Collective tunneling transitions take place in the case that a system has two nearly degenerate ground states with a slight energy splitting, which provides the time scale of the tunneling. The Liouville equation determines the evolution of the density matrix, while the Schrödinger equation determines that of a state. The Liouville equation seems to be more powerful for calculating accurately the energy splitting of two nearly degenerate eigenstates. However, no method to exactly solve the Liouville eigenvalue equation has been established. The usual projection operator method for the Liouville equation is not feasible. We analytically solve the Liouville evolution equation for nuclear collective tunneling from one Hartree minimum to another, proposing a simple and solvable model Hamiltonian for the transition. We derive an analytical expression for the splitting of energy eigenvalues from a spectral function of the Liouville evolution using a half-projected operator method. A full-order analytical expression for the energy splitting is obtained. We define the collective tunneling path of a microscopic Hamiltonian for collective tunneling, projecting the nuclear ground states onto n-particle n-hole state spaces. It is argued that the collective tunneling path sector of a microscopic Hamiltonian can be transformed into the present solvable model Hamiltonian.
ASJC Scopus subject areas
- Physics and Astronomy (miscellaneous)