We develop and test an ab initio time-dependent Basis Function (tBF) method to solve non-perturbative and time-dependent problems in quantum mechanics. For our test problem, we apply this method to the Coulomb excitation of the deuteron by an impinging heavy ion. In the tBF method applied to deuterium, we employ wave functions for its bound and excited states to calculate its transition probabilities and the r.m.s. radius during the scattering process. For comparison, corresponding results based on first-order perturbation theory are also provided. For the Coulomb excitation process in a weak and time-varying Coulomb field, where higher-order effects are negligible, we obtain good agreement of the results based on these two methods. The tBF method is then applied to the Coulomb excitation process with stronger external field. The higher-order effects, such as those appearing in the reorientation of the polarization of the deuteron system, are analyzed.
We develop and test an ab initio time-dependent Basis Function (tBF) method to solve non-perturbative and time-dependent problems in quantum mechanics. For our test problem, we apply this method to the Coulomb excitation of the deuteron by an impinging heavy ion. In the tBF method applied to deuterium, we employ wave functions for its bound and excited states to calculate its transition probabilities and the r.m.s. radius during the scattering process. For comparison, corresponding results based on first-order perturbation theory are also provided. For the Coulomb excitation process in a weak and time-varying Coulomb field, where higher-order effects are negligible, we obtain good agreement of the results based on these two methods. The tBF method is then applied to the Coulomb excitation process with stronger external field. The higher-order effects, such as those appearing in the reorientation of the polarization of the deuteron system, are analyzed.
We have developed a new theory to study excited states and dynamics of atomic nuclei, which we call the Canonical-basis time-dependent Hartree-Fock-Bogoliubov (Cb-TDHFB). The CbTDHFB is a simplified theory of the full TDHFB theory, treating pairing energy functional with a BCS-like approximation which is assumed to be diagonal in the canonical basis. Implementing the theory with a real-space and real-time method, we can describe nuclear excitations and dynamics taking account of deformation effects and pairing correlations, even for heavy nuclei with a reasonable computational costs. In this thesis, we first present a derivation of the CbTDHFB theory and we apply the method to linear-response calculations for even-even light to heavy nuclei and demonstrate its capability and accuracy by comparing our results with recent calculations of the quasi-particle random-phase approximation (QRPA) which is equivalent to the small amplitude limit of TDHFB, with Skyrme functionals. We derived the Cb-TDHFB equations starting with the full TDHFB equations, introducing a simplified form for the pairing energy functional. In the derivation, we start with the ordinary TDHFB equation written in terms of the time-dependent generalized density matrix R(t) which is composed of the time-dependent normal density matrix ρ(t) and the pairing tensor κ(t). We then rewrite the equation in the canonical basis, which diagonalize ρ(t). The derived TDHFB equations in the canonical basis has a very simple form. However, they are not useful practically for a general case since it costs much to obtain the canonical basis themselves at each time. Solving the time-dependent equation for the canonical basis is as difficult as solving the full TDHFB equation. This difficulty disappears if we employ a simplified pairing energy density functional, which is equivalent to that employed in the BCS approximation for ground state calculations. For linear response calculations, we present results of the Cb-TDHFB calculations for excited states of some spherical and deformed nuclei to show performance of the method. We show strength functions of isovector dipole and isoscalar quadrupole excitations for some spherical and deformed nuclei. First, we show the comparison results of Cb-TDHFB with the QRPA in order to confirm the justification of canonical-basis formulation and the reduction of computational cost with this new framework. We show isoscalar quadrupole strength functions of neutron-rich nucleus 34Mg for confirmation of our new method, and also discuss full self-consistent calculations of 24Mg as an example of light nucleus and 208Pb and 154Sm as an example of heavy nuclei, in order to be clear, especially, the effects of spin-orbit and Coulomb residual interaction. Next, in order to show typical strength functions for spherical and deformed nuclei, we discuss isovector dipole and isoscalar quadrupole mode for 34Si as an example of spherical nucleus, for 34Mg as an example of prolate deformed nucleus and for 24Ne as an example of oblate deformed nucleus. For a confirmation of computational cost of the calculation using Cb-TDHFB, we isovector dipole strength functions of deformed heavy nuclei 172Yb and 236−240U. We show then the comparison our results with experimental data for photo-nuclear reaction cross sections. And we also show systematic calculations low-energy E1 strength which is often called Pygmy dipole resonance and which is well-known as a important strength to understand nucleosynthesis. In this thesis, we discuss the appearance of low-energy E1 strength functions of C, O, Ne, Mg, Si, S, Ar and Ca isotopes which are relatively light nuclei.
