Fractional quantum mechanics with topological constraint
-
Alexander Iomin
Abstract
An application of fractional integrodifferentiation in quantum processes is presented. We considered two examples of Lévy flights in finite configuration space, which are the examples of the application of the fractional space derivatives in quantum dynamics with topological constraint. These are Lévy flights in an infinite well potential and a quantum damping dynamics of a fractional kicked rotor. It is shown that the Riesz fractional derivative corresponds to the Hamiltonian dynamics. We present a detailed analysis of this quantization in the framework of the path integral approach construction for the system with topological constraint. We also show that the Weyl fractional derivative can quantize an open system. To this end, a fractional kicked rotor is studied in the framework of the fractional Schrödinger equation. The system is described by the non-Hermitian Hamiltonian by virtue of the Weyl fractional derivative. Violation of space symmetry leads to acceleration of the orbital momentum. Quantum localization saturates this acceleration, such that the average value of the orbital momentum can be a direct current and the system behaves like a ratchet. The classical counterpart is a nonlinear kicked rotor with absorbing boundary conditions.
Abstract
An application of fractional integrodifferentiation in quantum processes is presented. We considered two examples of Lévy flights in finite configuration space, which are the examples of the application of the fractional space derivatives in quantum dynamics with topological constraint. These are Lévy flights in an infinite well potential and a quantum damping dynamics of a fractional kicked rotor. It is shown that the Riesz fractional derivative corresponds to the Hamiltonian dynamics. We present a detailed analysis of this quantization in the framework of the path integral approach construction for the system with topological constraint. We also show that the Weyl fractional derivative can quantize an open system. To this end, a fractional kicked rotor is studied in the framework of the fractional Schrödinger equation. The system is described by the non-Hermitian Hamiltonian by virtue of the Weyl fractional derivative. Violation of space symmetry leads to acceleration of the orbital momentum. Quantum localization saturates this acceleration, such that the average value of the orbital momentum can be a direct current and the system behaves like a ratchet. The classical counterpart is a nonlinear kicked rotor with absorbing boundary conditions.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- Fractional electromagnetics 1
- Fractional electrodynamics with spatial dispersion 25
- Fractional-calculus tools applied to study the nonexponential relaxation in dielectrics 53
- Fractional diffusion-wave phenomena 71
- Fractional diffusion and parametric subordination 99
- The fractional advection-dispersion equation for contaminant transport 129
- Anomalous diffusion in interstellar medium 151
- Fractional kinetics in random/complex media 183
- Nonlocal quantum mechanics: fractional calculus approach 207
- Fractional quantum fields 237
- Fractional quantum mechanics of open quantum systems 257
- Fractional quantum mechanics with topological constraint 279
- Fractional time quantum mechanics 299
- Index 317
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- Fractional electromagnetics 1
- Fractional electrodynamics with spatial dispersion 25
- Fractional-calculus tools applied to study the nonexponential relaxation in dielectrics 53
- Fractional diffusion-wave phenomena 71
- Fractional diffusion and parametric subordination 99
- The fractional advection-dispersion equation for contaminant transport 129
- Anomalous diffusion in interstellar medium 151
- Fractional kinetics in random/complex media 183
- Nonlocal quantum mechanics: fractional calculus approach 207
- Fractional quantum fields 237
- Fractional quantum mechanics of open quantum systems 257
- Fractional quantum mechanics with topological constraint 279
- Fractional time quantum mechanics 299
- Index 317
