Complex absorbing potential is usually required in a time-dependent wave packet method to accomplish the calculation in a truncated region.Usually it works effectively but becomes inefficient when the wave function in...Complex absorbing potential is usually required in a time-dependent wave packet method to accomplish the calculation in a truncated region.Usually it works effectively but becomes inefficient when the wave function involves translational energy of broad range,particularly involving ultra-low energy.In this work,a new transparent boundary condition(TBC)is proposed for the time-dependent wave packet method.It in principle is of spectral accuracy when typical discrete variable representations are applied.The prominent merit of the new TBC is that its accuracy is insensitive to the translational energy distribution of the wave function,in contrast with the complex absorbing potential.Application of the new TBC is given to one-dimensional particle wave packet scatterings from a barrier with a potential well,which supports resonances states.展开更多
We present a fully time-dependent quantum wave packet evolution method for investigating molecular dynamics in intense laser fields.This approach enables the simultaneous treatment of interactions among multiple elect...We present a fully time-dependent quantum wave packet evolution method for investigating molecular dynamics in intense laser fields.This approach enables the simultaneous treatment of interactions among multiple electronic states while simultaneously tracking their time-dependent electronic,vibrational,and rotational dynamics.As an illustrative example,we consider neutral H_(2)molecules and simulate the laser-induced excitation dynamics of electronic and rotational states in strong laser fields,quantitatively distinguishing the respective contributions of electronic dipole transitions(within the classical-field approximation)and non-resonant Raman processes to the overall molecular dynamics.Furthermore,we precisely evaluate the relative contributions of direct tunneling ionization from the ground state and ionization following electronic excitation in the strong-field ionization of H_(2).The developed methodology shows strong potential for performing high-precision theoretical simulations of electronic-vibrational-rotational state excitations,ionization,and dissociation dynamics in molecules and their ions under intense laser fields.展开更多
We consider the gravitational effect of quantum wave packets when quantum mechanics, gravity, and thermodynamics are simultaneously considered. Under the assumption of a thermodynamic origin of gravity, we propose a g...We consider the gravitational effect of quantum wave packets when quantum mechanics, gravity, and thermodynamics are simultaneously considered. Under the assumption of a thermodynamic origin of gravity, we propose a general equation to describe the gravitational effect of quantum wave packets. In the classical limit, this equation agrees with Newton＇s law of gravitation. For quantum wave packets, however, it predicts a repulsive gravitational effect. We propose an experimental scheme using superfluid helium to test this repulsive gravitational effect. Our studies show that, with present technology such as superconducting gravimetry and gravitational effect for superfluid helium are within cold atom interferometry, tests of the repulsive experimental reach.展开更多
Spatial, temporal and coherent superposition of quantum states is considered. A consistent interpretation of the simultaneous superposition of stationary quantum states within material wave packets is proposed.
The traveling wave group that is defined on conserved physical values is the vehicle of transmission for a unidirectional photon or free particle having a wide wave front. As a stable wave packet, it expresses interna...The traveling wave group that is defined on conserved physical values is the vehicle of transmission for a unidirectional photon or free particle having a wide wave front. As a stable wave packet, it expresses internal periodicity combined with group localization. An uncertainty principle is precisely derived that differs from Heisenberg’s. Also derived is the phase velocity beyond the horizon set by the speed of light. In this space occurs the reduction of the wave packet which is represented by comparing phase velocities in the direction of propagation with the transverse plane. The new description of the wave function for the stable free particle or antiparticle contains variables that were previously ignored. Deterministic physics must always appear probabilistic when hidden variables are bypassed. Secondary hidden variables always occur in measurement. The wave group turns out to be both uncertain and probabilistic. It is ubiquitous in physics and has many consequences.展开更多
Quantum dynamics for the D＋OD＋ reaction at the collision energy range of 0.0-1.0 eV was studied on an accurate ab initio potential energy surface. Both of the endothermic abstraction （D＋OD＋-→O＋＋D2） and thermo...Quantum dynamics for the D＋OD＋ reaction at the collision energy range of 0.0-1.0 eV was studied on an accurate ab initio potential energy surface. Both of the endothermic abstraction （D＋OD＋-→O＋＋D2） and thermoneutral exchange （D＋OD＋--＊D＋OD＋） channels were investigated from the same set of time-dependent quantum wave packets method under cen- trifugal sudden approximation. The reaction probability dependence with collision energy, the integral cross sections, and the thermal rate constant of the both channels are calculated. It is found that there is a convex structure in the reaction path of the exchange reaction. The calculated time evolution of the wave packet distribution at J=0 clearly indicates that the convex structure significantly influences the dynamics of the exchange and abstraction channels of title reaction.展开更多
