Anomalous Relaxation and ThreeLevel System: A Fractional Schrödinger Equation Approach
Abstract
:1. Introduction
2. Schrödinger Equation without Kinetic Terms
3. Schrödinger Equation with Kinetic Dependences
4. Discussions and Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Fractional Ordinary Differential Equations
Appendix B. Fractional Partial Differential Equations
Algorithm A1 Algorithm to solve space FPDE 

Appendix C. List of Symbols
Symbol  Meaning 

i  Imaginary unity. 
ℏ  Reduced Planck constant. 
$\alpha $  Time fractional order. 
$\psi $  Wave function. 
$\hat{H}$  Hamiltonian operator. 
${}_{c}^{\mathrm{C}}{D}_{t}^{\alpha}$  Time Caputo derivative. 
${\partial}^{\alpha}/\partial {t}^{\alpha}$  Time Caputo derivative. 
${E}_{i}$  Eigenvalue of energy. 
${\gamma}_{ij}$  Strenght coupling between i and j. 
${\omega}_{ij}$  Frequency of external perturbation. 
$\hat{p}$  Momentum operator. 
$\mu $  Space fractional order. 
$\hat{p}{}^{\phantom{\rule{0.166667em}{0ex}}\mu}$  Fractional momentum operator. 
m  Mass. 
$\mathcal{F}$  Fourier transformation. 
$\mathcal{L}$  Laplace transformation. 
${\psi}_{i}{}^{2}$  Probability density. 
${h}_{t}$  Time step. 
${h}_{x}$  Space step. 
$\sigma $  Standard deviation. 
$\langle {(\Delta x)}^{2}\rangle $  Mean square displacement. 
$\mathcal{G}$  Green function. 
${\mathrm{H}}_{p,q}^{m,n}$  H Fox function. 
$\Gamma \left(x\right)$  Gamma function. 
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Lenzi, E.K.; Gabrick, E.C.; Sayari, E.; de Castro, A.S.M.; Trobia, J.; Batista, A.M. Anomalous Relaxation and ThreeLevel System: A Fractional Schrödinger Equation Approach. Quantum Rep. 2023, 5, 442458. https://doi.org/10.3390/quantum5020029
Lenzi EK, Gabrick EC, Sayari E, de Castro ASM, Trobia J, Batista AM. Anomalous Relaxation and ThreeLevel System: A Fractional Schrödinger Equation Approach. Quantum Reports. 2023; 5(2):442458. https://doi.org/10.3390/quantum5020029
Chicago/Turabian StyleLenzi, Ervin K., Enrique C. Gabrick, Elaheh Sayari, Antonio S. M. de Castro, José Trobia, and Antonio M. Batista. 2023. "Anomalous Relaxation and ThreeLevel System: A Fractional Schrödinger Equation Approach" Quantum Reports 5, no. 2: 442458. https://doi.org/10.3390/quantum5020029