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Some Selected Applications of Bohmian Mechanics

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Bohmian Mechanics, Open Quantum Systems and Continuous Measurements

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Abstract

In this chapter, our purpose is to simply show that Bohmian mechanics is a powerful route to bring about new solutions to problems discussed by conventional quantum mechanical approaches, apart from allowing some striking correspondence between both frameworks. This goal is carried out by choosing some key quantum mechanical problems in the framework of Bohmian mechanics such as, for example, the so-called Ermakov–Bohm invariants, boundary conditions and uncertainty principle in tunneling, the quantum traversal time, Airy wave packets and Airy slits, the detection of inertial and gravitational masses with Airy wave packets, the geometric phase analyzing the Aharonov–Bohm effect and quantum vortices, the reformulation of the Gross–Pitaevskii equation within the hydrodynamical framework and, finally, the study of simple dissipative dynamics by using the well-known Caldirola-Kanai Hamiltonian. In this dissipative scenario, the motion of a free particle, the quantum interference of two wave packets and the dynamics in a linear potential as well as the corresponding of a damped harmonic oscillator (within the underdamped, critically damped and overdamped regimes) are finally analyzed for ulterior references.

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References

  1. Bernstein, J.: More about Bohm’s quantum. Am. J. Phys. 79, 601–606 (2011)

    Article  ADS  Google Scholar 

  2. Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  3. Goldstein, S.: Quantum theory without observers - Part I. Phys. Today, 51(3), 42–46 (1998); Part II. Phys. Today 51(4), 38–42 (1998)

    Google Scholar 

  4. Wyatt, R.E.: Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics. Springer, New York (2005)

    MATH  Google Scholar 

  5. Sanz, A.S., Miret-Artés, S.: A trajectory description of quantum processes. Part I. Fundamentals. Lect. Notes Phys. 850, 1–299 (2012)

    Article  MATH  Google Scholar 

  6. Sanz, A.S., Miret-Artés, S.: A trajectory description of quantum processes. Part II. Applications. Lect. Notes Phys. 831, 1–333 (2014)

    Article  MATH  Google Scholar 

  7. Nassar, A.B.: Time-dependent harmonic oscillator: An Ermakov-Nelson process. Phys. Rev. A 32, 1862–1863 (1985) (See references there in)

    Google Scholar 

  8. Lewis, H.R.: Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians. Phys. Rev. Lett. 18, 510–512 (1967)

    Article  ADS  Google Scholar 

  9. Ray, J.R.: Quantum invariants. Phys. Rev. 28, 2603–2605 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  10. Ermakov, V.P.: Second-order differential equations. Conditions of complete integrability, Univ. Izv. Kiev Ser. III 9 1–25 (1880)

    Google Scholar 

  11. Milne, W.E.: The numerical determination of characteristic numbers. Phys. Rev. 35, 863–867 (1930)

    Article  ADS  Google Scholar 

  12. Pinney, E.: The nonlinear differential equation \(y^{\prime \prime }+p(x)y+c{{y}^{-3}}=0\). Proc. Am. Math. Soc. 1, 681–681 (1950)

    MathSciNet  MATH  Google Scholar 

  13. Nassar, A.B.: New quantum squeezed states for the time-dependent harmonic oscillator. J. Opt. B: Quantum Semiclassical Opt. 3, S226–S228 (2002) (See references therein)

    Google Scholar 

  14. Davydov, A.S.: Quantum Mechanics, 2nd edn. Pergamon Press, Oxford (1965)

    Google Scholar 

  15. Merzbacher, E.: Quantum Mechanics, 2nd edn. Wiley, New York (1970)

    MATH  Google Scholar 

  16. Schiff, L.I.: Quantum Mechanics, 3rd edn. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  17. French, A.P., Taylor, E.F.: Quantum Mechanics. MIT Series. W. W. Norton, New York (1978)

    Google Scholar 

  18. Nassar, A.B.: Boundary conditions in tunneling via quantum hydrodynamics. NASA Conf. Publ. 3197, 149–154 (1993)

    Google Scholar 

  19. Nassar, A.B.: Quantum traversal time. Phys. Rev. A 38, 683–687 (1988)

    Article  ADS  Google Scholar 

  20. Koonin, S.E.: Computational Physics. Benjamin/Cummings Publication, New York (1985) (See in part B, example 7)

    Google Scholar 

  21. Goldberg, A., Schey, H.M., Schwartz, J.L.: Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena. Am. J. Phys. 35, 177–186 (1967)

