A formulation of the complete dynamics of electrons and nuclei is presented. The dynamical equations are derived using the time-dependent variational principle (TDVP). The approximate electronic state vectors are antisymmetrized geminal power (AGP) states parameterized as projected coherent states, while the nuclei are treated as classical point particles. This leads to a formulation of time-dependent AGP theory that generalizes time-dependent Hartree–Fock (TDHF) theory and explicitly includes the dynamics of the nuclei. The linear approximation to the evolution equations (the classical harmonic approximation) which corresponds to a generalized random phase approximation (RPA) based on an AGP electronic reference state and which explicitly includes the dynamics of the nuclei, is studied and presented in this paper. The equations are formulated in terms of the primitive nonorthogonal electronic atomic basis thus avoiding any transformation to orthonormal molecular orbitals during the evolution.
A formulation of the complete dynamics of electrons and nuclei is presented. The dynamical equations are derived using the time-dependent variational principle (TDVP). The approximate electronic state vectors are antisymmetrized geminal power (AGP) states parameterized as projected coherent states, while the nuclei are treated as classical point particles. This leads to a formulation of time-dependent AGP theory that generalizes time-dependent Hartree–Fock (TDHF) theory and explicitly includes the dynamics of the nuclei. The linear approximation to the evolution equations (the classical harmonic approximation) which corresponds to a generalized random phase approximation (RPA) based on an AGP electronic reference state and which explicitly includes the dynamics of the nuclei, is studied and presented in this paper. The equations are formulated in terms of the primitive nonorthogonal electronic atomic basis thus avoiding any transformation to orthonormal molecular orbitals during the evolution.
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.
We investigate the validity of the time-dependent Hartree-Fock (TDHF) approximation for a finite fermion model system. With the help of coherent states we derive equations of motion for the one-particle density, which are of classical Hamiltonian form and can be interpreted in terms of generalized coordinates and momenta. The corresponding trajectories in phase space show transitions between regular and chaotic regimes depending on the parameters of the model. A chaotic behavior indicates a failure of the TDHF approximation because the exact solutions are always regular due to the linearity of the Schr\"odinger equation.
This paper presents the equations-of-motion method as a useful and flexible tool in the study of nuclear spectroscopy. It is partly a review, but also it introduces a new and much more powerful equations-of-motion technique which supercedes the older linearization methods. The older methods worked with operator equations. To obtain closed expressions they had to be linearized in a rather arbitrary manner. The present approach works with the ground-state expectation of operator equations and thereby avoids all problems of linearization. Thus, like the Green's function method, the equations-of-motion method becomes potentially exact. It has many advantages over Green's function methods, however, among which are its greater compactness, simplicity, and the physical insight it yields.The method is first applied to rederive the random phase approximation (RPA) and the quasi-particle RPA (QRPA) and to show precisely what terms they neglect. It is demonstrated that some of these terms have coherent phases. A higher RPA and QRPA are then derived to include these terms. The corrections have some interesting effects: notably, there is a reduction of the effective interaction strength and a stabilization of the nucleus against sudden phase transitions. The equations-of-motion method is also used to generalize, in a very simple and natural way, the Hartree-Fock (HF) and Hartree-Bogolyubov (HB) concepts of independent particles and quasi-particles to nonsimple ground states.The equations-of-motion method is presented as a simple extension of the shell model to the treatment of excitationg of a correlated ground state. By concentrating on the quantities of direct physical interest, the complexity of workins with correlated wavefunctions is avoided.
A theory of large amplitude collective motion of a many-particle system is presented, which is relevant, for example, to nuclear fission. The theory is a combination of techniques used in many areas of physics and mathematics. The starting point is the application of the time-dependent Schrödinger equation to generate invariant subspaces of the Hamiltonian in the Hartree–Fock approximation. This is a generalization of the group-theoretical device of generating orbits of a group in the construction of reduced representations. It is shown how solutions of the time-dependent Schrödinger equation can be expressed as instantaneous stationary states of a constrained static Hamiltonian. Thus contact is made with the traditional cranking models and constrained Hartree–Fock theories of large amplitude collective motion. The collective motion is quantized using the Hill–Wheeler–Griffin method of generator coordinates in a basis of generalized coherent states. One is thereby able to exploit much of the theory of harmonic oscillator coherent states, which have been so successfully used in the quantum theory of the laser. The resulting Schrödinger equation for the collective dynamics is expressed both in the Bargmann representation and in the more familiar Schrödinger representation. It is shown that solution of the Schrödinger equation in the small amplitude harmonic approximation reproduces the well-known RPA result. A pilot calculation for 28Si shows that application in large amplitude is also feasible.
The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied by transforming the time-dependent Hartree-Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the Hartree-Fock energy surface in the phase space. It is shown that as the interaction strength increases these time-dependent Hartree-Fock solutions undergo a qualitative change associated with the ground state phase transition previously studied in terms of coherent states. For twobody interactions stronger than the critical value, two types of time-dependent Hartree-Fock solutions (the "librations" and "rotations" in Hamilton's mechanics) exist simultaneously, while for weaker interactions only the rotations persist. It is also shown that the coherent states with the maximum total pseudospin value are determinants, so that time-dependent Hartree-Fock analysis is equivalent to the coherent state method.NUCLEAR STRUCTURE Lipkin-Meshkov-Glick model, TDHF, transformation of TDHF equations to Hamilton's canonical form, coherent states, ground state phase transition.