Since its inception Bohmian mechanics has been generally regarded as a hidden-variable theory aimed at providing an objective description of quantum phenomena. To date, this rather narrow conception of Bohm’s proposal has caused it more rejection than acceptance. Now, after 65 years of Bohmian mechanics, should still be such an interpretational aspect the prevailing appraisal? Why not favoring a more pragmatic view, as a legitimate picture of quantum mechanics, on equal footing in all respects with any other more conventional quantum picture? These questions are used here to introduce a discussion on an alternative way to deal with Bohmian mechanics at present, enhancing its aspect as an efficient and useful picture or formulation to tackle, explore, describe and explain quantum phenomena where phase and correlation (entanglement) are key elements. This discussion is presented through two complementary blocks. The first block is aimed at briefly revisiting the historical context that gave rise to the appearance of Bohmian mechanics, and how this approach or analogous ones have been used in different physical contexts. This discussion is used to emphasize a more pragmatic view to the detriment of the more conventional hidden-variable (ontological) approach that has been a leitmotif within the quantum foundations. The second block focuses on some particular formal aspects of Bohmian mechanics supporting the view presented here, with special emphasis on the physical meaning of the local phase field and the associated velocity field encoded within the wave function. As an illustration, a simple model of Young’s two-slit experiment is considered. The simplicity of this model allows to understand in an easy manner how the information conveyed by the Bohmian formulation relates to other more conventional concepts in quantum mechanics. This sort of pedagogical application is also aimed at showing the potential interest to introduce Bohmian mechanics in undergraduate quantum mechanics courses as a working tool rather than merely an alternative interpretation.
. [J]. Frontiers of Physics, 2019, 14(1): 11301.
A. S. Sanz. Bohm’s approach to quantum mechanics: Alternative theory or practical picture?. Front. Phys. , 2019, 14(1): 11301.
A. Einstein, Kinetic equilibrium of absorption and emission of blackbody radiation by an atom, Phys. Z. 18, 121 (1917)
2
D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables (I), Phys. Rev. 85(2), 166 (1952) https://doi.org/10.1103/PhysRev.85.166
3
D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables (II), Phys. Rev. 85(2), 180 (1952) https://doi.org/10.1103/PhysRev.85.180
4
D. Bohm, Wholeness and the Implicate Order, Routledge, New York, 1980
5
P. Pylkkänen, B. J. Hiley, and I. Pättiniemi, Individuals across the Sciences, Oxford University Press, New York, 2015
6
R. E. Wyatt, Quantum Dynamics with Trajectories, Springer, New York, 2005
7
P. K. Chattaraj, Ed., Quantum Trajectories, CRC Taylor and Francis, New York, 2010
8
K. H. Hughes and G. Parlant, eds., Quantum Trajectories, CCP6, Daresbury, UK, 2011
9
A. S. Sanz and S. Miret-Artés, A Trajectory Description of Quantum Processes (II): Applications, vol. 831 of Lecture Notes in Physics, Springer, Berlin, 2014
10
X. Oriols and J. Mompart, Eds., Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology, Pan Standford Publishing, Singapore, 2012
11
A. Benseny, G. Albareda, A. S. Sanz, J. Mompart, and X. Oriols, Applied Bohmian mechanics, Eur. Phys. J. D 68(10), 286 (2014) https://doi.org/10.1140/epjd/e2014-50222-4
12
Following Schiff [82] (see p. 171), here the term “picture” will also be used instead of “representation”, leaving the latter “for designation of the choice of axes in Hilbert space or, equivalently, the choice of the complete orthonormal set of functions, with respect to which the states and dynamical variables are specified.”
13
D. F. Styer, M. S. Balkin, K. M. Becker, M. R. Burns, C. E. Dudley, S. T. Forth, J. S. Gaumer, M. A. Kramer, D. C. Oertel, L. H. Park, M. T. Rinkoski, C. T. Smith, and T. D. Wotherspoon, Nine formulations of quantum mechanics, Am. J. Phys. 70(3), 288 (2002) https://doi.org/10.1119/1.1445404
14
For a more detailed account on David Bohm’s life and work the interested reader is encouraged to consult the bibliographical work published in 1997 by his colleague Hiley [Biogr. Mems. Fell. R. Soc. 43, 107 (1997)].
