Collective Motions in Nuclei by the Method of Generator Coordinates

Abstract

For an $A$-particle system, a trial wave function is constructed of the form $\ensuremath{\Psi}({\mathrm{x}}_{1}, \ensuremath{\cdots}, {\mathrm{x}}_{A})=\ensuremath{\int}\ensuremath{\phi}({\mathrm{x}}_{1}, \ensuremath{\cdots}, {\mathrm{x}}_{A}; \ensuremath{\alpha})f(\ensuremath{\alpha})d\ensuremath{\alpha}.$ The preliminary nucleonic wave function, $\ensuremath{\phi}$, solves the problem in a "construction potential." This potential depends upon a "deformation parameter" or "generator coordinate," $\ensuremath{\alpha}$. The collective wave function, $f(\ensuremath{\alpha})$, or "generator function," is folded into $\ensuremath{\phi}$ to produce a system wave function that depends only upon the coordinates, ${\mathrm{x}}_{i}$, of the particles. In the integration, the deformation parameters dissolve away. They do not appear in the final state function; they only generate it. No collective coordinates ever come into use nor do such coordinates ever have to be defined. In typical cases when the generator function contains one or more nodes, it generates nodes in the system wave function $\ensuremath{\Psi}$ of the kind that describe collective kinetic energy. The energy of the system is extremized with respect to choice of the generator function, $f(\ensuremath{\alpha})$. No Hamiltonian ever appears except the $A$-particle Hamiltonian. All nucleons are treated on the same basis whether in or above closed shells. The appropriate variational calculation leads to an integral equation or "generator wave equation" for $f(\ensuremath{\alpha})$. This equation is solved in two limiting cases: the quadratic approximation, and the $\ensuremath{\delta}$-function approximation. An analysis is made of the Peierls-Yoccoz procedure to calculate the effectivemass parameter in cases where the forces acting in the system are invariant with respect to translation or rotation. There is no external machinery to drive the construction potential. The effective inertia constant does not appear likely to agree in general with that calculated for the essentially different problem of particles in such a machine-driven potential, though the latter value is presumably more nearly correct for physical applications. The trial wave function in the method of generator coordinates is designed for simplicity, not for precision. It is applied in the following paper to the dilatational and shape oscillations of ${\mathrm{O}}^{16}$.