Quantum trajectory dynamics in imaginary time with the momentum-dependent quantum potential

Abstract

The quantum trajectory dynamics is extended to the wave function evolution in imaginary time. For a nodeless wave function a simple exponential form leads to the classical-like equations of motion of trajectories, representing the wave function, in the presence of the momentum-dependent quantum potential in addition to the external potential. For a Gaussian wave function this quantum potential is a time-dependent constant, generating zero quantum force yet contributing to the total energy. For anharmonic potentials the momentum-dependent quantum potential is cheaply estimated from the global Least-squares Fit to the trajectory momenta in the Taylor basis. Wave functions with nodes are described in the mixed coordinate space/trajectory representation at little additional computational cost. The nodeless wave function, represented by the trajectory ensemble, decays to the ground state. The mixed representation wave functions, with lower energy contributions projected out at each time step, decay to the excited energy states. The approach, illustrated by computing energy levels for anharmonic oscillators and energy level splitting for the double-well potential, can be used for the Boltzmann operator evolution.

Trajectory approach to the Schrödinger–Langevin equation with linear dissipation for ground states

“…(15). [70][71][72][73][74], these Bohmian trajectories follow the main features of the evolving probability density, and they finally become stationary. In Bohmian mechanics [9,[17][18][19][20][21], the particle is usually at rest for stationary states because the particle velocity turns out to be zero everywhere.…”

“…Recently, dissipative Bohmian trajectories have been analyzed within the Caldirola-Kanai framework [69]. Furthermore, the quantum trajectory dynamics has been extended to the wave function evolution in imaginary time for the ground state energies and wave functions of quantum systems [70][71][72][73][74].…”

Efficient quantum trajectory representation of wavefunctions evolving in imaginary time

“…(12), (15), and (16) has been shown 11,20 to give accurate ZPE estimates for anharmonic systems, including the double well, and for the triatomic molecules with a reasonably small (quadratic) fitting basis determining U and to converge to the QM result for larger bases (3-6 functions). For multidimensional bound systems, however, the sampling, or the trajectory representation problem was identified: in bound classical potentials (−V is a barrier) the Lagrangian trajectories fall off the barrier top and leave the region of high wavefunction density.…”

“…(4), the last term in Eq. (13) is interpreted as the momentum-dependent quantum potential (MDQP), 11,20 …”

“…[9][10][11] Here we will consider how the wavefunction representation affects the character of the imaginary-time quantum trajectory dynamics and the accuracy of its numerical implementation.…”