J. Mehra, The Beat of a Different Drum: The Life and Science of Richard Feynman, Oxford University Press, Oxford, 1994
19
Throughout this work, the concept of quantum particle or system is used to denote the degrees of freedom of interest necessary to characterize and study a real physical system. If such degrees of freedom are translational, they can be associated without loss of generality with the particle itself, since they account for its center of mass, as in classical mechanics. However, this is a particular case, since they may also describe vibrations, rotations, etc., and although Schrodinger’s equation and its Bohmian reformulation are still valid in these cases, they no longer refer to particles in the usual sense.
A. S. Sanz and S. Miret-Artés, Quantum phase analysis with quantum trajectories: A step towards the creation of a Bohmian thinking, Am. J. Phys. 80, 525 (2012) https://doi.org/10.1119/1.3698324
23
M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill, New York, 1966
24
W. H. Zurek and J. A. Wheeler, Quantum Theory of Measurement, Princeton University Press, Princeton, NJ, 1983
25
S. J. Hawking, The Dreams that Stuff is Made of, Running Press, Philadephia, PA, 2011
26
P. Marage and G. Wallenborn, Eds., The Solvay Councils and the Birth of Modern Physics, Vol. 22 of Science Networks. Historical Studies, Birkhäuser, Basel, 1999
27
J. T. Cushing, Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony, University of Chicago Press, Chicago, 1994
28
B. L.van der Waerden, Sources of Quantum Mechanics, Dover, New York, 1966
29
G. Bacciagaluppi and A. Valentini, Eds., Quantum Theory at the Crossroads. Reconsidering the 1927 Solvay Conference, Science Networks. Historical Studies, Cambridge University Press, Cambridge, 2009
30
J. von Neumann and R. T. B. (transl.), Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955
31
L. de Broglie, Non-Linear Wave Mechanics: A Causal Interpretation, Elsevier, Amsterdam, 1960
32
A. Einstein, B. Podolsky, and N. Rosen, Can quantummechanical description of physical reality be considered complete? Phys. Rev. 47(10), 777 (1935) https://doi.org/10.1103/PhysRev.47.777
33
E. Schrödinger and M. Born, Discussion of probability relations between separated systems, Math. Proc. Camb. Philos. Soc. 31(04), 555 (1935) https://doi.org/10.1017/S0305004100013554
34
E. Schrödinger and P. A. M. Dirac, Probability relations between separated systems, Math. Proc. Camb. Philos. Soc. 32(03), 446 (1936) https://doi.org/10.1017/S0305004100019137
L. Gilder, The Age of Entanglement. When Quantum Physics was Reborn, Alfred A. Knopf, New York, 2008
38
J. P. Dowling and G. J. Milburn, Quantum technology: the second quantum revolution, Phil. Trans. Roy. Soc. Lond. A 361(1809), 1655 (2003)
39
C. B. Parker, McGraw Hill Encyclopaedia of Physics, 2nd Ed., McGraw Hill, Princeton, 1994
40
A. Aspect, P. Grangier, and G. Roger, Experimental tests of realistic local theories via Bell’s theorem, Phys. Rev. Lett. 47(7), 460 (1981) https://doi.org/10.1103/PhysRevLett.47.460
41
A. Aspect, P. Grangier, and G. Roger, Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedanken experiment: A new violation of Bell’s inequalities, Phys. Rev. Lett. 49(2), 91 (1982) https://doi.org/10.1103/PhysRevLett.49.91
42
A. Aspect, J. Dalibard, and G. Roger, Experimental Test of Bell’s Inequalities Using Time- Varying Analyzers, Phys. Rev. Lett. 49(25), 1804 (1982) https://doi.org/10.1103/PhysRevLett.49.1804
43
The three experiments performed of Aspects and coworkers in these years set precedent for future experimental tests of Bell inequalities. However, it is worth noticing that they were not the first empirical tests. For an account on entanglement, from its appearance on stage to the theoretical and experimental works around Bell-type tests, the interested reader may find a fresh historical overview in Ref. [37] [see also: A. D. Aczel, Entanglement. The Greatest Mystery in Physics (Four Walls Eight Windows, New York, 2001)].
