Solution of the “Classical” Schrödinger equation for a driven symmetric triple well: A comparison with its classical counterpart

Self Cite

For a symmetric triple well potential, driven by the forces associated with the bifurcation diagram of a logistic map, the tunneling and quantum localization are studied using quantum theory of motion and time‐dependent Fourier grid Hamiltonian methods. Detailed analysis reveals that application of only asymmetric or symmetric perturbation results into either quantum localization or over‐barrier transition and no tunneling while application of mixed symmetry perturbation gives either tunneling or over‐barrier transition, depending on temporal nature and initial position of the particle. For bifurcative and chaotic symmetric‐asymmetric perturbation, with truncation of mixed symmetry perturbation, a sudden jump in energy causes a transition from the tunneling phenomenon to the over‐barrier transition. With particle located initially near to either of the minima of the unperturbed well, quantum localization, or over‐barrier transition is observed, depending on types of perturbation used.

v = \frac{italicℏ}{m true({normalψ}_{real}^{2} + {normalψ}_{img}^{2} true)} true({normalψ}_{real} \nabla {normalψ}_{img} - {normalψ}_{img} \nabla {normalψ}_{real} true)

where

ψ = {normalψ}_{real} + i {normalψ}_{img}

.…”

Section: Methodsmentioning

confidence: 99%

v = \frac{italicℏ}{m true({normalψ}_{real}^{2} + {normalψ}_{img}^{2} true)} true({normalψ}_{real} \nabla {normalψ}_{img} - {normalψ}_{img} \nabla {normalψ}_{real} true)

where

ψ = {normalψ}_{real} + i {normalψ}_{img}

. Again, considering the complex nature of

ψ true(x, t true)

, TDSE becomes

ℏ \frac{\partial {normalψ}_{real}}{\partial t} = H {normalψ}_{img}

and

ℏ \frac{\partial {normalψ}_{img}}{\partial t} = - H {normalψ}_{real}

that is, a set of first order coupled differential equations in time.…”

Section: Methodsmentioning

confidence: 99%

“…Here, we must note that, in Equation ,

g true(t_{n} true)

Section: Numerical Detailsmentioning

confidence: 99%

“…Here, we have taken the system parameters same as in our previous paper, that is,

L = 8.0 a . u .

N = 151

a = 0.95 a . u .

{normalω}_{0} = 0.016498 a . u .

, and

m = 1837 a . u .

For perturbation,

λ = 0.1 a . u .

and the time‐step for TDFGH calculation is taken as

Δ t = 0.0001 a . u .

Also, for classical dynamical calculation, we have chosen the initial position of the particle to be either of the three well‐minima and the initial momentum has been chosen to be same as the

true〈 p true〉

of the particle at

t = 0

, determined from corresponding Ehrenfest dynamics. All of the calculations and results, reported here, are considered in atomic unit

true(ℏ = m_{e} = true| e true| = 1 true)

.…”

Section: Numerical Detailsmentioning

confidence: 99%

See 2 more Smart Citations

Tunneling and quantum localization in chaos‐driven symmetric triple well potential: An approach using quantum theory of motion

Kar

Chattaraj

2017

Self Cite

“…Bohmian mechanics as a hydrodynamic formulation of quantum mechanics has revealed new physical insights concerning dynamical processes . For example, a detailed analysis has been carried out for a driven triple‐well potential using quantum trajectories . Complex‐valued quantum trajectories have been employed to study quantum systems, including Coulombic systems, the hydrogen molecule ion, the interference fringes in slit experiments, the electric, magnetic, and thermal effects on a quantum dot, and chaotic trajectories in complex space .…”

Section: Introductionmentioning

confidence: 99%

Quantum‐classical transition of the dissipative wave packet dynamics for barrier scattering

Chou

2018

A nonlinear equation based on the Schrödinger–Langevin equation is employed to analyze the quantum‐classical transition of dissipative systems for wave packet barrier scattering. The transition equation is obtained by subtracting a modified quantum potential term depending on a degree of quantumness in the Schrödinger–Langevin equation, and it provides a continuous description for the transition process of dissipative systems from purely quantum to purely classical regimes. Important properties possessed by the equation are derived. The energy dissipation rate of the dissipative system does not depend on the degree of quantumness. It is shown that the equations of motion for the expectation values of the position and momentum operators for dissipative systems hold for all the degree of quantumness. This feature establishes the correspondence between classical and quantum dissipative systems. Computational results for the transition process of dissipative wave packet dynamics and transition trajectories are presented and analyzed for model systems involving the propagation of a free Gaussian wave packet and the wave packet scattering from an Eckart barrier. Computational results demonstrate that the agreement between the classical‐limit transition trajectories and classical trajectories is excellent, except in some regions. The single‐valued transition wave function implies that transition trajectories cannot pass through the same configuration space point at the same time.

Moving boundary truncated grid method: Multidimensional quantum dynamics

Lee

Chou

2019

The moving boundary truncated grid (TG) method is used to study wave packet dynamics of multidimensional quantum systems. As time evolves, appropriate Eulerian grid points required for propagating a wave packet are activated and deactivated with no advance information about the dynamics. This method is applied to the Henon-Heiles potential and wave packet barrier scattering in two, three, and four dimensions. Computational results demonstrate that the TG method not only leads to a great reduction in the number of grid points needed to perform accurate calculations but also is computationally more efficient than the full grid calculations.adaptive grid, Henon-Heiles potential, moving boundary truncated grid method, quantum barrier scattering, wave packet dynamics