We propose a solution method of Time Dependent Schr?dinger Equation (TDSE) and the advection equation by quantum walk/quantum cellular automaton with spatially or temporally variable parameters. Using numerical method...We propose a solution method of Time Dependent Schr?dinger Equation (TDSE) and the advection equation by quantum walk/quantum cellular automaton with spatially or temporally variable parameters. Using numerical method, we establish the quantitative relation between the quantum walk with the space dependent parameters and the “Time Dependent Schr?dinger Equation with a space dependent imaginary diffusion coefficient” or “the advection equation with space dependent velocity fields”. Using the 4-point-averaging manipulation in the solution of advection equation by quantum walk, we find that only one component can be extracted out of two components of left-moving and right-moving solutions. In general it is not so easy to solve an advection equation without numerical diffusion, but this method provides perfectly diffusion free solution by virtue of its unitarity. Moreover our findings provide a clue to find more general space dependent formalisms such as solution method of TDSE with space dependent resolution by quantum walk.展开更多
Tools have been designed obtain information about chemical bonds from quantum mechanical calculations.They work well for solutions of the stationary Schr?dinger equation, but it is not clear whether Lewis electron pai...Tools have been designed obtain information about chemical bonds from quantum mechanical calculations.They work well for solutions of the stationary Schr?dinger equation, but it is not clear whether Lewis electron pairs they aim to reproduce survive in time-dependent processes, in spite of the underlying Pauli principle being obeyed in this regime. A simple model of two same-spin non-interacting fermions in a one-dimensional box with an opaque wall, is used to study this problem, because it allows presenting the detailed structure of the wave function. It is shown that i)oscillations persisting after the Hamiltonian stopped changing produce for certain time intervals states where Lewis electron pairs are spatially separated, and ii) methods(like density analysis, or the electron localization function) that are widely used for describing bonding in the stationary case, have limitations in such situations. An exception is provided by the maximum probability domain(the spatial domain that maximizes the probability to find a given number of particles in it). It is conceptually simple, and satisfactorily describes the phenomenon.展开更多
Time evaluation of wave functions for any quantum mechanical system/particle is essential nevertheless quantum mechanical counterpart of the time dependant classical wave equation does simply not appear. Epistemologic...Time evaluation of wave functions for any quantum mechanical system/particle is essential nevertheless quantum mechanical counterpart of the time dependant classical wave equation does simply not appear. Epistemologically and ontologically considered time dependant momentum operator is initially defined and an Alternative Time Dependant Schrodinger Wave Equation (ATDSWE) is plainly derived. Consequent equation is primarily solved for the free particles, in a closed system, signifying a good agreement with the outcomes of the ordinary TDSWE. Free particle solution interestingly goes further possibly tracing some signs of new pathways to resolve the mysterious quantum world.展开更多
A general finite element solution of the Schrodinger equation for a onedimensional problem is presented.The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term....A general finite element solution of the Schrodinger equation for a onedimensional problem is presented.The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term.Furthermore,it is possible to include external magnetic or electric fields,as well as spin-orbital and spinmagnetic interactions.We use analytically soluble problems to validate the solver.The predicted numerical auto-states are compared with the analytical ones,and selected mean values are used to validate the auto-functions.In order to analyze the performance of the time-dependent Schrodinger equation,a traveling wave package benchmark was reproduced.In addition,a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages.Other general problems,related to periodic potentials,are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions.Some simple cases of known periodic potential solutions are reported.展开更多
The performances of absorbing boundary conditions （ABCs） in four widely used finite difference time domain （FDTD） methods, i.e. explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the t...The performances of absorbing boundary conditions （ABCs） in four widely used finite difference time domain （FDTD） methods, i.e. explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the time-dependent Schrodinger equation are assessed and compared. The computation efficiency for each approach is also evaluated. A typical evolution problem of a single Gaussian wave packet is chosen to demonstrate the performances of the four methods combined with ABCs. It is found that ABCs perfectly eliminate reflection in implicit and explicit staggered-time methods. However, small reflection still exists in explicit and Chebyshev methods even though ABCs are applied.展开更多