44
It is worth highlighting the special role of Chris Dewdney in this story, specifically in the rekindling of Bohm’s formulation, which arose as a direct consequence when he was looking for research problems for his PhD at Birkbeck. While doing such a search, he came across Bohm’s formulation and thought about the possibility to compute the trajectories and quantum potential for the twoslit experiment. His calculations of the trajectories plus Chris Philippidis’ calculations of the quantum potential gave rise to the 1979 paper “Quantum interference and the quantum potential” [45], after having “pressurized” Basil Hiley to take an interest on the issue. Then, after completing his PhD, other papers followed, extending the model neutron interferometry, spin measurement and superposition, EPR, etc. [15].
45
C. Philippidis, C. Dewdney, and B. J. Hiley, Quantum interference and the quantum potential, Nuovo Cim. B 52(1), 15 (1979) https://doi.org/10.1007/BF02743566
46
C. Dewdney and B. J. Hiley, A quantum potential description of one-dimensional time-dependent scattering from square barriers and square wells, Found. Phys. 12(1), 27 (1982) https://doi.org/10.1007/BF00726873
47
B. J. Hiley, Quantum mechanics: Historical contingency and the Copenhagen hegemony by James T. Cushing, Stud. Hist. Phil. Mod. Phys. 28(2), 299 (1997) https://doi.org/10.1016/S1355-2198(97)00005-1
48
D. Bohm and B. J. Hiley, The Undivided Universe, Routledge, New York, 1993
49
D. Dürr, S. Goldstein, and N. Zanghì, Quantum equilibrium and the origin of absolute uncertainty, J. Stat. Phys. 67(5–6), 843 (1992) https://doi.org/10.1007/BF01049004
50
D. Dürr and S. Teufel, Bohmian Mechanics: The Physics and Mathematics of Quantum Theory, Springer, Berlin, 2009
W. Bierter and H. L. Morrison, Derivation of the Landau quantum hydrodynamics for interacting Bose systems, J. Low Temp. Phys. 1(2), 65 (1969) https://doi.org/10.1007/BF00628262
56
E. A. Jr McCullough and R. E. Wyatt, Quantum dynamics of the collinear (H, H2) reaction, J. Chem. Phys. 51(3), 1253 (1969) https://doi.org/10.1063/1.1672133
57
E. A. Jr McCullough and R. E. Wyatt, Dynamics of the collinear H+H2 reaction (I): Probability density and flux, J. Chem. Phys. 54(8), 3578 (1971) https://doi.org/10.1063/1.1675384
58
E. A. Jr McCullough and R. E. Wyatt, Dynamics of the collinear H+H2 reaction (II): Energy analysis, J. Chem. Phys. 54(8), 3592 (1971) https://doi.org/10.1063/1.1675385
59
J. O. Hirschfelder, A. C. Christoph, and W. E. Palke, Quantum mechanical streamlines (I): Square potential barrier, J. Chem. Phys. 61(12), 5435 (1974) https://doi.org/10.1063/1.1681899
60
J. O. Hirschfelder, C. J. Goebel, and L. W. Bruch, Quantized vortices around wavefunction nodes (II), J. Chem. Phys. 61(12), 5456 (1974) https://doi.org/10.1063/1.1681900
61
J. O. Hirschfelder and K. T. Tang, Quantum mechanical streamlines (III): Idealized reactive atom–diatomic molecule collision, J. Chem. Phys. 64(2), 760 (1976) https://doi.org/10.1063/1.432223
62
J. O. Hirschfelder and K. T. Tang, Quantum mechanical streamlines (IV): Collision of two spheres with square potential wells or barriers, J. Chem. Phys. 65(1), 470 (1976) https://doi.org/10.1063/1.432790
63
J. O. Hirschfelder, The angular momentum, creation, and significance of quantized vortices, J. Chem. Phys. 67(12), 5477 (1977) https://doi.org/10.1063/1.434769
