Evaluating solid angles via geometric phase

Riccardo Borghi and Turi Maria Spinozzi
Dipartimento di Ingegneria, Università degli Studi “Roma Tre”, Rome, Italy

The concept of geometric phase is employed to express the solid angle subtended by a closed curve in the three-dimensional space at a given point. To this aim, the general formalism developed for the geometric phase during the eighties is applied to the 2×2 matrix representing the Hamiltonian associated to the unit vector evolving along the projection of the closed path on the unit sphere. The solid angle can be expressed by means of a simple contour integral, which provides a direct implementation of the general expression of the anholonomy associated to the parallel transport of a vector along a closed path on a sphere. A nontrivial example of application of this formula is shown. © Anita Publications. All rights reserved.
Keywords: Quantum Mechnaics, Geometric Phase, Mathematical Methods in Physics.

Peer Review Information
Method: Single- anonymous; Screened for Plagiarism? Yes
Buy this Article in Print © Anita Publications. All rights reserve

References

1. Maxwell J C, A Treatise on Electricity and Magnetism, Vol II, 3rd edn, (Oxford University Press, London), 1892, p. 36.
2. Born M, Wolf E, Principles of Optics, 7th edn, (Cambridge University Press, Cambridge), 1999.
3. Friberg A T, Thompson B J, (eds), Selected Papers on Coherence and Radiometry, SPIE Milestone Series MS 69, 1993.
4. Masket A V, Solid Angle Contour Integrals, Series, and Tables, Rev Sci Instr, 28(1957)191–197.
5. Tryka S, Angular distribution of the solid angle at a point subtended by a circular disk, Opt Commun, 137(1997)317–333.
6. Prata M J, Analytical calculation of the solid angle subtended by a circular disc detector at a point cosine source, Nuclear Instr Meth Phys Res, A 521(2004)576–585.
7. Berry M V, Quantal phase factors accompanying adiabatic changes, Proc R Soc A, 392(1984)45–57.
8. Hannay J H, Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian, J Phys A, 18(1985) 221–230.
9. Berry M V, The adiabatic phase and Pancharatnam’s phase for polarized light, J Mod Opt, 34(1987)1401–1407.
10. Simon R, Mukunda N, Bargmann invariant and the geometry of Guoy phase, Phys Rev Lett, 70(1993)880–883 .
11. See, for instance, the introductory discussion, in Berry M V, The quantum phase, five years after, in Geometric Phases in Physics, Shapere A, Wilczek F, (eds), (World Scientific), 1989, pp 7–28.
12. The proof reported here corresponds to the approach reported, in Aharonov Y, Anandan J, Phase change during a cyclic quantum evolution, Phys Rev Lett, 58 (1987)1593. Such approach is also well summarized in Sec. 6.3.1 of Schleich W, Quantum Optics in Phase Space, (Wiley, Berlin), 2001.
13. To prove this invariance, one could use the classical decomposition of a generic 3D rotation in terms of Euler angles. See for instance H Goldstein, C Poole, J Safko, Classical Mechanics, 3rd edn, (Addison Wesley, New York), 2002.