Quantum dynamics calculations for the title reaction H（2S） ＋ S2（X3∑g） → SH（X2П） ＋S（3P） are performed byusing a globally accurate double many-body expansion potential energy surface [J. Phys. Chem. A 115 5...Quantum dynamics calculations for the title reaction H（2S） ＋ S2（X3∑g） → SH（X2П） ＋S（3P） are performed byusing a globally accurate double many-body expansion potential energy surface [J. Phys. Chem. A 115 5274 （2011）]. The Chebyshev real wave packet propagation method is employed to obtain the dynamical information, such as reaction probability, initial state-specified integral cross section, and thermal rate constant. It is found not only that there is a reaction threshold near 0.7 eV in both reaction probabilities and integral cross section curves, but also that both the probability and cross section increase firstly and then decrease as the collision energy increases. The existence of the resonance structure in both the probability and cross section curves is ascribed to the deep potential well. The calculation of the rate constant reveals that the reaction occurring on the potential energy surface of the ground-state HS2 is slow to take place.展开更多
We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe ...We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe the particle dynamics, are in agreement with the Hamilton equations only if in the time dependent phases one considers the Lagrangian instead of the Hamiltonian which leads to the conventional Schr?dinger equation. We define a relativistic quantum principle asserting that a quantum particle has a finite frequency spectrum, with a cutoff propagation velocity c as a universal constant not depending on the coordinate system, and that any time dependent phase variation is the same in any system of coordinates. From the time dependent phase invariance,the relativistic kinematics is obtained. We consider two types of possible interactions: 1) An interaction with an external field, by a modification of the time dependent phase differential with the terms proportional to the differentials of the space-time coordinates multiplied with the components of this field four-potential, and 2)an interaction by a deformation of the space-time coordinates, due to a gravitational field. From the invariance of the time dependent phase with field components, we obtain a mechanical force of the form of Lorentz’s force, and three Maxwell equations: The Gauss-Maxwell equations for the electric and magnetic fluxes, and the Faraday-Maxwell equation for the electromagnetic induction. When the fourth equation,Ampère-Maxwell, is considered, the interaction field takes the form of the electromagnetic field. For a low propagation velocity of the particle waves, we get a packet of waves with the time dependent phases proportional to the relativistic Hamiltonian, as in Dirac’s famous theory of spin, and a slowly-varying amplitude with a phase proportional to the momentum and this velocity. In the framework of our theory, the spin is obtained as an all quantum effect, without any additional assumption to the quantum theory. When a space-time deformation is considered in the time dependent phase of a quantum particle, from the group velocity we get the particle dynamics according to the general theory of relativity. In this way, the relativistic dynamics, the electromagnetic field, and the spin of a quantum particle are obtained only from the invariance of the time dependent phases of the particle wave functions.展开更多
According to a novel electronic ground-state potential energy surface of H2O^＋（X^4 A″）,we calculate the reaction probabilities and the integral cross section for the titled reaction O^＋＋ D2→OD^＋＋ D by the Che...According to a novel electronic ground-state potential energy surface of H2O^＋（X^4 A″）,we calculate the reaction probabilities and the integral cross section for the titled reaction O^＋＋ D2→OD^＋＋ D by the Chebyshev wave packet propagation method.The reaction probabilities in a collision-energy range of 0.0 e V–1.0 e V show an oscillatory structure for the O^＋＋ D2 reaction due to the existence of the potential well.Compared with the results of Martinez et al.,the present integral cross section is large,which is in line with experimental data.展开更多
The quantum state-to-state calculations of the D + ND→N + D_2 reaction are performed on a potential energy surface of 4 A'' state. The state-resolved integral and differential cross sections and product state...The quantum state-to-state calculations of the D + ND→N + D_2 reaction are performed on a potential energy surface of 4 A'' state. The state-resolved integral and differential cross sections and product state distributions are calculated and discussed. It is found that the rotational distribution, rather than the vibrational distribution, of the product has an obvious inversion. Due to the fact that it is a small-impact-parameter collision, its product D_2 is mainly dominated by rebound mechanism, which can lead to backward scattering at low collision energy. As the collision energy increases, the forward scattering and sideward scattering begin to appear. In addition, the backward collision is also found to happen at high collision energy, through which we can know that both the rebound mechanism and stripping mechanism exist at high collision energy.展开更多
