- The paper introduces a novel analytical method employing Heun confluent equations to derive a complete eigenspectrum for hyperbolic double well potentials.
- It avoids truncating the power series of conventional QES models, enabling accurate analysis of intra-well tunneling and quantum interference phenomena.
- Extensive numerical validation demonstrates its broad applicability to low-dimensional quantum systems, enhancing modeling of molecular and semiconductor structures.
Analysis of Wavepacket Dynamics in Hyperbolic Double Well Potentials
In this paper, the authors present a novel analytical method for computing the eigenspectrum and corresponding eigenstates of hyperbolic double well potentials. This approach transcends the customary methodologies employed with quasi-exactly solvable (QES) models by leveraging Heun confluent differential equations to accurately elucidate the dynamics of the system.
The authors begin by mapping the time-independent Schrödinger equation onto the Heun confluent differential equation, which is subsequently solved using an infinite power series. The coefficients of this series are expressed as polynomials in the quantisation parameter, and the roots of these polynomials represent the eigenenergies of the system. This approach diverges from typical QES models, which often yield only a subset of eigenvalues and necessitate specific constraints for solvability. Notably, the authors avoid truncating the power series at particular parameters of the potential, thereby enabling the determination of a complete spectrum without restrictive assumptions.
Strong numerical results are obtained from an extensive series of eigenstates calculated using the new quantisation condition, providing a robust validation of the methodology. The authors demonstrate their analytical framework through an exhaustive study of electronic wavepacket dynamics, with particular emphasis on intra-well tunneling and quantum interference phenomena, such as quantum bridges.
Key implications of this work include the methodology's applicability to a broader range of hyperbolic potentials, beyond those already explored. This condition is particularly relevant for defining quantisation conditions in systems where conventional methods prove insufficient. Additionally, the paper explores the behavior of wavepackets in various scenarios, offering significant insights into the temporal evolution of low-dimensional quantum systems.
The theoretical implications are profound, as the authors present a means of independently tuning variables like width and peak location of initial wavepackets, granting control over initial conditions and enabling precise identification of developments in the quantum state over time. Practically, such methods enhance our computational capabilities for modeling molecular systems, semi-conductor heterostructures, and optical lattices, among others.
Future advancements in this field could expand the model to investigate non-symmetric and time-dependent scenarios, further enriching our understanding of quantum tunneling and interference phenomena. The quantisation criterion proposed can be extended to additional classes of potentials, optimizing it even for those problems that might not be symmetrical or time-independent. Thus, the implications for both theoretical frameworks and practical applications in quantum physics are considerable.