64
P. Lazzeretti and R. Zanasi, Inconsistency of the ringcurrent model for the cyclopropenyl cation, Chem. Phys. Lett. 80(3), 533 (1981) https://doi.org/10.1016/0009-2614(81)85072-5
65
P. Lazzeretti, E. Rossi, and R. Zanasi, Singularities of magnetic-field induced electron current density: A study of the ethylene molecule, Int. J. Quantum Chem. 25(6), 929 (1984) https://doi.org/10.1002/qua.560250602
S. Pelloni and P. Lazzeretti, Stagnation graphs and topological models of magnetic-field induced electron current density for some small molecules in connection with their magnetic symmetry, Int. J. Quantum Chem. 111(2), 356 (2011) https://doi.org/10.1002/qua.22658
J. A. N. F. Gomes, Topological elements of the magnetically induced orbital current densities, J. Chem. Phys. 78(7), 4585 (1983) https://doi.org/10.1063/1.445299
70
R. J. F. Berger, H. S. Rzepa, and D. Scheschkewitz, Ring Currents in the Dismutational Aromatic Si6R6, Angew. Chem. Int. Ed. 49(51), 10006 (2010) https://doi.org/10.1002/anie.201003988
71
Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Walking and orbiting droplets, Nature 437(7056), 208 (2005) https://doi.org/10.1038/437208a
S. Protière, A. Boudaoud, and Y. Couder, Particle–wave association on a fluid interface, J. Fluid Mech. 554(–1), 85 (2006)
74
E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, and Y. Couder, Path-memory induced quantization of classical orbits, Proc. Natl. Acad. Sci. USA 107(41), 17515 (2010) https://doi.org/10.1073/pnas.1007386107
D. M. Harris, J. Moukhtar, E. Fort, Y. Couder, and J. W. M. Bush, Wavelike statistics from pilot-wave dynamics in a circular corral, Phys. Rev. E 88, 011001(R) (2013)
The traditional definition for quantum observable in terms of expectation values is not appealed here, because the purpose is to offer a physical, real-lab view, where one deals with detectors that register individual signals (regardless of the inherent complexity involved in the technicalities of the detection process, and not with abstract algebraic objects.
81
I. Kant, Critique of Pure Reason, The Cambridge Edition of the Works of Immanuel Kant, Cambridge University Press, Cambridge, 1998), translated by P. Guyer and A. W. Wood
82
L. I. Schiff, Quantum Mechanics, 3rd Ed., McGraw-Hill, Singapore, 1968
83
W. Braunbek and G. Laukien, Optik (Stuttg.) 9, 174 (1952)
84
M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th Ed. Cambridge University Press, Cambridge, 1999 https://doi.org/10.1017/CBO9781139644181
85
R. D. Prosser, The interpretation of diffraction and interference in terms of energy flow, Int. J. Theor. Phys. 15(3), 169 (1976) https://doi.org/10.1007/BF01807089
T. Wünscher, H. Hauptmann, and F. Herrmann, Which way does the light go? Am. J. Phys. 70(6), 599 (2002) https://doi.org/10.1119/1.1450570
90
E. Hesse, Modelling diffraction during ray tracing using the concept of energy flow lines, J. Quant. Spectrosc. Radiat. Transf. 109(8), 1374 (2008) https://doi.org/10.1016/j.jqsrt.2007.11.002
91
A. S. Sanz, M. Davidović, M. Božić, and S. Miret-Artés, Understanding interference experiments with polarized light through photon trajectories, Ann. Phys. 325(4), 763 (2010) https://doi.org/10.1016/j.aop.2009.12.005
92