Quantum state-to-state dynamics of the N(4S) + H-2(X1+Σ) → NH(X3Σ) + H(2S) reaction is reported in an accurate novel potential energy surface constructed by Zhai et al.(2011 J. Chem. Phys. 135 104314). The time-dep...Quantum state-to-state dynamics of the N(4S) + H-2(X1+Σ) → NH(X3Σ) + H(2S) reaction is reported in an accurate novel potential energy surface constructed by Zhai et al.(2011 J. Chem. Phys. 135 104314). The time-dependent wave packet method, which is implemented on graphics processing units, is used to calculate the differential cross sections. The influences of the collision energy on the product state-resolved integral cross sections and total differential cross sections are calculated and discussed. It is found that the products NH are predominated by the backward scattering due to the small impact parameter collisions, with only minor components being forward and sideways scattered, and have an inverted rotational distribution and no inversion in vibrational distributions; both rebound and stripping mechanisms exist in the case of high collision energies.展开更多
A time dependent quantum wave packet method was used to study the dynamics of dissociative adsorption of H 2 and D 2 on a flat and static surface. The molecule surface interaction is described using a modified London ...A time dependent quantum wave packet method was used to study the dynamics of dissociative adsorption of H 2 and D 2 on a flat and static surface. The molecule surface interaction is described using a modified London Eyring Polanyi Sato (LEPS) type potential for the H 2/Ni(100) system. The three dimensional (3 D) dissociation probabilities were calculated for different initial rovibrational states as a function of initial incident energies. Our results show that the dissociation of the diatomic rotational states whose quantum numbers satisfy j+m =odd is forbidden at low energies for the homonuclear H 2 and D 2 due to the selection rule. The effect of the rotational orientation of diatoms on adsorption predicts that the in plane rotation (m=j) is more favorable for dissociation than the out of plane rotation (m=0) . Enhanced dissociation for vibrationally excited molecules and the significant enhancement of the dissociation probability of H 2 when compared to D 2 were explained reasonably in terms of quantum mechanical zero point energies, the tunneling effect and the reflection from an activation barrier.展开更多
In the special theory of relativity, massive particles can travel at neither the speed of light c nor faster. Meanwhile, since the photon was quantized, many have thought of it as a point particle. How pointed? The id...In the special theory of relativity, massive particles can travel at neither the speed of light c nor faster. Meanwhile, since the photon was quantized, many have thought of it as a point particle. How pointed? The idea could be a mathematical device or physical simplification. By contrast, the preceding notion of wave-group duality has two velocities: a group velocity vg and a phase velocity vp. In light vp = vg = c;but it follows from special relativity that, in massive particles, vp > c. The phase velocity is the product of the two best measured variables, and so their product constitutes internal motion that travels, verifiably, faster than light. How does vp then appear in Minkowski space? For light, the spatio-temporal Lorentz invariant metric is s2=c2t2−x2−y2−z2, the same in whatever frame it is viewed. The space is divided into 3 parts: firstly a cone, symmetric about the vertical axis ct > 0 that represents the world line of a stationary particle while the conical surface at s = 0 represents the locus for light rays that travel at the speed of light c. Since no real thing travels faster than the speed of light c, the surface is also a horizon for what can be seen by an observer starting from the origin at time t = 0. Secondly, an inverted cone represents, equivalently, time past. Thirdly, outside the cones, inaccessible space. The phase velocity vp, group velocity vg and speed of light are all equal in free space, vp = vg = c, constant. By contrast, for particles, where causality is due to particle interactions having rest mass mo > 0, we have to employ the Klein-Gordon equation with s2=c2t2−x2−y2−z2+mo2c2. Now special relativity requires a complication: vp.vg = c2 where vg c and therefore vp > c. In the volume outside the cones, causality due to light interactions cannot extend beyond the cones. However, since vp > c and even vp >> c when wavelength λ is long, extreme phase velocities are then limited in their causal effects by the particle uncertainty σ, i.e. to vgt ± σ/ω, where ω is the particle angular frequency. This is the first time the phase range has been described for a massive particle.展开更多
The method of time-dependent quantum wave packet dynamics is used to calculate the femtosecond pump-probe photoelectron spectra and study the wave packet dynamic processes of the double-minimum potential state 6^1∑^...The method of time-dependent quantum wave packet dynamics is used to calculate the femtosecond pump-probe photoelectron spectra and study the wave packet dynamic processes of the double-minimum potential state 6^1∑^＋ of NaK in intense laser fields. The evolutions of the wave packet and the photoelectron energy spectra with time and internuclear distance are described in detail. The wave packet dynamic information of the 6^1∑^＋ state can be extracted from the photoelectron energy spectra.展开更多
基金supported by the National Natural Science Foundation of China (No.21733006,No.21825303 and No.21688102)the Strategic Priority Research Program of Chinese Academy of Sciences (No.XDB17010200).
