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.
Quantum tunneling in many-body systems is the subject of many experimental and theoretical studies in fields ranging from cold atoms to nuclear physics. However, theoretical description of quantum tunneling with strongly interacting particles, such as nucleons in atomic nuclei, remains a major challenge in quantum physics. An initial-value approach to tunneling accounting for the degrees of freedom of each interacting particle is highly desirable. Inspired by existing methods to describe instantons with periodic solutions in imaginary time, we investigate the possibility to use an initial value approach to describe tunneling at the mean-field level. Real-time and imaginary-time Hartree dynamics are compared to the exact solution in the case of two particles in a two-well potential. Whereas real-time evolutions exhibit a spurious self-trapping effect preventing tunneling in strongly interacting systems, the imaginary-time-dependent mean-field method predicts tunneling rates in excellent agreement with the exact solution. Being an initial-value method, it could be more suitable than approaches requiring periodic solutions to describe realistic systems such as heavy-ion fusion.
A self-consistent method for evaluation of mass parameters is presented in the framework of the adiabatic limit of the time-dependent Hartree-Fock approximation, reduced to a single collective variable. The corresponding collective path is assumed to be given either by solving a constrained Hartree-Fock problem with a given time-even constraining operator
Q, or by scaling a static Hartree-Fock equilibrium solution. In the former case, once the path is given, a method for solving the equation of motion (of the Hamilton type) is provided, which reduces to a double-constrained Hartree-Fock problem with the time-even constraint
Q and with a time-odd constraining operator
P. In the case of the scaling path, an analytical solution of the Hamilton equation is discussed and the adiabatic mass for the particular case of an isoscalar quadrupole
Q20 mode is given. The operator
P, which is uniquely determined from the knowledge of
Q, has the physical meaning of a momentum operator; it satisfies, together with
Q, a weak quantal conjugation relation. Finally, the connection between the two paths is discussed in terms of generalized random-phase approximation sum rules.NUCLEAR STRUCTURE Evaluation of mass parameters within time-dependent Hartree-Fock approximation in adiabatic limit for single collective variable motion. Discussion and comparison of constrained Hartree-Fock and scaling paths. Expression as a doubly constrained Hartree-Fock problem with momentum operator uniquely defined from coordinate operator. Connection with sum rule and Inglis cranking approaches.
To describe quantal collective phenomena, it is useful to requantize the time-dependent mean-field dynamics. We study the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory for the two-level pairing Hamiltonian, and compare results of different quantization methods. The one constructing microscopic wave functions, using the TDHFB trajectories fulfilling the Einstein-Brillouin-Keller quantization condition, turns out to be the most accurate. The method is based on the stationary-phase approximation to the path integral. We also examine the performance of the collective model which assumes that the pairing gap parameter is the collective coordinate. The applicability of the collective model is limited for the nuclear pairing with a small number of single-particle levels, because the pairing gap parameter represents only a half of the pairing collective space.
The starting point for most studies of large amplitude collective motion in nuclear physics, the time-dependent Hartree-Fock equations, can be mapped to a problem in classical Hamiltonian mechanics, which is the form of the problem studied in this work. For a system with N degrees of freedom, collective motion is identified with motion completely confined to a surface in K<N dimensions. Conditions for the existence of such decoupled surfaces are worked out and a procedure for constructing them is formulated. For most practical problems, where such surfaces do not strictly exist, a concept of approximate decoupling is developed and a test of its accuracy is described. Our results generalize the method of maximum global decoupling found in the earlier literature.