Ground and excited states of spherically symmetric potentials through an imaginary-time evolution method: application to spiked harmonic oscillators

Abstract

Starting from a time-dependent Schrödinger equation, stationary states of 3D central potentials are obtained. An imaginary-time evolution technique coupled with the minimization of energy expectation value, subject to the orthogonality constraint leads to ground and excited states. The desired diffusion equation is solved by means of a finite-difference approach to produce accurate wave functions, energies, probability densities and other expectation values. Applications in case of 3D isotropic harmonic oscillator, Morse as well the spiked harmonic oscillator are made. Comparison with literature data reveals that this is able to produce high-quality and competitive results. The method could be useful for this and other similar potentials of interest in quantum mechanics. Future and outlook of the method is briefly discussed.

Quantum confinement in 1D systems through an imaginary-time evolution method

“…In this section, we give an overview of the ITP method as employed here for a particle under confinement. More complete account could be found in the references [45][46][47][48][49][50][51]. Our starting point is TDSE, which for a particle under the influence of a potential V (x) in 1D, is given by (atomic unit employed unless otherwise mentioned),…”

“…This procedure was initially proposed several decades ago and thereafter was successfully applied to a number of systems invoking several different implementation schemes [38][39][40][41][42][43][44]. The present implementation has been successfully used in a few free systems, such as ground states in atoms, diatomic molecules within a quantum fluid dynamical density functional theory [45,46], low-lying states in harmonic, anharmonic potentials in 1D, 2D, as well as spiked oscillator [47][48][49][50][51]. However, this scheme has never been attempted in confinement situations.…”

Information analysis in free and confined harmonic oscillator

“…After the original proposal that came several decades ago, a number of successful implementations [80,81,82,83,84,85,86] have been reported in the literature since then. In this work we have adopted an implementation, which has been successfully applied to a number of physical systems, such as atoms, diatomic molecules within a quantum fluid dynamical density functional theory (DFT) [87,88], as well as some model (harmonic, anharmonic, self-interacting, double-well, spiked oscillators) potentials [89,90,91,92,93], in both 1D, 2D and 3D. Recently this was also extended to confinement (SCHO and ACHO) problems [34] with very good success.…”

“…This method is, in principle, exact. Here we have provided the equations for 1D SCHO; however this has been easily extended to higher dimensions (see, e.g., [92,93]. The confinement condition can be achieved by reducing the boundary from infinity to finite region, as expressed in the following equation (symmetric box of length 2R),…”

The finite difference time domain (FDTD) method to determine energies and wave functions of two-electron quantum dot

“…To solve the time-independence Schrödinger equation, the FDTD method can be devided into two methods: realtime FDTD (R-FDTD) method [3,5] and imaginary-time FDTD (I-FDTD) method [6,7,8,9]. The R-FDTD method uses evolution of a wavefunction by the discretized time-dependence Schrödinger equation and Fourier transformation procedure to obtain eigen energies and wavefunctions.…”