A New Proof for the Harmonic-Potential Theorem

Generally "exact" quantum Monte Carlo computations for the ground state of many bosons make use of importance sampling. The importance sampling is based either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of path integral ground state (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a nonzero overlap with the ground state, which gives rise to a discrete path which is sampled via a Metropolis-like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid (4)He with PIGS by considering as initial wave function a constant, i.e., the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e., there is no importance sampling. For both phases we obtain results converging to those obtained by considering the best available variational wave function (the shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance, a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function, and the static structure factor, but also for off-diagonal ones, such as the one-body density matrix. This robustness of PIGS can be traced back to the fact that the chosen initial wave function acts only at the beginning of the path without affecting the imaginary time propagator. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature path integral Monte Carlo. On the other hand, a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.

Motivated by an unexpected experimental observation from the Cambridge group, [Eigen etal., Nature 563, 221 (2018)], we study the evolution of the momentum distribution of a degenerate Bose gas quenched from the weakly interacting regime to the unitary regime. For the two-body problem, we establish a relation that connects the momentum distribution at a long time to a subleading term in the initial wave function. For the many-body problem, we employ the time-dependent Bogoliubov variational wave function and find that, in certain momentum regimes, the momentum distribution at long times displays the same exponential behavior found by the experiment. Moreover, we find that this behavior is universal and is independent of the short-range details of the interaction potential. Consistent with the relation found in the two-body problem, we also numerically show that this exponential form is hidden in the same subleading term of the Bogoliubov wave function in the initial stages. Conceptually, our results show that, for quench to the universal regime and coherent quantum dynamics afterward, the universal longtime behavior is hidden in the initial state.

We used nonequilibrium many-body Green techniques for the time varying resonant phenomena in few-body systems. By relating the transition energy shift and width to the self-energy and by applying the functional variational techniques to the transition energy shift and width, the corresponding initial state resonant wave functions are derived. Then we obtained time-dependent decaying state wave functions, evolving from these initial states by solving the time-dependent Schr\"odinger equation for the decaying system numerically. Finally, we observed the plausible time decaying behaviors of our model as predicted by real few-body systems.

We have studied the accuracy and the numerical stability of several methods for solving time dependent quantum problems in which a localized system is disturbed for a finite time. The methods assume that the wave function can be represented throughout the interaction time as a sum of Gaussian wave packets; the time evolution of the wave function is given by the evolution of the parameters in each packet. One of the methods propagates the packets independently and the other couples them by using a variational principle. Applications were made to calculate the time evolution of the ground state of a Morse oscillator and of a linear combination of the first four Morse states, and to calculate the vibrational excitation by coupling to a strong laser field. We have found that the coupled packets method provides accurate, numerically stable solutions for these problems, while the decoupled packet method fails for some of them. We have investigated the dependence of the results on the method used to fit the initial wave function to the sum of Gaussians and found that the nonlinear, least square fitting is superior to other methods. A particularly important observation is that the results obtained by the coupled Gaussians method are improved as the number of packets is increased but the propagation equations become unstable if too many packets are used to fit the wave function. Methods for monitoring and avoiding such instabilities are discussed in the paper. Our overall conclusion is that the use of the coupled, frozen Gaussians method may play an important role in the study of time dependent quantum processes in which localized quantum degrees of freedom are disturbed for a moderately long time (i.e., ten times the time constant of the system).

FFT-split-operator code for solving the Dirac equation in 2+1 dimensions

For time (t)-dependent wave functions, we derive rigorous conjugate relations between analytic decompositions (in the complex t plane) of phases and log moduli. We then show that reciprocity, taking the form of Kramers-Kronig integral relations (but in the time domain), holds between observable phases and moduli in several physically important instances. These include the nearly adiabatic (slowly varying) case, a class of cyclic wave functions, wave packets, and noncyclic states in an ``expanding potential''. The results define a unique phase through its analyticity properties, and exhibit the interdependence of geometric phases and related decay probabilities. Several known quantum-mechanical applications possess the reciprocity property obtained in the paper.

We describe the time evolution of a wave function in the infinite square well using a fractional revival formalism, and show that at all times the wave function can be described as a superposition of translated copies of the initial wave function. Using the model of a wave form propagating on a dispersionless string from classical mechanics to describe these translations, we connect the reflection symmetry of the square-well potential to a reflection symmetry in the locations of these translated copies, and show that they occur in a ``parity-conserving'' form. The relative phases of the translated copies are shown to depend quadratically on the translation distance along the classical path. We conclude that the time-evolved wave function in the infinite square well can be described in terms of translations of the initial wave-function shape, without approximation and without any reference to its energy eigenstate expansion. That is, the set of translated initial wave functions forms a Hilbert space basis for the time-evolved wave functions.

