Abstract
In the last years new interest has been addressed to the fluid dynamical formulation of quantum mechanics of Madelung-De Broglie-Bohm. Various attempts to implement numerically this formulation have been presented recently concerning photodissociation and scattering, sharing the use of the lagrangian and semi-lagrangian method to discretize the spatial independent variables. The possibility of using moving numerical grids suited for wave packet propagation, in general computationally cheaper than fixed grids, is attractive but difficult when QFD is applied to molecular scattering. This aspect is discussed in the paper, and a particular solution of the problem is proposed. The result for transmission probability through the Eckart barrier as a function of momentum is obtained by Fourier transforming the final wave packet; the agreement with analytical results is good on a large range, the computational time is reasonable and the algorithm used is relatively simple in comparison to other ones proposed for solving this problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Madelung, V.E.: Z. Phys. 40, 332 (1926)
De Broglie, L.: C. R. Acad. Sci. Paris, 183, 447 (1926)
Bohm, D.: A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I. Phys. Rev. 85, 166 (1952)
Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, New York (1993)
Selleri, F.: The Wave-Particle Duality. Plenum, New York (1992)
Burant, J.C., Tully, J.C.: Nonadiabatic dynamics via the classical limit Schroedinger equation. Journal of Chemical Physics 112, 6097 (2000)
Lopreore, C.L., Wyatt, R.E.: Quantum Wave Packet Dynamics with Trajectories. Physical Review Letters 313, 189 (1999)
Bittner, E.R., Wyatt, R.E.: Integrating the quantum Hamilton–Jacobi equations by wavefront expansion and phase space analysis. Journal of Chemical Physics 113, 8888 (2000)
Trahan, C.J., Wyatt, R.E.: An arbitrary Lagrangian-Eulerian approach to solving the quantum hydrodynamic equations of motion: equidistribution with “smart” springs. Journal of Chemical Physics 118, 4784–4790 (2003); and references therein
Pagano, D.: Application of the Hydrodynamic Formulation of Quantum Mechanics to Atom-Surface Dynamics. Ph.D Thesis in Chimica dei Materiali Innovativi-XV Ciclo, University of Bari (2003)
Leforestier, C., Bergeron, G., Hiberty, P.C.: A quantum mechanical investigation of a collinear model for collision-induced dissociation. Chemical Physics Letters 84, 385–389 (1981)
Hu, X., Ho, T., Rabitz, H.: Multivariate Radial Basis Interpolation for Quantum Fluid Dynamical Equations. Computers and mathematics with applications 43, 525–537 (2002)
Billing, G.D., Mikkelsen, K.V.: Advanced Molecular Dynamics and Chemical Kinetics, ch. 5
Kuppermann, A.: A Bohmian view of quantum reaction dynamics, oral presentation. In: Vth Workshop on Quantum Reaction Scattering, Dept. of Chemistry, University of Perugia (Italy), June 25-27 (1999)
Oriols, X., Martin, F., Sune, J.: Oscillatory Bohm trajectories in resonant tunneling structures. Solid State Communications 99, 123–128 (1996)
Kendrick, B.K.: A new method for solving the quantum hydrodynamic equations of motion. Journal of Chemical Physics 119, 5805–5817 (2003)
Baines, M.J.: Grid adaptation via node movement. Applied Numerical Mathematics 26, 77–96 (1998)
Esposito. F: work in preparation
Manolopoulos, D.E., Light, J.C.: A log derivative formulation of reaction rate theory. Chemical Physics Letters 216, 18–26 (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Esposito, F. (2004). Numerical Implementation of Quantum Fluid Dynamics: A Working Example. In: Laganá, A., Gavrilova, M.L., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds) Computational Science and Its Applications – ICCSA 2004. ICCSA 2004. Lecture Notes in Computer Science, vol 3044. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24709-8_32
Download citation
DOI: https://doi.org/10.1007/978-3-540-24709-8_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22056-5
Online ISBN: 978-3-540-24709-8
eBook Packages: Springer Book Archive