Above-threshold ionization （ATI） of a hydrogen atom exposed to chirped laser fields is investigated theoretically by solving the time-dependent Schrodinger equation. By comparing the energy spectra, the two-dimensio...Above-threshold ionization （ATI） of a hydrogen atom exposed to chirped laser fields is investigated theoretically by solving the time-dependent Schrodinger equation. By comparing the energy spectra, the two-dimensional momentum spectra, and the angular distributions of photoelectron for the laser pulses with different chirp rates, we show a very clear chirp dependence both in the multiphoton and tunneling ionization processes but no chirp dependence in the single-photon ionization. We find that the chirp dependence in the multiphoton ionization based ATI can be attributed to the excited bound states. In the single-photon and tunneling ionization regimes, the electron can be removed directly from the ground state and thus the excited states may not be very important. It indicates that the chirp dependence in the tunneling ionization based ATI processes is mainly due to the laser pulses with different chirp rates,展开更多
This paper studies the multiphoton resonant ionization by two-colour laser pulses in the hydrogen atom by solving the time-dependent Schroedinger equation. By fixing the parameters of fundamental laser field and scann...This paper studies the multiphoton resonant ionization by two-colour laser pulses in the hydrogen atom by solving the time-dependent Schroedinger equation. By fixing the parameters of fundamental laser field and scanning the frequency of second laser field, it finds that the ionization probability shows several resonance peaks and is also much larger than the linear superposition of probabilities by applying two lasers separately. The enhancement of the ionization happens when the system is resonantly pumped to the excited states by absorbing two or more colour photons non-sequentially.展开更多
Two-dimensional （2D） electron momentum distributions and energy spectra of a hydrogen in an intense laser field are calculated by solving the time-dependent SchrSdinger equation combined with the window-operator tec...Two-dimensional （2D） electron momentum distributions and energy spectra of a hydrogen in an intense laser field are calculated by solving the time-dependent SchrSdinger equation combined with the window-operator technique. Compared with the standard projection technique, the window-operator technique has the advantage that the continuum states of atoms can be avoided in the calculation. We show that the 2D electron momentum distributions and the energy spectra from those two techniques accord quite well with each other if an appropriate energy width is used in the window operator.展开更多
基金supported in part by TUT Programs on Advanced Simulation Engineering,Toyohashi University of Technology.
文摘We propose a solution method of Time Dependent Schr?dinger Equation (TDSE) and the advection equation by quantum walk/quantum cellular automaton with spatially or temporally variable parameters. Using numerical method, we establish the quantitative relation between the quantum walk with the space dependent parameters and the “Time Dependent Schr?dinger Equation with a space dependent imaginary diffusion coefficient” or “the advection equation with space dependent velocity fields”. Using the 4-point-averaging manipulation in the solution of advection equation by quantum walk, we find that only one component can be extracted out of two components of left-moving and right-moving solutions. In general it is not so easy to solve an advection equation without numerical diffusion, but this method provides perfectly diffusion free solution by virtue of its unitarity. Moreover our findings provide a clue to find more general space dependent formalisms such as solution method of TDSE with space dependent resolution by quantum walk.
文摘Tools have been designed obtain information about chemical bonds from quantum mechanical calculations.They work well for solutions of the stationary Schr?dinger equation, but it is not clear whether Lewis electron pairs they aim to reproduce survive in time-dependent processes, in spite of the underlying Pauli principle being obeyed in this regime. A simple model of two same-spin non-interacting fermions in a one-dimensional box with an opaque wall, is used to study this problem, because it allows presenting the detailed structure of the wave function. It is shown that i)oscillations persisting after the Hamiltonian stopped changing produce for certain time intervals states where Lewis electron pairs are spatially separated, and ii) methods(like density analysis, or the electron localization function) that are widely used for describing bonding in the stationary case, have limitations in such situations. An exception is provided by the maximum probability domain(the spatial domain that maximizes the probability to find a given number of particles in it). It is conceptually simple, and satisfactorily describes the phenomenon.
