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Physics Procedia

Quantum Approach to One-body Dissipation

Margarit Rizea, N. Cârjan

Published Jan 1, 2012• 5 citations
Publisher

Abstract

The nuclear dissipation, i.e. the conversion of collective energy into intrinsic energy is investigated in the frame of quantum mechanics. Using appropiate numerical procedures, we follow the motion of individual nucleons according to the time-dependent Schr̈odinger equation with time-dependent potential. In particular we study the transition from the saddle to the scission point during the low energy fission of 236U. Different rates T of change of the nuclear shape along this path were considered. The overlap integrals between the static solutions of the bi-dimensional Schr̈odinger equation and the time-dependent wave packets yield the transition probabilities and hence the singleparticle excitations during the saddle-to-scission descent. Using the numerical solutions other relevant pre-scission properties have been evaluated as well.

DOI
10.1016/j.phpro.2012.04.011

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The nuclear dissipation, i.e. the conversion of collective energy into intrinsic energy is investigated in the frame of quantum mechanics. Using appropiate numerical procedures, we follow the motion of individual nucleons according to the time-dependent Schr̈odinger equation with time-dependent potential. In particular we study the transition from the saddle to the scission point during the low energy fission of 236U. Different rates T of change of the nuclear shape along this path were considered. The overlap integrals between the static solutions of the bi-dimensional Schr̈odinger equation and the time-dependent wave packets yield the transition probabilities and hence the singleparticle excitations during the saddle-to-scission descent. Using the numerical solutions other relevant pre-scission properties have been evaluated as well.
The general problem of dissipation in macroscopic large-amplitude collective motion and its relation to energy diffusion of intrinsic degrees of freedom of a nucleus is studied. By applying the cranking approach to the nuclear many-body system, a set of coupled dynamical equations for the collective classical variable and the quantum mechanical occupancies of the intrinsic nuclear states is derived in the limit of weak coupling of the collective and intrinsic subsystems. Different dynamical regimes of the intrinsic nuclear motion and its consequences on time properties of collective dissipation are discussed.
A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave-packet dynamics of chemical reaction rates.
Chemical dynamics, in principle, should be understood by solving the time-dependent Schrödinger equation for a molecular system, describing motion of the nuclei and electrons. However, the computational efforts to solve this partial second-order differential equation scales exponentially with the system size, which prevents us from getting exact numerical solutions for systems larger than 4-5 atoms. Thus, approximations simplifying the picture are necessary. The so-called Born-Oppenheimer approximation, separating motion of the electrons and nuclei is the central one: solution to the electronic Schrödinger equation defines the potential energy surface on which the nuclear motion unfolds, and there are standard quantum chemistry software packages for solving the electronic Schrödinger equation. For the nuclear Schrödinger equation, however, there are no widely applicable quantum-mechanical approaches, and most simulations are performed using classical Newtonian mechanics which is often adequate due to large nuclear masses. However, the nuclear quantum effects are significant for chemical processes involving light nuclei at low energies, and including these effects into simulation, even approximately, is highly desirable. In this dissertation, an approximate methodology of including quantum-mechanical effects within the quantum trajectory or the de Broglie-Bohm formulation of the Schrödinger equations is developed. Use of the trajectory framework makes the approach scalable to hundreds of degrees of freedom. The methodology is applied to study high-dimensional systems (solid He4 and others) relevant to chemistry.
A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed.
A procedure is developed for treating a nuclear collective motion via the path integral formalism by defining a collective propagator in the "phase space" of the collective subspace. This collective subspace is projected out of the full many-body Hilbert space by an approach of the Generator Coordinate Method. Under the approximation of quadratic "momentum" dependence we write the propagator in the coordinate representation from which we derive the collective Schrödinger equation in terms of a canonical pair and .
At present there are two vastly different ab initio approaches to the description of the many-body dynamics: the density functional theory (DFT) and the functional integral (path integral) approaches. On one hand, if implemented exactly, the DFT approach can allow in principle the exact evaluation of arbitrary one-body densities. However, when applied to large amplitude collective motion (LACM), this approach needs to be extended in order to accommodate the phenomenon of surface-hopping, when adiabaticity is strongly violated and the description of a system using a single (generalized) Slater determinant is not valid anymore. The functional integral approach on the other hand does not appear to have such restrictions, but its implementation does not appear to be a straightforward endeavor. However, within a functional integral approach one seems to be able to evaluate in principle any kind of observable, such as the fragment mass and energy distributions in nuclear fission. These two radically different approaches can likely be brought together by formulating a stochastic time-dependent DFT approach to many-body dynamics.