received

3 January 1994

Manuscript revised

24 February 1994