Over the last three decades numerous numerical methods for solving the time-dependent Schr\"{o}dinger equation within the single-active electron approximation have been developed for studying ionization of atomic targets exposed to an intense laser field. In addition, various numerical techniques for extracting the photoelectron spectra from the time-dependent wave function have emerged. In this paper we compare photoelectron spectra obtained by either projecting the time-dependent wave function at the end of the laser pulse onto the continuum state having proper incoming boundary condition or by using the window-operator method. Our results for three different atomic targets show that the boundary condition imposed onto the continuum states plays a crucial role for obtaining correct spectra accurate enough to resolve fine details of the interference structures of the photoelectron angular distribution.

We develop the continuum mechanics of quantum many-body systems in the linear response regime. The basic variable of the theory is the displacement field, for which we derive a closed equation of motion under the assumption that the time-dependent wave function in a locally comoving reference frame can be described as a geometric deformation of the ground-state wave function. We show that this equation of motion is exact for systems consisting of a single particle, and for all systems at sufficiently high frequency, and that it leads to an excitation spectrum that has the correct integrated strength. The theory is illustrated by simple model applications to one- and two-electron systems.

The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics.

A method is presented that reconstructs the amplitude and phase of an unknown time-dependent pure-state wave function entirely from experimental position data. No assumptions about the wave function are needed. At a series of times, a large collection of position data is taken, but no measurements of momentum or energy are necessary. The momenta, however, are required for the reconstruction of the wave function and are supplied by analysis of the position data via quantum particle trajectories. Simulations of the method are provided for helium atoms in single and double slit experiments.

We report on the nonlinear tunneling effects of spatial solitons of the generalized nonlinear Schrödinger equation with distributed coefficients in an external harmonic potential. By using the homogeneous balance principle and the F-expansion technique we find the spatial bright and dark soliton solutions. We then display tunneling effects of such solutions occurring under special conditions; specifically when the spatial solitons pass unchanged through the potential barriers and wells affected by special choices of the diffraction and/or the nonlinearity coefficients. Our results show that the solitons display tunneling effects not only when passing through the nonlinear potential barriers or wells but also when passing through the diffractive barriers or wells. During tunneling the solitons may also undergo a controllable compression.

We have calculated the effect of a magnetic field on the evolution of angular momentum eigenfunctions of a charged particle. An additional harmonic potential is supplemented to trap the wave packet. We find the probability density of the wave function is oscillating in the radial direction with a time period determined by the strength of the effective harmonic potential. When the magnetic field is along the z direction, if the initial wave function is an eigenfunction of Lˆz , the probability density of the particle remains axis-symmetric. While for the case of an eigenfunction of Lˆx , it is anisotropic in the x−y plane and rotates with a time period inverse proportional to the strength of the external magnetic field. We also extend the results in a phenomenological way to the case with an external magnetic field that varies harmonically in time.

Electron capturing and de-capturing dynamics in a single semiconductor quantum well due to the interaction with longitudinal optical phonons are studied by directly solving the time-dependent Schrodinger equation. Time-dependent wave functions are calculated with an initial electron wave packet impinging on the quantum well, being reflected and transmitted, captured into and de-captured from the quantum well. We have obtained new information on electron capture and de-capture times as function of electron-phonon coupling strength. The red energy shift of ground state energy is also found to be linearly proportional to the electron- phonon interaction strength. These results can not be simply obtained using Fermi Golden Rule.

We investigate the exclusive semileptonic B{sub c}{yields}(D,{eta}{sub c},B,B{sub s})l{nu}{sub l}, {eta}{sub b}{yields}B{sub c}l{nu}{sub l} (l=e,{mu},{tau}) decays using the light-front quark model constrained by the variational principle for the QCD-motivated effective Hamiltonian. The form factors f{sub +}(q{sup 2}) and f{sub -}(q{sup 2}) are obtained from the analytic continuation method in the q{sup +}=0 frame. While the form factor f{sub +}(q{sup 2}) is free from the zero mode, the form factor f{sub -}(q{sup 2}) is not free from the zero mode in the q{sup +}=0 frame. We quantify the zero-mode contributions to f{sub -}(q{sup 2}) for various semileptonic B{sub c} decays. Using our effective method to relate the non-wave-function vertex to the light-front valence wave function, we incorporate the zero-mode contribution as a convolution of the zero-mode operator with the initial and final state wave functions. Our results are then compared to the available experimental data and the results from other theoretical approaches. Since the prediction on the magnetic dipole B{sub c}*{yields}B{sub c}+{gamma} decay turns out to be very sensitive to the mass difference between B{sub c}* and B{sub c} mesons, the decay width {gamma}(B{sub c}*{yields}B{sub c}{gamma}) may help in determining the mass of B{sub c}* experimentally. Furthermore, we compare the resultsmore » from the harmonic oscillator potential and the linear potential and identify the decay processes that are sensitive to the choice of confining potential. From the future experimental data on these sensitive processes, one may obtain more realistic information on the potential between the quark and antiquark in the heavy meson system.« less