    Article  ADS  Google Scholar 

  22. Huang, Z.H., Cutler, P.H., Feuchtwang, T.E., Good Jr., R.H., Kazes, E., Nguyen, H.Q., Park, S.K.: Computer simulation of a wave packet tunneling through a square barrier. IEEE 36, 2665–2670 (1989)

    Google Scholar 

  23. https://phet.colorado.edu/en/simulation/quantum-tunneling

  24. Hartman, T.E.: Tunneling of a wave packet. J. Appl. Phys. 33, 3427–3433 (1962)

    Article  ADS  Google Scholar 

  25. Winful, H.G.: Delay time and the Hartman effect in quantum tunneling. Phys. Rev. Lett. 91, 260401(1–4) (2003)

    Google Scholar 

  26. Nassar, A.B.: Scattering via invariants of quantum hydrodynamics. Phys. Lett. 146, 89–92 (1990)

    Article  MathSciNet  Google Scholar 

  27. Reid, J.R., Ray, J.R.: Ermakov systems, nonlinear superposition, and solutions of nonlinear equations of motion. J. Math. Phys. 21, 1583–1587 (1980)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Lutzky, M.: Noether’s theorem and the time-dependent harmonic oscillator. Phys. Lett. A 68, 3–4 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  29. Büttiker, M.: Larmor precession and the traversal time for tunneling. Phys. Rev. 27, 6178–6188 (1983)

    Article  Google Scholar 

  30. Nassar, A.B., Bassalo, J.M.F., Alencar, P.T.S., Cancela, L.S.G.: Wave propagator via quantum fluid dynamics. Phys. Rev. 56, 1230–1233 (1997)

    ADS  Google Scholar 

  31. Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)

    MATH  Google Scholar 

  32. Berry, M.V., Balazs, N.L.: Nonspreading wave packets. Am. J. Phys. 47, 264–267 (1979)

    Article  ADS  Google Scholar 

  33. Vallée, O., Soares, M.: Airy Function and Applications to Physics. World Scientific, Hackensack (2004)

    Book  MATH  Google Scholar 

  34. Siviloglou,G.A., Broky, J., Dogariu, A., Christodoulides, D.N.: Observation of accelerating Airy beams. Phys. Rev. Lett. 99, 213901(1–4) (2007) (See references therein)

    Google Scholar 

  35. Stützle, R., Gø:bel, M.C., Hø:rner, Th., Kierig, E., Mourachko, I., Oberthaler, M.K., Fedorov, M.V., Yakovlev, V.P., van Leeuwen, K.A.H., Schleich, W.P.: Observation of nonspreading wave packets in an imaginary potential. Phys. Rev. Lett. 95, 110405(1–4) (2005)

    Google Scholar 

  36. Gutiérrez-Vega, J.C., Iturbe-Castillo, M.D., Chávez-Cerda, S.: Alternative formulation for invariant optical fields: Mathieu beams. Opt. Lett. 25, 1493–1495 (2000)

    Article  ADS  Google Scholar 

  37. Bandres, M.A., Gutiérrez-Vega, J.C., Chavez-Cerda, S.: Parabolic nondiffracting optical wave fields. Opt. Lett. 29, 44–46 (2004)

    Article  ADS  Google Scholar 

  38. Durnin, J.: Exact solutions for nondiffracting beams. I. The scalar theory. J. Opt. Soc. Am. A 4, 651–654 (1987)

    Article  ADS  Google Scholar 

  39. Durnin, J., Miceli, J.J., Eberly, J.H.: Diffraction-free beams. Phys. Rev. Lett. 58, 1499–1501 (1987)

    Article  ADS  Google Scholar 

  40. Durnin, J., Miceli, J.J., Eberly, J.H.: Comment on Bessel and Gaussian beams. Phys. Rev. Lett. 66, 838–838 (1991)

    Article  ADS  Google Scholar 

  41. Besieris, I.M., Shaarawi, A.M.: Accelerating Airy wave packets in the presence of quadratic and cubic dispersion. Phys. Rev. E 78, 046605(1–6) (2008)

    Google Scholar 

  42. Besieris, I.M., Shaarawi, A.M., Ziolkowski, R.W.: Nondispersive accelerating wave packets. Am. J. Phys. 62, 519–521 (1994)