Non-relativistic quantum scattering theory informs our broad understanding of nuclear collision processes. However, detailed theoretical insights into the dynamics of fusion, whereby the colliding nuclei form a compound nucleus, remain elusive with conventional approaches. Fundamentally, understanding the complex dissipative processes inherent to compound nucleus formation and when they occur during the collision remains challenging. This, for example, prevents a consistent description of heavy-ion fusion over a wide range of collision energies and hinders searches for new super-heavy elements. To understand these dynamics, we need time-dependent approaches to nuclear collisions. In this thesis, a non-relativistic, time-dependent wave packet theory for potential scattering of charged quantum particles (such as nuclei) has been developed. It formally extends the application of Tannor and Weeks's (1993) formulation of the scattering matrix theory to systems containing the long-range Coulomb potential, by utilising Dollard's (1964) Møller operators. Using the asymptotic localisation of wave packets, analytical theorems have been formulated that allow convenient numerical application of this method. Numerical tests illustrating the theory mark the first rigorous application of this method in nuclear physics. Further insights are gained through a novel analysis of the wave packet time-correlation function (which underlies the theory) to understand how different dynamical processes, such as the formation of long-lived, quasi-bound states, during scattering contribute to the scattering matrix. This provides guidance on using the time-correlation function to obtain fusion observables, along with the associated challenges. Furthermore, this analysis showed that time-correlation functions can be obtained from static calculations, opening numerous new possibilities for studying the dynamics of nuclear reactions, in general, using this approach. This work opens new avenues for studying the dynamics of compound nucleus formation and other dissipative processes in nuclear reactions.
The Green function formalism introduced by Keldysh for irreversible processes is applied to the calculation of the time-dependent charge exchange probabilities in scattering problems. Considering the initial state of two non-interacting subsystems described by an independent particle model, their interaction is taken in a first stage, as evolving in time according to a time-dependent Hartree - Fock (TDHF) scheme. The equations of motion for the two-times Green functions are subsequently obtained in Dyson-like form by including the residual electronic correlations expanded perturbatively up to second order. It is shown that, by solving these equations of motion, this in turn allows one to calculate the two-particle correlation functions at equal times, as required to describe the fractions of charge in each subsystem. This procedure contrasts with that of considering expectation values of the correlation functions taken between the corrected TDHF state up to an equivalent order in the perturbation. In order to test the advantages of this method we have applied this scheme to the scattering of an atom from a three-substrate atom chain described by an Anderson Hamiltonian, where comparisons with exact solutions can be easily established. We found that, in this case, the fractions of charges carried by the scattered particle obtained with our proposal compare fairly well with exact results, within an ample range of parameter selection.
A direct method to derive the expression for the polarization potential between a charged particle and a two-fragment quantum system is developed. This method is based on the asymptotic properties of the two-body Coulomb Green function. The explicit form of the polarization potential constant is obtained for low-energy deuteron-nucleus scattering and for ion-hydrogen scattering. The properties of the polarization potential at high collision energies are applied to obtain an expression for the ion-atom scattering total cross section.
We present a time-dependent quantum algorithm for nuclear inelastic scattering in the time-dependent basis function on qubits approach. This algorithm aims to quantum simulate a subset of the nuclear inelastic scattering problems that are of physical interest in which the internal degrees of freedom of the reaction system are excited by time-dependent external interactions. We expect that our algorithm will enable an exponential speedup in simulating the dynamics of the subset of the inelastic scattering problems, which would also be advantageous for the applications to more complicated scattering problems. For a demonstration problem, we solve for the Coulomb excitation of the deuteron, where the quantum simulations are performed with IBM Qiskit.
Abstract. Using the time-dependent theory of quantum mechanics, we investigate nuclear electric dipole responses. The time evolution of a wave function is explicitly calculated in the coordinatespace representation. The particle continuum is treated with the absorbing boundary condition. Calculated time-dependent quantities are transformed into those of familiar energy representation. We apply the method to a three-body model for 11 Li and to the mean-field model for 22 O, then discuss properties of E1 response. TIME-DEPENDENT METHOD FOR NUCLEAR RESPONSES Atomic nuclei exhibit a variety of responses to different experimental probes; Coulomb and nuclear excitations, spin- and isospin-dependent probes, high- and low-energy reactions. In order to investigate properties of these nuclear excitations, it is useful to study response function for a probe (external) operator of interest. Normally, these quantum many-body problems are theoretically studied using the energy representation, in which calculations are carried out by either diagonalizing the Hamiltonian matrices or solving differential equations with a fixed energy. These approaches are advantageous when we are interested in states within a limited energy range. However, when we calculate