K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, Photon trajectories, anomalous velocities and weak measurements: A classical interpretation, New J. Phys. 15(7), 073022 (2013) https://doi.org/10.1088/1367-2630/15/7/073022
93
R. V. Waterhouse, T. W. Yates, D. Feit, and Y. N. Liu, Energy streamlines of a sound source, J. Acoust. Soc. Am. 78(2), 758 (1985) https://doi.org/10.1121/1.392445
E. A. Skelton and R. V. Waterhouse, Energy streamlines for a spherical shell scattering plane waves, J. Acoust. Soc. Am. 80(5), 1473 (1986) https://doi.org/10.1121/1.394402
96
R. V. Waterhouse, D. G. Crighton, and J. E. Ffowcs-Williams, A criterion for an energy vortex in a sound field, J. Acoust. Soc. Am. 81(5), 1323 (1987) https://doi.org/10.1121/1.394537
M. Davidović, Á. S. Sanz, and M. Božić, Description of classical and quantum interference in view of the concept of flow line, J. Russ. Laser Res. 36(4), 329 (2015) https://doi.org/10.1007/s10946-015-9507-y
G. H. Yuan, S. Vezzoli, C. Altuzarra, E. T. F. Rogers, C. Couteau, C. Soci, and N. I. Zheludev, Quantum superoscillation of a single photon, Light Sci. Appl. 5(8), e16127 (2016) https://doi.org/10.1038/lsa.2016.127
101
Z. Y. Zhou, Z. H. Zhu, S. L. Liu, Y. H. Li, S. Shi, D. S. Ding, L. X. Chen, W. Gao, G. C. Guo, and B. S. Shi, Quantum twisted double-slits experiments: Confirming wavefunctions’ physical reality,Sci. Bull. 62(17), 1185 (2017) https://doi.org/10.1016/j.scib.2017.08.024
102
G. L. Long, W. Qin, Z. Yang, and J. L. Li, Realistic interpretation of quantum mechanics and encounter-delayedchoice experiment, Sci. China Phys. Mech. Astron. 61(3), 030311 (2018) https://doi.org/10.1007/s11433-017-9122-2
A. S. Sanz, M. Davidović, and M. Božić, Full quantum mechanical analysis of atomic three-grating Mach– Zehnder interferometry, Ann. Phys. 353, 205 (2015) https://doi.org/10.1016/j.aop.2014.11.012
105
A. S. Sanz, F. Borondo, and S. Miret-Artés, Particle diffraction studied using quantum trajectories, J. Phys.: Condens. Matter 14(24), 6109 (2002) https://doi.org/10.1088/0953-8984/14/24/312
106
A. S. Sanz, Investigating Puzzling Aspects of the quantum theory by means of its hydrodynamic formulation, Found. Phys. 45(10), 1153 (2015) https://doi.org/10.1007/s10701-015-9917-2
S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Observing the average trajectories of single photons in a two-slit interferometer, Science 332(6034), 1170 (2011) https://doi.org/10.1126/science.1202218
111
B. Braverman and C. Simon, Proposal to observe the nonlocality of Bohmian trajectories with entangled photons, Phys. Rev. Lett. 110(6), 060406 (2013) https://doi.org/10.1103/PhysRevLett.110.060406
112
W. P. Schleich, M. Freyberger, and M. S. Zubairy, Reconstruction of Bohm trajectories and wave functions from interferometric measurements, Phys. Rev. A 87(1), 014102 (2013) https://doi.org/10.1103/PhysRevA.87.014102
113
M. H. Shamos (Ed.), Great Experiments in Physics, Dover, New York, 1987
114
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge, 1987
115
Concerning this aspect, it is worth mentioning that some recent models [116] conclude that phase correlations arise when a point-like treatment is assumed for quantum particles, particularly for electrons. If such particles are understood as extended objects, then local variables are claimed to be sufficient to explain such correlations.