    Article  ADS  Google Scholar 

  43. Unnikrishnan, K., Rau, A.R.P.: Uniqueness of the Airy packet in quantum mechanics. Am. J. Phys. 64, 1034–1036 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Greenberger, D.M.: Nonspreading wave packets. Am. J. Phys. 48, 256–256 (1980)

    Article  ADS  Google Scholar 

  45. Nassar, A.B., Bassalo, J.F., Alencar, P.T.S.: Dispersive Airy packets. Am. J. Phys. 63, 849–852 (1995)

    Article  ADS  Google Scholar 

  46. Ray, J.R.: Exact solutions to the time-dependent Schrödinger equation. Phys. Rev. A 26, 729–733 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  47. Burgan, J.R., Feix, M.R., Fijalkow, E., Munier, A.: Solution of the multidimensional quantum harmonic oscillator with time-dependent frequencies through Fourier, Hermite and Wigner transforms. Phys. Lett. A 74, 11–14 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. These space-time transformations used here are a generalization of those found in references [45, 46]. However, Equations (2.166) and (2.167) have clear physical meaning; they are essential in determining the trajectories of the Airy packet

    Google Scholar 

  49. Ruby, L.: Applications of the Mathieu equation. Am. J. Phys. 64, 39–44 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  50. Nassar, A.B., Machado, F.L.A.: Solvable Hill and Pø:schl-Teller equations. Phys. Rev. A 35, 3159–3160 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  51. Bessel, F.W.: Versuche ber die Kraft, mit welcher die Erde Krper von verschiedener Beschaffenheit anzieht. Ann. Phys. 101, 401–417 (1832)

    Article  Google Scholar 

  52. Eötvös, R.V., Pekár, D., Fekete, E.: Beitrge zum Gesetze der Proportionalitt von Trgheit und Gravitt. Ann. Phys. 373, 11–66 (1922)

    Article  Google Scholar 

  53. Roll, P.G., Krotkov, R., Dicke, R.H.: The equivalence of inertial and passive gravitational mass. Ann. Phys. (N.Y.) 26, 442–517 (1964)

    Google Scholar 

  54. Kajari, E., Harshman, N.L., Rasel, E.M., Stenholm, S., Süssmann, G., Schleich, W.P.: Inertial and gravitational mass in quantum mechanics. Appl. Phys. B 100, 4360 (2010) (See references therein)

    Google Scholar 

  55. Colella, R., Overhauser, A.W., Werner, S.A.: Observation of gravitationally induced quantum interference. Phys. Rev. Lett. 34, 1472–1474 (1975)

    Article  ADS  Google Scholar 

  56. Greenberger, D.M., Overhauser, A.W.: Coherence effects in neutron diffraction and gravity experiments. Rev. Mod. Phys. 51, 43–78 (1979)

    Article  ADS  Google Scholar 

  57. Bonse, U., Wroblewski, T.: Measurement of neutron quantum interference in noninertial frames. Phys. Rev. Lett. 51, 1401–1404 (1983)

    Article  ADS  Google Scholar 

  58. Peters, A., Chung, K.Y., Chu, S.: Measurement of gravitational acceleration by dropping atoms. Nature 400, 849–852 (1999)

    Article  ADS  Google Scholar 

  59. Müller, H., Peters, A., Chu, S.: A precision measurement of the gravitational redshift by the interference of matter waves. Nature 463, 926–930 (2010)

    Article  ADS  Google Scholar 

  60. Lämmerzahl, C.: On the equivalence principle in quantum theory. Gen. Relativ. Gravit. 28, 1043–1070 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. Viola, L., Onofrio, R.: Testing the equivalence principle through freely falling quantum objects. Phys. Rev. D 55, 455–462 (1997)

    Article  ADS  Google Scholar 

  62. Hughes, K.J., Burke, J.H.T., Sackett, C.A.: Suspension of atoms using optical pulses, and application to gravimetry. Phys. Rev. Lett. 102, 150403(1–4) (2009)

    Google Scholar 

  63. Fray, S., Alvarez Diez, C., H\({\ddot{\rm n}}\)sch, T.W., Weitz, M.: Atomic interferometer with amplitude gratings of light and its applications to atom based tests of the equivalence principle. Phys. Rev. Lett. 93, 240404(1–4) (2004)

