Quantum dynamics in low-dimensional systems with position-dependent mass and product-like fractal ge скачать в хорошем качестве

Quantum dynamics in low-dimensional systems with position-dependent mass and product-like fractal ge

The provided text explores the development of a generalized Schrödinger equation specifically designed for anisotropic fractal media where particles exhibit position-dependent mass (PDM). By integrating the Li and Ostoja-Starzewski approach with quantum dynamics, the research introduces a modified momentum operator to investigate how complex geometries and varying mass distributions influence physical properties. The author derives energy eigenvalues for several potential types, demonstrating that while the uncertainty principle remains intact, the resulting energy levels are uniquely shaped by fractal anisotropy. A practical application to Cadmium selenide nanocrystals reveals that these systems possess a significantly lower density of states and higher group velocities than traditional models suggest. Ultimately, this work provides a sophisticated mathematical framework for understanding the electronic behavior of low-dimensional semiconductors and porous nanomaterials. The research synthesized in this document explores a novel quantum mechanical framework that integrates Li and Ostoja-Starzewski Approach (LOSA) fractal geometry with Position-Dependent Mass (PDM) . By applying this framework to low-dimensional systems, specifically semiconductor nanocrystals, the study identifies significant deviations from standard quantum mechanical models.Key Takeaways: ● Modified Framework: The study successfully derives a modified Schrödinger equation and a generalized momentum operator ( $p̂_f$ ) that incorporates fractal anisotropy and spatial mass variation. ● Physical Consistency: Despite the modification of the momentum operator, the fundamental commutation relations and the Heisenberg uncertainty principle remain unaltered under specific parameter constraints. ● Energy Level Shifts: The energy eigenvalues for various effective potentials (inverse square root, harmonic oscillator-like) are found to be strongly influenced by the fractal parameter $\alpha$ , generally resulting in reduced energy levels compared to standard approaches. Nanoscale Implications: Applied to Cadmium selenide (CdSe) nanocrystals, the model predicts a density of states significantly lower than standard results and an enhanced group velocity, both of which align with observed behaviors in anisotropic homogenized dielectric media. Rami Ahmad El-Nabulsi, "Quantum dynamics in low-dimensional systems with position-dependent mass and product-like fractal geometry," Physica E: Low-dimensional Systems and Nanostructures, 134, 2021, 114827, ISSN 1386-9477, https://doi.org/10.1016/j.physe.2021....