文摘Time evaluation of wave functions for any quantum mechanical system/particle is essential nevertheless quantum mechanical counterpart of the time dependant classical wave equation does simply not appear. Epistemologically and ontologically considered time dependant momentum operator is initially defined and an Alternative Time Dependant Schrodinger Wave Equation (ATDSWE) is plainly derived. Consequent equation is primarily solved for the free particles, in a closed system, signifying a good agreement with the outcomes of the ordinary TDSWE. Free particle solution interestingly goes further possibly tracing some signs of new pathways to resolve the mysterious quantum world.
文摘A general finite element solution of the Schrodinger equation for a onedimensional problem is presented.The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term.Furthermore,it is possible to include external magnetic or electric fields,as well as spin-orbital and spinmagnetic interactions.We use analytically soluble problems to validate the solver.The predicted numerical auto-states are compared with the analytical ones,and selected mean values are used to validate the auto-functions.In order to analyze the performance of the time-dependent Schrodinger equation,a traveling wave package benchmark was reproduced.In addition,a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages.Other general problems,related to periodic potentials,are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions.Some simple cases of known periodic potential solutions are reported.
基金supported by the State Key Development Program for Basic Research of China (No. 2006CB932404)
文摘The performances of absorbing boundary conditions （ABCs） in four widely used finite difference time domain （FDTD） methods, i.e. explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the time-dependent Schrodinger equation are assessed and compared. The computation efficiency for each approach is also evaluated. A typical evolution problem of a single Gaussian wave packet is chosen to demonstrate the performances of the four methods combined with ABCs. It is found that ABCs perfectly eliminate reflection in implicit and explicit staggered-time methods. However, small reflection still exists in explicit and Chebyshev methods even though ABCs are applied.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11664035,11465016,11764038,11364038,and 11564033)
文摘Above-threshold ionization （ATI） of a hydrogen atom exposed to chirped laser fields is investigated theoretically by solving the time-dependent Schrodinger equation. By comparing the energy spectra, the two-dimensional momentum spectra, and the angular distributions of photoelectron for the laser pulses with different chirp rates, we show a very clear chirp dependence both in the multiphoton and tunneling ionization processes but no chirp dependence in the single-photon ionization. We find that the chirp dependence in the multiphoton ionization based ATI can be attributed to the excited bound states. In the single-photon and tunneling ionization regimes, the electron can be removed directly from the ground state and thus the excited states may not be very important. It indicates that the chirp dependence in the tunneling ionization based ATI processes is mainly due to the laser pulses with different chirp rates,
基金supports by the Beijing Key Laboratory for Nano-Photonics and Nano-Structure
文摘This paper studies the multiphoton resonant ionization by two-colour laser pulses in the hydrogen atom by solving the time-dependent Schroedinger equation. By fixing the parameters of fundamental laser field and scanning the frequency of second laser field, it finds that the ionization probability shows several resonance peaks and is also much larger than the linear superposition of probabilities by applying two lasers separately. The enhancement of the ionization happens when the system is resonantly pumped to the excited states by absorbing two or more colour photons non-sequentially.
基金supported by the National Natural Science Foundation of China(Grant Nos.11044007 and 11064013)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20096203110001)the Foundation of Northwest Normal University,China(Grant Nos.NWNU-KJCXGC-03-62 and NWNU-KJCXGC-03-70)
文摘Two-dimensional （2D） electron momentum distributions and energy spectra of a hydrogen in an intense laser field are calculated by solving the time-dependent SchrSdinger equation combined with the window-operator technique. Compared with the standard projection technique, the window-operator technique has the advantage that the continuum states of atoms can be avoided in the calculation. We show that the 2D electron momentum distributions and the energy spectra from those two techniques accord quite well with each other if an appropriate energy width is used in the window operator.