    Google Scholar 

  64. Kasevich, M., Chu, S.: Atomic interferometry using stimulated Raman transitions. Phys. Rev. Lett. 67, 181–184 (1991)

    Google Scholar 

  65. Aminoff, C.G., Steane, A.M., Bouyer, P., Desbiolles, P., Dalibard, J., Cohen-Tannoudji, C.: Cesium atoms bouncing in a stable gravitational cavity. Phys. Rev. Lett. 71, 3083–3086 (1993)

    Google Scholar 

  66. Ovchinnikov, Yu.B., Manek, I., Grimm, R.: Surface trap for Cs atoms based on evanescent-wave cooling. Phys. Rev. Lett. 79, 2225–2228 (1997)

    Google Scholar 

  67. Wallis, H., Dalibard, J., Cohen-Tannoudji, C.: Sisyphus cooling of a bound atom. Appl. Phys. B 54, 407–419 (1992)

    Article  ADS  Google Scholar 

  68. Shapere, A., Wilczek, F. (eds.): Geometric Phases in Physics. World Scientific, Singapore (1989)

    MATH  Google Scholar 

  69. Markowski, B., Vinitskii, S.I.: Topological Phases in Quantum Theory. World Scientific, Singapore (1989)

    Google Scholar 

  70. Bohm, A., Mostafazadeh, A., Koizumi, H., Niu, Q., Zwanziger, J.: The Geometric Phase in Quantum Systems. Springer, Heidelberg (2003)

    Book  MATH  Google Scholar 

  71. Chruscinski, D., Jamiolkpwski, A.: Geometric Phases in Classical and Quantum Mechanics. Birkhäuser, Boston (2004)

    Book  Google Scholar 

  72. Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. Roy. Soc. Lond. A 392, 45–57 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  73. Aharonov, Y., Anandan, A.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  74. Samuel, J., Bhandari, R.: General setting for Berry’s phase. Phys. Rev. Lett. 60, 2339–2342 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  75. Fock, V.: ber die Beziehung zwischen den Integralen der quantenmechanischen Bewegungsgleichungen und der Schrdingerschen Wellengleichung. Z. Phys. 49, 323–338 (1928)

    Article  ADS  MATH  Google Scholar 

  76. Bortolotti, F.: Sulle rappresentazioni conformi, e su di una interpretazione fisica del parallelismo di Levi-Civita. Rend. R. Naz. Lincei IV, 552–556 (1926)

    Google Scholar 

  77. Rytov, S.M.: On the transition from wave to geometric optics. Dokl. Akad. Nauk. USSR 18, 263–266 (1938)

    MATH  Google Scholar 

  78. Pancharatnam, S.: Generalized theory of interference and its applications. Proc. Ind. Acad. Sci. Ser. A 44, 247–262 (1956)

    MathSciNet  Google Scholar 

  79. Longuet-Higgins, H.C., Öpik, U., Pryce, M.H.L., Sack, R.A.: Studies of the Jahn-Teller effect. II. The dynamical problem. Proc. Roy. Soc. Lond. A 244, 1–16 (1958)

    Google Scholar 

  80. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959)

    Google Scholar 

  81. Mead, C.A., Truhlar, D.G.: On the determination of Born-Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei. J. Chem. Phys. 70, 2284–2296 (1979)

    Article  ADS  Google Scholar 

  82. Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry‘s phase. Phys. Rev. Lett. 51, 2167–2170 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  83. Mukunda, N., Simon, R.: Quantum kinematic approach to the geometric phase I. General formalism. Ann. Phys. (NY) 228, 205–268 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  84. Sjöqvist, E., Carlsen, H.: Geometric phase, quantum measurements, and the de Broglie-Bohm model. Phys. Rev. A 56, 1638–1641 (1997)

    Article  ADS  Google Scholar 

  85. Parmenter, R.H., Valentine, R.W.: Properties of the geometric phase of a de Broglie-Bohm causal quantum mechanical trajectory. Phys. Lett. A 219, 7–14 (1996)

    Article  ADS  Google Scholar 

  86. Chou, C.-C., Wyatt, R.E.: Geometric phase in Bohmian mechanics. Ann. Phys. (NY) 325, 2234–2250 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  87. Bohm, D., Hiley, B.J.: The Undivided Universe. Routledge, London (1993)

