Fluid-like properties of the probability densities in quantum mechanics

American Journal of Physics

Volumn 69, Issue 4, 2001, Pages 470-475

  Mita, Katsunori ^a  

^a ST MARY'S COLLEGE OF MARYLAND   (United States)

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EID: 23044528184     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (23)

1. Except, for instance, C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977), pp. 824-827.
2. E. Madelung, "Quantum theory in a hydrodynamical form," Z. Phys. 40, 322-326 (1926).
3. T. Takabayashi, "On the formulation of quantum mechanics associated with classical pictures," Prog. Theor. Phys. 8, 143-182 (1952);
4. "Remarks on the formulation of quantum mechanics with classical pictures and on relations between linear scalar fields and hydrodynamical fields," Prog. Theor. Phys. ibid. 9, 187-222 (1953).
5. D. Bohm, "A suggested interpretation of the quantum theory in terms of 'hidden' variables. I," Phys. Rev. 85, 166-179 (1952);
6. "A suggested interpretation of the quantum theory in terms of 'hidden' variables. II," Phys. Rev. ibid. 85, 180-193 (1952).
7. S. K. Ghosh and B. M. Deb, "Densities, density-functionals and electron fluids," Phys. Rep. 92, 1-44 (1982). This report contains extensive literature on the hydrodynamic model.
8. M. Bunge, "Survey of the interpretations of quantum mechanics," Am. J. Phys. 24, 272-286 (1956).
9. Also, see M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), p. 291.
10. In its spirit, our approach is similar to that of Robinett who examined probability densities of various classical and quantum systems. In this paper, we are comparing the flows of probability densities with the flows of the classical, ideal fluid. See R. W. Robinett, "Quantum and classical probability distributions for position and momentum," Am. J. Phys. 63, 823-832 (1995).
11. The left operator frequently appears in the context of the adjoint equation in quantum field theory. See, for example, J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965), p. 54.
12. Cohen-Tannoudji et al. provide the probability density operator |r〉〈r| and probability current density operator K=(1/2m)[|r〉〈r|P+P|r〉〈r|]. Note that the second term in K acts as the left operator. One could rewrite the content of this paper in terms of these local operators. See Ref. 1, p. 239.
13. See Ref. 1, p. 238.
14. J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Redwood City, CA, 1985), pp. 101-103.
15. In the hydrodynamic model, the mass flux density is factored into the mass density and local velocity. However, this "velocity" is stripped of any sense of probability, and thus it is hard to give it a quantum mechanical interpretation.
16. R. W. Robinett, Quantum Mechanics (Oxford Univ., New York, 1997), p. 90.
17. G. Arfken, Mathematical Methods for Physicists (Academic, Orlando, 1985), p. 41.
18. See Ref. 1, p. 826, Eq. (11).
19. Compare Eq. (38) with ∂u/∂t + ▽·S=-J·E, where H is the energy density of the electromagnetic field, S is the Poynting vector, and J·E is the work done on charged particles per unit volume by the local field. With correspondences ψ*Tψ⇔u, ψ*ΘTψ⇔S, and (ψ*Θψ)·(-▽V) ⇔-J·E, these two equations are identical inform. This is not surprising since both equations express local conservation of energy.
20. See, for example, Ref. 1, p. 827.
21. There are two formulations of fluid dynamics. In the Lagrangian formulation, one describes the path of a fluid particle. In the Eulerian formulation, one describes the behavior of the fluid at a given spatial location.
22. See, for example, L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959), pp. 10-14.
23. See, for example, R. A. Granger, Fluid Mechanics (Dover, New York, 1995), p. 31.