文摘Complex absorbing potential is usually required in a time-dependent wave packet method to accomplish the calculation in a truncated region.Usually it works effectively but becomes inefficient when the wave function involves translational energy of broad range,particularly involving ultra-low energy.In this work,a new transparent boundary condition(TBC)is proposed for the time-dependent wave packet method.It in principle is of spectral accuracy when typical discrete variable representations are applied.The prominent merit of the new TBC is that its accuracy is insensitive to the translational energy distribution of the wave function,in contrast with the complex absorbing potential.Application of the new TBC is given to one-dimensional particle wave packet scatterings from a barrier with a potential well,which supports resonances states.
基金supported by the National Key Research and Development Program of China(Grant No.2022YFA1602502)the National Natural Science Foundation of China(Grant No.12450404)。
文摘We present a fully time-dependent quantum wave packet evolution method for investigating molecular dynamics in intense laser fields.This approach enables the simultaneous treatment of interactions among multiple electronic states while simultaneously tracking their time-dependent electronic,vibrational,and rotational dynamics.As an illustrative example,we consider neutral H_(2)molecules and simulate the laser-induced excitation dynamics of electronic and rotational states in strong laser fields,quantitatively distinguishing the respective contributions of electronic dipole transitions(within the classical-field approximation)and non-resonant Raman processes to the overall molecular dynamics.Furthermore,we precisely evaluate the relative contributions of direct tunneling ionization from the ground state and ionization following electronic excitation in the strong-field ionization of H_(2).The developed methodology shows strong potential for performing high-precision theoretical simulations of electronic-vibrational-rotational state excitations,ionization,and dissociation dynamics in molecules and their ions under intense laser fields.
文摘We consider the gravitational effect of quantum wave packets when quantum mechanics, gravity, and thermodynamics are simultaneously considered. Under the assumption of a thermodynamic origin of gravity, we propose a general equation to describe the gravitational effect of quantum wave packets. In the classical limit, this equation agrees with Newton＇s law of gravitation. For quantum wave packets, however, it predicts a repulsive gravitational effect. We propose an experimental scheme using superfluid helium to test this repulsive gravitational effect. Our studies show that, with present technology such as superconducting gravimetry and gravitational effect for superfluid helium are within cold atom interferometry, tests of the repulsive experimental reach.
文摘Spatial, temporal and coherent superposition of quantum states is considered. A consistent interpretation of the simultaneous superposition of stationary quantum states within material wave packets is proposed.
文摘The traveling wave group that is defined on conserved physical values is the vehicle of transmission for a unidirectional photon or free particle having a wide wave front. As a stable wave packet, it expresses internal periodicity combined with group localization. An uncertainty principle is precisely derived that differs from Heisenberg’s. Also derived is the phase velocity beyond the horizon set by the speed of light. In this space occurs the reduction of the wave packet which is represented by comparing phase velocities in the direction of propagation with the transverse plane. The new description of the wave function for the stable free particle or antiparticle contains variables that were previously ignored. Deterministic physics must always appear probabilistic when hidden variables are bypassed. Secondary hidden variables always occur in measurement. The wave group turns out to be both uncertain and probabilistic. It is ubiquitous in physics and has many consequences.
