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Number of results

Journal

2010 | 8 | 1 | 42-48

Article title

On some hydrodynamical aspects of quantum mechanics

Authors

Content

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Languages of publication

EN

Abstracts

EN
In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the complex polynomial (i.e. Borel-Weil) realization of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dimensional geometric quantum mechanics, the Schrödinger velocity field on projective Hilbert space is divergence-free (being Killing with respect to the Fubini-Study metric) and fulfils the stationary Euler equation, with pressure proportional to the Hamiltonian uncertainty (squared). We explicitly determine the critical points of the pressure of this “Schrödinger fluid”, together with its vorticity, which turns out to depend on the spacings of the energy levels. These results follow from hydrodynamical properties of Killing vector fields valid in any (finite dimensional) Riemannian manifold, of possible independent interest.

Publisher

Journal

Year

Volume

8

Issue

1

Pages

42-48

Physical description

Dates

published
1 - 2 - 2010
online
15 - 11 - 2009

Contributors

author
  • Dipartimento di Informatica, Università degli Studi di Verona, Ca’ Vignal 2, Strada le Grazie 15, I-37134, Verona, Italia

References

  • [1] R. Abraham, J. E. Marsden, Foundations of Mechanics (Benjamin/Cummings, Reading, MA, 1978)
  • [2] V. I. Arnol’d, B. Khesin, Topological Methods in Hydrodynamics (Springer, Berlin, 1998)
  • [3] A. Ashtekar, T. A. Schilling, In: On Einstein’s path (Springer, New York, 1999) 23
  • [4] A. Benvegnu, N. Sansonetto, M. Spera, J. Geom. Phys. 51, 229 (2004) http://dx.doi.org/10.1016/j.geomphys.2003.10.008[Crossref]
  • [5] A. Benvegnu, M. Spera, Rev. Math. Phys. 18, 1075 (2006) http://dx.doi.org/10.1142/S0129055X06002863[Crossref]
  • [6] R. Bott, L. T. Tu, Differential Forms in Algebraic Topology (Springer, Berlin, 1982)
  • [7] D. C. Brody, L. P. Hughston, J. Geom. Phys. 38, 19 (2001) http://dx.doi.org/10.1016/S0393-0440(00)00052-8[Crossref]
  • [8] D. Chruscinski, A. Jamiołkowski, Geometric Phases in Classical and Quantum Mechanics (Birkhauser, Boston, 2004)
  • [9] R. Cirelli, M. Gatti, A. Manià, J. Geom. Phys. 45, 267 (2003) http://dx.doi.org/10.1016/S0393-0440(01)00031-6[Crossref]
  • [10] R. Cirelli, A. Mania, L. Pizzocchero, J. Math. Phys. 31, 2891 (1990) http://dx.doi.org/10.1063/1.528941[Crossref]
  • [11] R. Cirelli, L. Pizzocchero, Nonlinearity 3, 1057 (1990) http://dx.doi.org/10.1088/0951-7715/3/4/006[Crossref]
  • [12] M. do Carmo, Riemannian Geometry (Birkhäuser, Boston, 1992)
  • [13] D. G. Ebin, J. E. Marsden, Ann. Math. 92, 102 (1970) http://dx.doi.org/10.2307/1970699[Crossref]
  • [14] S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, (Springer, Heidelberg, 1987)
  • [15] G. Gaeta, M. Spera, Lett. Math. Phys. 16, 187 (1988) http://dx.doi.org/10.1007/BF00398955[Crossref]
  • [16] P. Griffiths, J. Harris, Principles of Algebraic Geometry (J. Wiley & Sons, New York, 1978)
  • [17] T. Guenault, Basic Superfluids (Taylor & Francis, London, 2003)
  • [18] R. C. Gunning, Lectures on Riemann Surfaces (Princeton University Press, Princeton, New Jersey, 1966)
  • [19] A. Heslot, Phys. Rev. D 31, 1341 (1985) http://dx.doi.org/10.1103/PhysRevD.31.1341[Crossref]
  • [20] E. Joos et al., Decoherence and the Appearance of a Classical World in Quantum Theory (Springer, Berlin, 2003)
  • [21] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry I (Wiley - Interscience Publishers, New York, 1963)
  • [22] L. D. Landau, M. E. Lifšits, Quantum Mechanics (Pergamon, London, 1960)
  • [23] J. E. Marsden, A. Weinstein, Physica 7 D, 305 (1983)
  • [24] D. McDuff, D. Salamon, Introduction to Symplectic Topology (Clarendon Press, Oxford, 1998)
  • [25] R. Narasimhan, Lectures on Riemann surfaces (Birkhauser, Basel, 1994)
  • [26] V. Penna, M. Spera, J. Geom. Phys. 27, 99 (1998) http://dx.doi.org/10.1016/S0393-0440(97)00070-3[Crossref]
  • [27] A. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin, 1986)
  • [28] M. Spera, J. Geom. Phys. 12, 165 (1993) http://dx.doi.org/10.1016/0393-0440(93)90032-A[Crossref]
  • [29] M. Spera, Milan Journal of Mathematics 74, 139 (2006) http://dx.doi.org/10.1007/s00032-006-0061-5[Crossref]
  • [30] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations (Springer, New York, 1996)
  • [31] A. Tyurin, Quantization, Classical and Quantum Field Theory and Theta Functions, CRM Monograph Series (AMS, Providence, RI, 2003)
  • [32] N. Woodhouse, Geometric Quantization (Clarendon Press, Oxford, 1992)

Document Type

Publication order reference

Identifiers

YADDA identifier

bwmeta1.element.-psjd-doi-10_2478_s11534-009-0070-4
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