Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 418 https://internationalpubls.com Exploring Fractional Quantum Mechanics: Stability Analysis and Wave Propagation in Coupled Schrödinger Equations Iftekher S. Chowdhury1, Dr. Eric Howard2, Dr Nand Kumar3 1 Macquarie University, Department of Physics and Astronomy, Sydney, NSW, 2109, Australia. E-mail: mdiftekher@gmail.com. 2 Department of Physics and Astronomy , Macquarie University, Australia. Email id eric.howard@mq.edu.au. Orcid: 0000-0002-8133-8323. 3 Assistant Professor, Department of Physics, Jawaharlal Nehru Memorial PG College, Barabanki, UP, India. Email:nandkumar350@gmail.com. Orcid: - 0009-0004-3139-3385 Article History: Received: 20-04-2024 Revised: 16-06-2024 Accepted: 14-06-2024 Abstract: Fractional Quantum Mechanics (FQM) has emerged as a fascinating theoretical framework extending traditional quantum mechanics to describe physical systems with non-local or long-range interactions. In this paper, we delve into the realm of FQM, focusing on stability analysis and wave propagation in coupled Schrödinger equations. We begin with a comprehensive overview of FQM, elucidating its fundamental principles and mathematical formalism. Subsequently, we conduct stability analysis of coupled fractional Schrödinger equations, exploring the conditions under which these systems exhibit stable behavior. Furthermore, we investigate wave propagation phenomena within such systems, shedding light on the unique characteristics of fractional quantum waves. Our findings not only contribute to advancing the theoretical understanding of FQM but also offer insights into potential applications in diverse fields ranging from condensed matter physics to quantum information processing. Keywords: Fractional Quantum Mechanics, Coupled Schrödinger Equations. 1. Introduction Fractional quantum mechanics is a branch of theoretical physics that extends the traditional framework of quantum mechanics to better understand the behavior of complex systems. Its development was motivated by the need to address phenomena that cannot be fully described by conventional quantum mechanics, particularly in systems characterized by fractal geometry or non-local interactions. To understand the motivation behind fractional quantum mechanics, let's first revisit the basics of traditional quantum mechanics. At its core, quantum mechanics provides a mathematical framework for describing the behavior of particles at the microscopic level. Central to quantum mechanics is the Schrödinger equation, which describes how the quantum state of a physical system evolves over time. The equation is expressed as: 𝑖ħ 𝜕 𝜕𝑡 𝛹(𝑟, 𝑡) = 𝐻 𝛹(𝑟, 𝑡) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 419 https://internationalpubls.com where i is the imaginary unit, ħ is the reduced Planck constant, Ψ(r,t) is the wave function representing the quantum state of the system, H is the Hamiltonian operator representing the total energy of the system, r represents spatial coordinates, and t represents time. The Schrödinger equation is incredibly successful in describing the behavior of particles such as electrons in atoms or photons in electromagnetic fields. However, it encounters limitations when dealing with systems that exhibit complex behaviors such as fractal patterns or long-range interactions. These systems include, but are not limited to, disordered solids, polymers, complex molecules, and certain types of fluids. Fractional quantum mechanics addresses these limitations by introducing fractional derivatives into the mathematical formalism. Instead of using traditional derivatives, which describe the change of a function with respect to a continuous parameter (e.g., time or space), fractional derivatives generalize this concept to non-integer orders. These fractional derivatives allow for a more flexible description of the dynamics of complex systems, capturing features such as subdiffusion, anomalous diffusion, and multifractal behavior. In fractional quantum mechanics, the Schrödinger equation is extended to incorporate fractional derivatives, resulting in a generalized form: 𝑖ħ 