文摘Quantum dynamics for the D＋OD＋ reaction at the collision energy range of 0.0-1.0 eV was studied on an accurate ab initio potential energy surface. Both of the endothermic abstraction （D＋OD＋-→O＋＋D2） and thermoneutral exchange （D＋OD＋--＊D＋OD＋） channels were investigated from the same set of time-dependent quantum wave packets method under cen- trifugal sudden approximation. The reaction probability dependence with collision energy, the integral cross sections, and the thermal rate constant of the both channels are calculated. It is found that there is a convex structure in the reaction path of the exchange reaction. The calculated time evolution of the wave packet distribution at J=0 clearly indicates that the convex structure significantly influences the dynamics of the exchange and abstraction channels of title reaction.
基金supported by the National Natural Science Foundation of China(Grant Nos.11074151 and 11304185)
文摘Quantum dynamics calculations for the title reaction H（2S） ＋ S2（X3∑g） → SH（X2П） ＋S（3P） are performed byusing a globally accurate double many-body expansion potential energy surface [J. Phys. Chem. A 115 5274 （2011）]. The Chebyshev real wave packet propagation method is employed to obtain the dynamical information, such as reaction probability, initial state-specified integral cross section, and thermal rate constant. It is found not only that there is a reaction threshold near 0.7 eV in both reaction probabilities and integral cross section curves, but also that both the probability and cross section increase firstly and then decrease as the collision energy increases. The existence of the resonance structure in both the probability and cross section curves is ascribed to the deep potential well. The calculation of the rate constant reveals that the reaction occurring on the potential energy surface of the ground-state HS2 is slow to take place.
文摘We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe the particle dynamics, are in agreement with the Hamilton equations only if in the time dependent phases one considers the Lagrangian instead of the Hamiltonian which leads to the conventional Schr?dinger equation. We define a relativistic quantum principle asserting that a quantum particle has a finite frequency spectrum, with a cutoff propagation velocity c as a universal constant not depending on the coordinate system, and that any time dependent phase variation is the same in any system of coordinates. From the time dependent phase invariance,the relativistic kinematics is obtained. We consider two types of possible interactions: 1) An interaction with an external field, by a modification of the time dependent phase differential with the terms proportional to the differentials of the space-time coordinates multiplied with the components of this field four-potential, and 2)an interaction by a deformation of the space-time coordinates, due to a gravitational field. From the invariance of the time dependent phase with field components, we obtain a mechanical force of the form of Lorentz’s force, and three Maxwell equations: The Gauss-Maxwell equations for the electric and magnetic fluxes, and the Faraday-Maxwell equation for the electromagnetic induction. When the fourth equation,Ampère-Maxwell, is considered, the interaction field takes the form of the electromagnetic field. For a low propagation velocity of the particle waves, we get a packet of waves with the time dependent phases proportional to the relativistic Hamiltonian, as in Dirac’s famous theory of spin, and a slowly-varying amplitude with a phase proportional to the momentum and this velocity. In the framework of our theory, the spin is obtained as an all quantum effect, without any additional assumption to the quantum theory. When a space-time deformation is considered in the time dependent phase of a quantum particle, from the group velocity we get the particle dynamics according to the general theory of relativity. In this way, the relativistic dynamics, the electromagnetic field, and the spin of a quantum particle are obtained only from the invariance of the time dependent phases of the particle wave functions.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11674198 and 11304185)
文摘According to a novel electronic ground-state potential energy surface of H2O^＋（X^4 A″）,we calculate the reaction probabilities and the integral cross section for the titled reaction O^＋＋ D2→OD^＋＋ D by the Chebyshev wave packet propagation method.The reaction probabilities in a collision-energy range of 0.0 e V–1.0 e V show an oscillatory structure for the O^＋＋ D2 reaction due to the existence of the potential well.Compared with the results of Martinez et al.,the present integral cross section is large,which is in line with experimental data.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11674198 and 11504206)the Natural Science Foundation of Shandong Province,China(Grant No.ZR2016AP14)the Taishan Scholar Project of Shandong Province,China
文摘The quantum state-to-state calculations of the D + ND→N + D_2 reaction are performed on a potential energy surface of 4 A'' state. The state-resolved integral and differential cross sections and product state distributions are calculated and discussed. It is found that the rotational distribution, rather than the vibrational distribution, of the product has an obvious inversion. Due to the fact that it is a small-impact-parameter collision, its product D_2 is mainly dominated by rebound mechanism, which can lead to backward scattering at low collision energy. As the collision energy increases, the forward scattering and sideward scattering begin to appear. In addition, the backward collision is also found to happen at high collision energy, through which we can know that both the rebound mechanism and stripping mechanism exist at high collision energy.
