# Time-dependent wave packet propagation using quantum hydrodynamics

- 337 Downloads
- 12 Citations

**Part of the following topical collections:**

## Abstract

A new approach for propagating time-dependent quantum wave packets is presented based on the direct numerical solution of the quantum hydrodynamic equations of motion associated with the de Broglie–Bohm formulation of quantum mechanics. A generalized iterative finite difference method (IFDM) is used to solve the resulting set of non-linear coupled equations. The IFDM is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a “smart” Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is generalized to treat higher-dimensional problems and anharmonic potentials. The method is applied to a one-dimensional Gaussian wave packet scattering from an Eckart barrier, a one-dimensional Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. The 2D scattering results represent the first successful application of an accurate direct numerical solution of the quantum hydrodynamic equations to an anharmonic potential energy surface.

## Keywords

Wave Packet Edge Point Arbitrary Lagrangian Eulerian Move Little Square Quantum Potential## Notes

### Acknowledgments

This work was done under the auspices of the US Department of Energy at Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the US Department of Energy under contract DE-AC52-06NA25396.

## Supplementary material

## References

- 1.Madelung E (1926) Z Phys 40:322Google Scholar
- 2.de Broglie L (1926) CR Acad Sci Paris 183:447Google Scholar
- 3.de Broglie L (1927) CR Acad Sci Paris 184:273Google Scholar
- 4.Bohm D (1952) Phys Rev 85:166CrossRefGoogle Scholar
- 5.Bohm D (1952) Phys Rev 85:180CrossRefGoogle Scholar
- 6.Holland PR (1993) The quantum theory of motion. Cambridge University Press, New YorkCrossRefGoogle Scholar
- 7.Lopreore C, Wyatt RE (1999) Phys Rev Lett 82:5190CrossRefGoogle Scholar
- 8.Hughes KH, Wyatt RE (2002) Chem Phys Lett 366:336CrossRefGoogle Scholar
- 9.Trahan CJ, Wyatt RE (2003) J Chem Phys 118:4784CrossRefGoogle Scholar
- 10.Kendrick BK (2003) J Chem Phys 119:5805CrossRefGoogle Scholar
- 11.Pauler DK, Kendrick BK (2004) J Chem Phys 120:603CrossRefGoogle Scholar
- 12.Kendrick BK (2004) J Chem Phys 121:2471CrossRefGoogle Scholar
- 13.Derrickson SW, Bittner ER, Kendrick BK (2005) J Chem Phys 123:54107-1CrossRefGoogle Scholar
- 14.Kendrick BK (2010) J Mol Struct Theochem 943:158CrossRefGoogle Scholar
- 15.Kendrick BK (2010) The direct numerical solution of the quantum hydrodynamic equations of motion. In: Chattaraj PK (eds) Quantum trajectories. CRC Press/Taylor & Francis Group, USA, p 325Google Scholar
- 16.Kendrick BK (2011) An iterative finite difference method for solving the quantum hydrodynamic equations of motion. In: Hughes KH, Parlant G (eds) Quantum trajectories. CCP6: Dynamics of Open Quantum Systems, Warrington, p 13Google Scholar
- 17.Wyatt RE, Bittner ER (2000) J Chem Phys 113:8898CrossRefGoogle Scholar
- 18.Rassolov VA, Garashchuk S (2004) J Chem Phys 120:6815CrossRefGoogle Scholar
- 19.Garashchuk S (2009) J Phys Chem A 113:4451CrossRefGoogle Scholar
- 20.Poirier B (2004) J Chem Phys 121:4501CrossRefGoogle Scholar
- 21.Babyuk D, Wyatt RE (2004) J Chem Phys 121:9230CrossRefGoogle Scholar
- 22.Burghardt I, Cederbaum LS (2001) J Chem Phys 115:10303CrossRefGoogle Scholar
- 23.Burghardt I, Moller KB, Hughes K (2007) In: Micha DA (eds) Springer series in chemical physics, p 391Google Scholar
- 24.Goldfarb Y, Degani I, Tannor DJ (2006) J Chem Phys 125:231103CrossRefGoogle Scholar
- 25.Rowland BA, Wyatt RE (2008) Chem Phys Lett 461:155CrossRefGoogle Scholar
- 26.Chou CC, Sanz AS, Miret-Artés S, Wyatt RE (2009) Phys Rev Lett 102:250401-1Google Scholar
- 27.Garashchuk S (2010) J Chem Phys 132:014112CrossRefGoogle Scholar
- 28.Garashchuk S (2010) Chem Phys Lett 491:96CrossRefGoogle Scholar
- 29.Wyatt RE (2005) Quantum dynamics with trajectories: introduction to quantum hydrodynamics. Springer, New YorkGoogle Scholar
- 30.Garashchuk S, Rassolov V, Prezhdo O (2011) Review in computational chemistry, vol 27. Wiley, London, pp 111–210Google Scholar
- 31.Scannapeico E, Harlow FH (1995) Los Alamos national laboratory report LA-12984Google Scholar
- 32.Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1986) Numerical recipes; the art of scientific computing. Cambridge University Press, New YorkGoogle Scholar
- 33.Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere Publishing Co., New YorkGoogle Scholar
- 34.Morse PM (1929) Phys Rev 34:57CrossRefGoogle Scholar
- 35.Kais S, Levine RD (1990) Phys Rev A 41:2301CrossRefGoogle Scholar
- 36.VonNeumann J, Richtmyer RD (1950) J Appl Phys 21:232CrossRefGoogle Scholar
- 37.Harlow FH (1960) Los Alamos scientific laboratory report LA-2412Google Scholar
- 38.Harlow FH, Welch JE (1965) Phys Fluids 8:2182CrossRefGoogle Scholar
- 39.Leonard BP (1979) Comput Meth Appl Mech Eng 19:59CrossRefGoogle Scholar
- 40.HSL (2011) A collection of Fortran codes for large scale scientific computation. http://www.hsl.rl.ac.u
- 41.Tannor DJ, Weeks DE (1993) J Chem Phys 98:3884Google Scholar
- 42.Wolfram Research, Inc (2010) Mathematica, version 8.0. Champaign, ILGoogle Scholar