We discuss an exact solution to the simplest nontrivial example of a geometrical phase in quantum mechanics. By means of this example: (1) we elucidate the fundamental distinction between rays and vectors in describing quantum mechanical states; (2) we show that superposition of quantal states is invalid; only decomposition is allowed—which is adequate for the measurement process. Our example also shows that the origin of singularities in the analog vector potential is to be found in the unavoidable breaking of projective symmetry caused by using the Schrödinger equation.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
M. Berry, “Anticipations of the geometrical phase,”Phys. Today 43, 34–40 (1990); see bottom of p. 34.
E. P. Wigner,Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, New York, 1959); cf. pp. 47–57 and 325–348.
For a review of this subject, see: A. Shapere and F. Wilczek,Geometric Phases in Physics (Advanced Series in Mathematical Physics, Vol. 9) (World Scientific, Singapore, 1989).
M. Berry, “Quantal phase factors accompanying adiabatic changes”,Proc. R. Soc. London A 392, 45–57 (1984).
L. J. Boya, “State space as projective space,”Found. Phys. 19, 1363 (1989).
L. C. Biedenharn and J. D. Louck,Angular Momentum in Quantum Physics (Encyclopedia of Mathematics and Its Applications, Vol. 8) (Addison-Wesley, Reading, Massachusetts, 1981), p. 459.
W. Greub and H.-R. Petry, “Minimal coupling and complex line bundles,”J. Math. Phys. 16, 1347–1351 (1975).
L. C. Biedenharn and J. D. Louck,The Racah-Wigner Algebra in Quantum Theory (Encyclopedia of Mathematics and Its Applications, Vol. 9) (Addison-Wesley, Reading, Massachusetts, 1981), p. 207.
P. A. M. Dirac, “Quantized Singularities in the Electromagnetic Field,”Proc. R. Soc. London A 133, 60 (1931).
For an introduction to this topic, see: N. Steenrod,The Topology of Fiber Bundles (Princeton Mathematical Series, Vol. 14) (Princeton University Press, Princeton, 1951).
To Professor Asim O. Barut on the occasion of his 65th birthday.
About this article
Cite this article
Solem, J.C., Biedenharn, L.C. Understanding geometrical phases in quantum mechanics: An elementary example. Found Phys 23, 185–195 (1993). https://doi.org/10.1007/BF01883623
- Exact Solution
- Quantum Mechanic
- Quantal State
- Mechanical State
- Vector Potential