基金supported by the National Natural Science Foundation of China(Grant No.11074151)the Natural Science Foundation of Shandong Province,China(Grant No.ZR2014AM022)
文摘Quantum state-to-state dynamics of the N(4S) + H-2(X1+Σ) → NH(X3Σ) + H(2S) reaction is reported in an accurate novel potential energy surface constructed by Zhai et al.(2011 J. Chem. Phys. 135 104314). The time-dependent wave packet method, which is implemented on graphics processing units, is used to calculate the differential cross sections. The influences of the collision energy on the product state-resolved integral cross sections and total differential cross sections are calculated and discussed. It is found that the products NH are predominated by the backward scattering due to the small impact parameter collisions, with only minor components being forward and sideways scattered, and have an inverted rotational distribution and no inversion in vibrational distributions; both rebound and stripping mechanisms exist in the case of high collision energies.
文摘A time dependent quantum wave packet method was used to study the dynamics of dissociative adsorption of H 2 and D 2 on a flat and static surface. The molecule surface interaction is described using a modified London Eyring Polanyi Sato (LEPS) type potential for the H 2/Ni(100) system. The three dimensional (3 D) dissociation probabilities were calculated for different initial rovibrational states as a function of initial incident energies. Our results show that the dissociation of the diatomic rotational states whose quantum numbers satisfy j+m =odd is forbidden at low energies for the homonuclear H 2 and D 2 due to the selection rule. The effect of the rotational orientation of diatoms on adsorption predicts that the in plane rotation (m=j) is more favorable for dissociation than the out of plane rotation (m=0) . Enhanced dissociation for vibrationally excited molecules and the significant enhancement of the dissociation probability of H 2 when compared to D 2 were explained reasonably in terms of quantum mechanical zero point energies, the tunneling effect and the reflection from an activation barrier.
文摘In the special theory of relativity, massive particles can travel at neither the speed of light c nor faster. Meanwhile, since the photon was quantized, many have thought of it as a point particle. How pointed? The idea could be a mathematical device or physical simplification. By contrast, the preceding notion of wave-group duality has two velocities: a group velocity vg and a phase velocity vp. In light vp = vg = c;but it follows from special relativity that, in massive particles, vp > c. The phase velocity is the product of the two best measured variables, and so their product constitutes internal motion that travels, verifiably, faster than light. How does vp then appear in Minkowski space? For light, the spatio-temporal Lorentz invariant metric is s2=c2t2−x2−y2−z2, the same in whatever frame it is viewed. The space is divided into 3 parts: firstly a cone, symmetric about the vertical axis ct > 0 that represents the world line of a stationary particle while the conical surface at s = 0 represents the locus for light rays that travel at the speed of light c. Since no real thing travels faster than the speed of light c, the surface is also a horizon for what can be seen by an observer starting from the origin at time t = 0. Secondly, an inverted cone represents, equivalently, time past. Thirdly, outside the cones, inaccessible space. The phase velocity vp, group velocity vg and speed of light are all equal in free space, vp = vg = c, constant. By contrast, for particles, where causality is due to particle interactions having rest mass mo > 0, we have to employ the Klein-Gordon equation with s2=c2t2−x2−y2−z2+mo2c2. Now special relativity requires a complication: vp.vg = c2 where vg c and therefore vp > c. In the volume outside the cones, causality due to light interactions cannot extend beyond the cones. However, since vp > c and even vp >> c when wavelength λ is long, extreme phase velocities are then limited in their causal effects by the particle uncertainty σ, i.e. to vgt ± σ/ω, where ω is the particle angular frequency. This is the first time the phase range has been described for a massive particle.
基金Project supported by the National Natural Science Foundation of China （Grant No 10374012）,
文摘The method of time-dependent quantum wave packet dynamics is used to calculate the femtosecond pump-probe photoelectron spectra and study the wave packet dynamic processes of the double-minimum potential state 6^1∑^＋ of NaK in intense laser fields. The evolutions of the wave packet and the photoelectron energy spectra with time and internuclear distance are described in detail. The wave packet dynamic information of the 6^1∑^＋ state can be extracted from the photoelectron energy spectra.
