Nonlocal quantum mechanics: fractional calculus approach
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Nick Laskin
Abstract
We show that a quantum system with long-range interparticle interaction can be described by invoking fractional calculus tools. The system under consideration is nonlocal exciton-phonon quantum dynamics on a 1D lattice. It has been shown that long-range power-law exciton-exciton interaction leads to a nonlocal integral term in the motion equation of an exciton subsystem if we go from discrete to continuous space. In some particular cases for power-law interaction with noninteger power, the nonlocal integral term can be expressed through a spatial derivative of fractional order. Considering exciton-phonon dynamics with long-range exciton-exciton interaction, we have obtained the system of two coupled equations, where one is the quantum fractional differential equation for the exciton subsystem while the other is a standard differential equation for the phonon subsystem. It has been found that the system of two coupled equations can be further simplified to come up with the following fractional nonlinear equations of motion: nonlinear fractional Schrödinger equation, nonlinear Hilbert-Schrödinger equation, fractional generalization of Zakharov system, and fractional Ginzburg-Landau equation. The appearance of fractional differential equations in the continuum limit of lattice dynamics allows us to apply powerful tools of fractional calculus to study nonlocal quantum phenomena.
Abstract
We show that a quantum system with long-range interparticle interaction can be described by invoking fractional calculus tools. The system under consideration is nonlocal exciton-phonon quantum dynamics on a 1D lattice. It has been shown that long-range power-law exciton-exciton interaction leads to a nonlocal integral term in the motion equation of an exciton subsystem if we go from discrete to continuous space. In some particular cases for power-law interaction with noninteger power, the nonlocal integral term can be expressed through a spatial derivative of fractional order. Considering exciton-phonon dynamics with long-range exciton-exciton interaction, we have obtained the system of two coupled equations, where one is the quantum fractional differential equation for the exciton subsystem while the other is a standard differential equation for the phonon subsystem. It has been found that the system of two coupled equations can be further simplified to come up with the following fractional nonlinear equations of motion: nonlinear fractional Schrödinger equation, nonlinear Hilbert-Schrödinger equation, fractional generalization of Zakharov system, and fractional Ginzburg-Landau equation. The appearance of fractional differential equations in the continuum limit of lattice dynamics allows us to apply powerful tools of fractional calculus to study nonlocal quantum phenomena.
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