    MATH  Google Scholar 

  88. Olariu, S., Popescu, I.I.: The quantum effects of electromagnetic fluxes. Rev. Mod. Phys. 57, 339–436 (1985)

    Article  ADS  Google Scholar 

  89. Aharonov, Y., Casher, A.: Topological quantum effects for neutral particles. Phys. Rev. Lett. 53, 319–321 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  90. Cimmino, A., Opat, G.I., Klein, A.G., Kaiser, H., Werner, S.A., Arif, M., Clothier, R.: Observation of the topological Aharonov-Casher phase shift by neutron interferometry. Phys. Rev. Lett. 63, 380–383 (1989)

    Article  ADS  Google Scholar 

  91. Elion, W.J., Wachters, J.J., Sohn, L.L., Mooij, J.D.: Observation of the Aharonov-Casher effect for vortices in Josephson-junction arrays. Phys. Rev. Lett. 71, 2311–2314 (1993)

    Article  ADS  Google Scholar 

  92. Sangster, K., Hinds, E.A., Barnett, S.M., Riis, E.: Measurement of the Aharonov-Casher phase in an atomic system. Phys. Rev. Lett. 71, 3641–3644 (1993)

    Article  ADS  Google Scholar 

  93. Konig, M., Tschetschetkin, A., Hankiewicz, E.M., Sinova, J., Hock, V., Daumer, V., Schäfer, M., Becker, C.R., Buhmann, H., Molenkamp, L.W.: Direct observation of the Aharonov-Casher phase. Phys. Rev. Lett. 96, 076804(1–4) (2006)

    Google Scholar 

  94. Pitaesvskii, L.P.: Vortex lines in an imperfect bose gas. Soviet Phys. JETP 13, 451–454 (1961)

    Google Scholar 

  95. Gross, E.P.: Structure of a quantized vortex in boson systems. Nuovo Cimento 20, 454–457 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  96. Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P.: Statistical Physics: Theory of the Condensed State. Course of Theoretical Physics. Butterworth-Heinemann, Oxford (1980)

    Google Scholar 

  97. Nassar, A.B., Bassalo, J.M.F., Alencar, P.T.S., de Souza, J.F., de Oliveira, J.E., Cattani, M.: Gaussian solitons in nonlinear Schrödinger equation. Il Nuovo Cimento 117, 941–946 (2002)

    Google Scholar 

  98. Goldstein, H.: Classical Mechanics. Addison-Wesley Publishing Company, Reading (1980)

    MATH  Google Scholar 

  99. Razavy, M.: Classical and Quantum Dissipative Systems. Imperial College Press, London (2005)

    MATH  Google Scholar 

  100. Caldirola, P.: Forze non conservative nella meccanica quantisitica. Nuovo Cimento 18, 393–400 (1941)

    Google Scholar 

  101. Kanai, E.: On the quantization of the dissipative systems. Prog. Theor. Phys. 3, 440–442 (1948)

    Article  ADS  Google Scholar 

  102. Sanz, A.S., Martinez-Casado, R., Peñate-Rodriguez, H.C., Rojas-Lorenzo, G., Miret-Artés, S.: Dissipative Bohmian mechanics within the Caldirola-Kanai framework: a trajectory analysis of wave-packet dynamics in viscid media. Ann. Phys. 347, 1–20 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  103. Sanz, A.S., Miret–Artés, S.: A trajectory-based understanding of quantum interference. J. Phys. A 41, 435303(1–23) (2008)

    Google Scholar 

  104. Sanz, A.S., Miret-Artés, S.: Setting-up tunneling conditions by means of Bohmian mechanics. J. Phys. A: Math. Theor. 44, 485301(1–17) (2011)

    Google Scholar 

  105. Heller, E.J.: Time-dependent approach to semiclassical dynamics. J. Chem. Phys. 62, 1544–1555 (1975)

    Google Scholar 

  106. Vandyck, M.A.: On the damped harmonic oscillator in the de Broglie-Bohm hidden-variable theory. J. Phys. A: Math. Gen. 27, 1743–1750 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Sciences, Harvard-Westlake School and UCLA Extension, Los Angeles, CA, USA

    Antonio B. Nassar

  2. Superior Council for Scientific Research, Institute of Fundamental Physics , Madrid, Spain

    Salvador Miret-Artés

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  1. Antonio B. Nassar
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Nassar, A.B., Miret-Artés, S. (2017). Some Selected Applications of Bohmian Mechanics. In: Bohmian Mechanics, Open Quantum Systems and Continuous Measurements. Springer, Cham. https://doi.org/10.1007/978-3-319-53653-8_2

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