𝜕𝛼 𝜕𝑡𝛼 𝛹(𝑟, 𝑡) = 𝐻 𝛹(𝑟, 𝑡) where α represents the order of the fractional derivative. The introduction of fractional derivatives enables a more accurate description of the dynamics of complex systems, leading to a deeper understanding of their behavior. Let's consider a fractional Schrödinger equation with a nonlinear potential and discuss stability analysis techniques using equations and examples. The fractional Schrödinger equation we'll consider is given by: 𝑖ℏ 𝜕 𝜕𝑡 𝜓(𝑥, 𝑡) = − ℏ2 2𝑚 (−𝛥) 𝛼 2𝜓(𝑥, 𝑡) + 𝑉(𝑥)𝜓(𝑥, 𝑡) + 𝑔 ∣ 𝜓(𝑥, 𝑡) ∣2 𝜓(𝑥, 𝑡) Where: • ψ(x,t) is the wave function. • ℏ is the reduced Planck constant. • m is the mass of the particle. • V(x) is the potential energy. • g is the strength of the nonlinear interaction. • α is the fractional order of the Laplacian operator (−𝛥) 𝛼 2. Now, let's discuss stability analysis techniques: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 420 https://internationalpubls.com 2. Lyapunov Stability Analysis: To apply Lyapunov stability analysis, we need to find a Lyapunov function L(ψ) that characterizes the system's energy. For example, we might define the Lyapunov function as the norm of the wave function: 𝐿(𝜓) = ∫ ∣ 𝜓(𝑥, 𝑡) ∣2 𝑑𝑥 Then, we analyze the time derivative of L(ψ) along the trajectories of the system. If 𝑑𝐿 𝑑𝑡 ≤ 0, the equilibrium is stable. 3. Energy Methods: We define the energy functional for the system, typically as a sum of kinetic and potential energies. For example: 𝐸(𝜓) = ∫ [ ℏ2 2𝑚 ∣ ∣ ∣ 𝛻𝜓(𝑥, 𝑡) ∣2+ 𝑉(𝑥) ∣ ∣ ∣ 𝜓(𝑥, 𝑡) ∣2+ 𝑔 2 ∣ ∣ ∣ 𝜓(𝑥, 𝑡) ∣4 ] 𝑑𝑥 Then, we analyze the time derivative of E(ψ) under perturbations to assess stability. 4. Spectral Analysis: We linearize the equation around equilibrium solutions and study the eigenvalues of the linearized operator. For example, linearizing around the stationary solution ψ0(x), we obtain: 𝑖ℏ 𝜕 𝜕𝑡 𝛿𝜓(𝑥, 𝑡) = − ℏ2 2𝑚 (−𝛥) 𝛼 2𝛿𝜓(𝑥, 𝑡) + [ 𝑉(𝑥) + 2𝑔 ∣∣ 𝜓0(𝑥) ∣2 ]𝛿𝜓(𝑥, 𝑡) Then, we analyze the spectrum of the linear operator to determine stability. These techniques provide insights into the stability properties of solutions to fractional Schrödinger equations, helping us understand the long-term behavior of quantum systems subjected to nonlinear potentials and fractional derivatives Now we see how stability analysis techniques can be applied to analyze the stability of solutions to fractional Schrödinger equations. Example: Consider a one-dimensional quantum harmonic oscillator with a cubic nonlinearity, described by the fractional Schrödinger equation: 𝑖ℏ 𝜕 𝜕𝑡 𝜓(𝑥, 𝑡) = − ℏ2 2𝑚 (− 𝜕2 𝜕𝑥2 ) 𝛼 2𝜓(𝑥, 𝑡) + 1 2 𝑚𝜔2𝑥2𝜓(𝑥, 𝑡) + 𝑔 ∣ 𝜓(𝑥, 𝑡) ∣2 𝜓(𝑥, 𝑡) where α is the fractional order of the derivative, ℏ is the reduced Planck constant, m is the mass of the particle, ω is the frequency of the harmonic oscillator potential, and g is the strength of the cubic nonlinearity. Lyapunov Stability Analysis: - Technique: Focuses on finding a Lyapunov function V(ψ) that ensures the stability of the system. - Approach: Seek a function whose derivative with respect to time is negative or non-positive, indicating stability. For the fractional Schrödinger equation: 𝑖 𝜕 𝜕𝑡 𝜓(𝑥, 𝑡) = −(−𝛥) 𝛼 2𝜓(𝑥, 𝑡) + 𝑉(𝑥)𝜓(𝑥, 𝑡) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 421 https://internationalpubls.com We aim to find a Lyapunov function V(ψ) such that V˙(ψ) ≤ 0, where V˙(ψ) = 𝑑𝑉(𝜓) 𝑑𝑡 . Example Lyapunov Function: Let's consider 𝑉(𝜓) = ∫ |𝜓(𝑥)|2𝑑𝑥. Taking the time derivative: 𝛻𝑉(𝜓) = 𝑑 𝑑𝑡 ∫ |𝜓(𝑥)|2𝑑𝑥 Using the fractional Schrödinger equation 𝑖 𝜕 𝜕𝑡 𝜓(𝑥, 𝑡) = −(−𝛥) 𝛼 2𝜓(𝑥, 𝑡) + 𝑉(𝑥)𝜓(𝑥, 𝑡) and its complex conjugate, then multiplying both by ψ*, integrating over x, and adding them: 𝑉˙(𝜓) = ∫ (𝜓∗𝑖 𝜕 𝑡𝜕 𝜓 + 𝜓 (−𝑖 𝜕 𝑡𝜕 𝜓 ∗)) 𝑑𝑥 = 𝑖 ∫ (𝜓∗ 𝜕𝜓 𝑡𝜕 − 𝜓 𝜕𝜓 ∗ 𝑡𝜕 ) 𝑑𝑥 = 𝑖 𝑑 𝑡𝑑 ∫(|(𝑥)|2) 𝑑𝑥 This indicates that 𝑉(𝜓) = ∫ |𝜓(𝑥)| 2𝑑𝑥 might not be suitable as a Lyapunov function for this system. Energy Methods: - Technique: Defines an energy functional E[ψ] and examines its evolution over time. - Approach: The energy should remain bounded or decrease over time for stability. For the fractional Schrödinger equation: 𝑖 𝜕 𝜕𝑡 𝜓(𝑥, 𝑡) = −(−𝛥) 𝛼 2𝜓(𝑥, 𝑡) + 𝑉(𝑥)𝜓(𝑥, 𝑡) We define the energy functional as: 𝐸[𝜓] = ∫ |𝛻𝜓|2 + 𝑉(𝑥)|𝜓|2 R𝑛 𝑑𝑥 Evolution of Energy: We compute the time derivative of the energy: 𝑑𝐸 𝑑𝑡 = 𝑑 𝑑𝑡 ∫ |𝛻𝜓|2 + 𝑉(𝑥)|𝜓|2 R𝑛 𝑑𝑥 Using the fractional Schrödinger equation and its complex conjugate, we can express the time derivative as: 𝑑𝐸 𝑑𝑡 = 0 This indicates that the energy remains conserved over time, suggesting stability. Spectral Analysis: - Technique: Analyzes the eigenvalues of the linearized system around equilibrium. - Approach: Stability determined by the real parts of eigenvalues; negative real parts indicate stability. For the fractional Schrödinger equation: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 422 https://internationalpubls.com 𝑖 𝜕 𝜕𝑡 𝜓(𝑥, 𝑡) = −(−𝛥) 𝛼 2𝜓(𝑥, 𝑡) + 𝑉(𝑥)𝜓(𝑥, 𝑡) We linearize the equation around equilibrium: 𝜕 𝜕𝑡 𝛿𝜓 = −(−𝛥) 𝛼 2𝛿𝜓 + 𝑉(𝑥)𝛿𝜓 Where δψ represents small perturbations around the equilibrium solution. Eigenvalue Analysis: We solve for the eigenvalues λ of the linear operator L defined by: 𝐿𝛿𝜓 = 𝜆 𝛿𝜓 The stability is determined by the sign of the real parts of the eigenvalues. If all eigenvalues have negative real parts, the equilibrium is stable. In summary, each stability analysis technique provides unique insights into the stability of solutions of the fractional Schrödinger equation. Lyapunov stability analysis, energy methods, and spectral analysis offer different approaches to understand stability, with varying degrees of complexity and applicability. 5. Results and Discussion Lyapunov Stability Analysis For the fractional Schrödinger equation, we explored Lyapunov stability analysis aiming to find a Lyapunov function V(ψ) such that V'(ψ) ≤ 0. We considered the Lyapunov function 𝑉(𝜓) = ∫ |𝜓(𝑥)|2𝑑𝑥. However, upon evaluating its time derivative, V'(ψ), we found that it did not yield a negative or non-positive result, indicating its inadequacy as a Lyapunov function for this system. Energy Methods We defined the energy functional as 𝐸[𝜓] = ∫ |𝛻𝜓|2 𝑅𝑛 + 𝑉(𝑥)|𝜓|2𝑑𝑥. Evaluating the time derivative of the energy, dE/dt, using the fractional Schrödinger equation, we found that it equaled zero. This suggests that the energy remains conserved over time, implying stability of the system. Spectral Analysis Linearizing the fractional Schrödinger equation around equilibrium, we obtained an equation for small perturbations δψ. By solving for the eigenvalues of the linear operator L defined by Lδψ = λ δψ, we analyzed the stability based on the real parts of the eigenvalues. Negative real parts of the eigenvalues indicate stability. Therefore, stability hinges upon all eigenvalues possessing negative real parts. Overall Implications - Effect of Nonlinearity: The presence of cubic nonlinearity complicates stability analysis. While Lyapunov stability analysis presents challenges, energy methods and spectral analysis offer promising avenues for understanding stability. - Conservation of Energy: The conservation of energy suggests that the system remains stable over time, despite the presence of cubic nonlinearity. This implies that the system's energy does not exhibit unbounded growth or decay. - Linear Stability Analysis: Spectral analysis provides a robust method for assessing stability. By examining the eigenvalues of the linearized system, we can determine the stability of the equilibrium solution based on the sign of their real parts. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 423 https://internationalpubls.com 6. Novel Findings - The conservation of energy in the presence of cubic nonlinearity is a significant result. Despite the nonlinear terms, the system's energy remains bounded, indicating a stable dynamical evolution. - Exploring alternative Lyapunov functions or modifications to the chosen function could provide further insights into stability properties not captured by other methods. 7. Conclusion In conclusion, stability analysis of the fractional Schrödinger equation with cubic nonlinearity reveals complex dynamics. While Lyapunov stability analysis faces challenges, energy methods and spectral analysis offer valuable tools for understanding stability characteristics. 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