Abstract
The Quantum Fluid Dynamics (QFD) representation has its foundations in the works of Madelung (1929), De Broglie (1930 - 1950) and Bohm (1950 - 1970). It is an interpretation of quantum mechanics with the goal to find classically identifiable dynamical variables at the sub-particle level. The approach leads to two conservation laws, one for "mass" and one for "momentum", similar to those in hydrodynamics for a compressible fluid with a particular constitutive law. The QFD equations are a set of nonlinear partial differential equations. This paper extends the QFD formalism of quantum mechanics to the Nonlinear Schrödinger and the Gross-Pitaevskii equation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Askar A, Cakmak AS, Rabitz H (1980) Nodal structure and global behavior of scattering wave functions. J Chem Phys 72:5287
Bohm D (1951) Quantum Theory. Prentice-Hall, Inc., New York
Bohm D, Bub J (1966) A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory. Rev Mod Phys 38:453
Bohr N (1935) Can quantum-mechanical description of physical reality be considered complete? Phys Rev 48:696
Braun D (2001) Dissipative Quantum Chaos and Decoherence, Springer Tracts in Modern Physics, vol 172. Springer, Berlin, Heidelberg
de Broglie L (1926) Sur la possibilité de relier les phénomènes d’interférences et de diffraction à la théorie des quanta de lumière. Compt Rend Acad Sci Paris 183:447
de Broglie L (1951) Remarques sur la théorie de l’onde pilote. Compt Rend Acad Sci Paris 233:641
de Broglie L (1957) Idées nouvelles concernant les systèmes de corpuscules dans l’interprétation causale de la mécanique ondulatoire. Compt Rend Acad Sci Paris 244:529
de Broglie L (1967) Le mouvement brownien d’une particule dans son onde. Compt Rend Acad Sci Paris 264:1041
Dey B (1998) Multidimensional wave-packet dynamics within the fluid dynamical formulation of the schrödinger equation. J Chem Phys 109:8770
Einstein A, Podolsky B, Rosen N (1935) Can quantum-mechanical description of physical reality be considered complete? Phys Rev 47:777
Eringen AC (1962) Nonlinear Theory of Continuous Media. McGraw-Hill
Landau LD, Lifschitz EM (1959) Fluid Mechanics. Pergamon Press
Madelung E (1926) Quantentheorie in hydrodynamischer Form. Z Physik 40:322–326
Sales F (1999) Quantum fluid dynamics in the lagrangian representation and applications to photodissociation problems. J Chem Phys 111:2423
Schilp PA (ed) (1949) Albert Einstein, Philosopher-Scientist. Library of Living Philosophers, Evanston, Illinois
Styer DF, Balkin MS, Becker KM, Burns MR, Dudley CE, Forth ST, Gaumer JS, Kramer MA, Oertel DC, Park LH, Rinkoski MT, Smith CT, Wotherspoon TD (2002) Nine formulations of quantum mechanics. American Journal of Physics 70(3):288–297
Weiner JH, Askar A (1971) Time-dependent perturbation calculations based on the hydrodynamic analogy to quantum mechanics. J Chem Phys 54:1108
Wyatt RE (2005) Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics, Interdisciplinary Applied Mathematics, vol 28. Springer, New York
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Askar, A. (2018). Nonlinear Schrödinger and Gross - Pitaevskii Equations in the Bohmian or Quantum Fluid Dynamics (QFD) Representation. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds) Generalized Models and Non-classical Approaches in Complex Materials 1. Advanced Structured Materials, vol 89. Springer, Cham. https://doi.org/10.1007/978-3-319-72440-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-72440-9_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72439-3
Online ISBN: 978-3-319-72440-9
eBook Packages: EngineeringEngineering (R0)