Efficient Quantum Trajectory Representation of Wavefunctions Evolving in Imaginary Time

Sophya V. Garashchuk, University of South Carolina - Columbia
J. Mazzuca
T. Vazhappilly

Copyright 2011 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

The following article appeared in

Garashchuk, S., Mazzuca, J., & Vazhappilly, T. (2011). Efficient quantum trajectory representation of wavefunctions evolving in imaginary time. The Journal of Chemical Physics, 135, 034104. http://dx.doi.org/10.1063/1.3610165

and may be found at



The Boltzmann evolution of a wavefunction can be recast as imaginary-time dynamics of the quantum trajectory ensemble. The quantum effects arise from the momentum-dependent quantum potential – computed approximately to be practical in high-dimensional systems – influencing the trajectories in addition to the external classical potential [S. Garashchuk, J. Chem. Phys.132, 014112 (2010)]. For a nodelesswavefunction represented as ψ(x, t) = exp ( −S(x, t)/ℏ) with the trajectory momenta defined by ∇S(x, t), analysis of the Lagrangian and Eulerian evolution shows that for bound potentials the former is more accurate while the latter is more practical because the Lagrangian quantum trajectories diverge with time. Introduction of stationary and time-dependent components into the wavefunction representation generates new Lagrangian-type dynamics where the trajectory spreading is controlled improving efficiency of the trajectory description. As an illustration, different types of dynamics are used to compute zero-point energy of a strongly anharmonic well and low-lying eigenstates of a high-dimensional coupled harmonic system.