The evolution of geometric Robertson–Schrödinger uncertainty principle for spin 1 system

: pp. 36–49
Received: July 07, 2021
Accepted: November 14, 2021

Mathematical Modeling and Computing, Vol. 9, No. 1, pp. 36–49 (2022)

Centre of Foundation Studies for Agricultural Science, Universiti Putra Malaysia
Faculty of Science, Universiti Putra Malaysia; Institute for Mathematical Research, University Putra Malaysia
Faculty of Science, Universiti Putra Malaysia
Faculty of Science, Universiti Putra Malaysia; Institute for Mathematical Research, University Putra Malaysia

Geometric Quantum Mechanics is a mathematical framework that shows how quantum theory may be expressed in terms of Hamiltonian phase-space dynamics.  The states are points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is specified by the Schrödinger equation in this framework.  The quest to express the uncertainty principle in geometrical language has recently become the focus of significant research in geometric quantum mechanics.  One has demonstrated that the Robertson–Schrödinger uncertainty principle, which is a stronger version of the uncertainty relation, can be defined in terms of symplectic form and Riemannian metric.  On the basis of this formulation, we study the dynamical behavior of the uncertainty relation for the spin 1 system in this work.  We show that under Hamiltonian flow, the Robertson–Schrödinger uncertainty principles are not invariant.  This is because, unlike the symplectic area, the Riemannian metric is not invariant under Hamiltonian flow throughout the evolution process.

  1. Heslot A.  Quantum mechanics as a classical theory.  Physical Review D. 31 (6), 1341–1348 (1985).
  2. Varadarajan V. S.  Boolean Algebras on a Classical Phase Space.  In: Geometry of Quantum Theory. Vol. 1.  Springer, New York (1968).
  3. Kibble T. W. B.  Geometrization of quantum mechanics.  Communications in Mathematical Physics. 65, 189–201 (1979).
  4. Cirelli R., Lanzavecchia P.  Hamiltonian vector fields in quantum mechanics. II Nuovo Cimento B. 79, 271–283 (1984).
  5. Ashtekar A., Schilling T. A.  Geometry of quantum mechanics.  AIP Conference Proceedings. 342 (1), 471–478 (1995).
  6. Anandan J.  A Geometric Approach to Quantum Mechanics.  Foundations of Physics. 21, 1265–1284 (1991).
  7. Brody D. C., Hughston L. P.  Geometric quantum mechanics.  Journal of Geometry and Physics. 38 (1), 19–53 (2001).
  8. Bengtsson I., Brannlund J., Zyczkowski K.  CPn, or, Entanglement Illustrated.  International Journal of Modern Physics A. 17 (31), 4675–4695 (2002).
  9. Chruściński D., Jamiołkowski A.  Geometric Phases in Classical and Quantum Mechanics.  Progress in Mathematical Physics. Vol. 36 (2004).
  10. Benvegnù A., Sansonetto N., Spera M.  Remarks on geometric quantum mechanics.  Journal of Geometry and Physics.  51 (2), 229–243 (2004).
  11. Chruściński D.  Geometric Aspects of Quantum Mechanics and Quantum Entanglement.  Journal of Physics: Conference Series. 30, 9–16 (2006).
  12. Bengtsson I., Zyczkowski K.  Geometry of Quantum States: An Introduction to Quantum Entanglement.  United Kingdom, Cambridge University Press (2006).
  13. Marmo G., Volkert G.  Geometrical Description of Quantum Mechanics – Transformation and Dynamics.  Physica Scripta. 82, 038117 (2010).
  14. Gallardo J. C.  The geometrical formulation of quantum mechanics.  Rev. Real Academia de Ciencias. Zaragoza. 67, 51–103 (2012).
  15. Heydari H.  Geometric formulation of quantum mechanics.  Preprint ArXiv: 1503.00238v2 (2016).
  16. Heisenberg W.  Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik.  Zeitschrift für Physik. 43 (3–4), 172–198 (1927), (in German).
  17. Robertson H. P.  The Uncertainty Principle.  Physical Review. 34 (1), 163–164 (1929).
  18. Schrödinger E.  Zum Heisenbergschen Unschärfeprinzip.  Sitzungsberichte der Preussischen Akademie der Wissenschaften.  Physikalisch-mathematische Klasse. 14, 296–303 (1930).
  19. Anandan J., Aharonov Y.  Geometry of quantum evolution.  Physical Review Letters. 65 (14), 1697–1700 (1990).
  20. de Gosson M.  The symplectic camel and phase space quantization.  Journal of Physics A: Mathematical and General. 34 (47), 10085–10096 (2001).
  21. de Gosson M.  Phase space quantization and the uncertainty principle.  Physics Letters A. 317 (5–6), 365–369 (2003).
  22. de Gosson M.  On the goodness of quantum blobs in phase space quantization.  Preprint ArXiv: quant-ph/0407129 (2004).
  23. de Gosson M.  Symplectic Geometry and Quantum Mechanics.  Birkhauser Verlag (2006).
  24. de Gosson M.  Symplectic Non-Squeezing Theorems, Quantization of Integrable Systems, and Quantum Uncertainty.  Preprint ArXiv: math-ph/0602055v1 (2006).
  25. de Gosson M.  The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?  Foundations of Physics. 99, 194–214 (2009).
  26. Andersson O., Heydari H.  Geometric uncertainty relation for mixed quantum states.  Journal of Mathematical Physics. 55 (4), 042110 (2014).
  27. Heydari H.  A geometric framework for mixed quantum states based on a Kähler structure.  Journal of Physics A: Mathematical and Theoretical. 48 (25), 255301 (2015).
  28. Sanborn B. A.  The uncertainty principle and the energy identity for holomorphic maps in geometric quantum mechanics.  Preprint ArXiv: 1710.09344 (2017).