Cascade calculation with schematic interactions

Abstract

In previous works we considered schematic Hamiltonians represented by simplified matrices. We defined 2 transition operators and calculated transition strengths from the ground state to all exited states.In many cases the strengths decreased nearly exponentially with excitation energy. Now we do the reverse We start with the highest energy state and calculate the cascade of transitions until the ground states is reached.On a log plot we show the average transition strength as a function of the number of energy intervals that were crossed. We give an analytic proof of exponential behavior for transition strength in the weak coupling limit for the T 2 transition operator.

Wave Functions of Pentadiagonal Matrices in the Weak Coupling Limit

“…In the previous papers the matrices that were addressed included tridiagonal, pentadiagonal and heptadiagonal [1][2][3][4].However most of the analytic work was performed for the tridiagonal case. Here we consider the more complex pentadiagonal case and address the problem of the ground state wave functions in the weak coupling limit.…”

“…. We have previously studies such matrices in works by A. Kingan and L. Zamick [1,2] and L.Wolfe and L.Zamick [3,4], but only for the case w=v.…”

“…In nuclear physics we typically use very complicated Hamiltonians which are often represented as a long string of 2 body matrix elements. Despite this the output is in some cases tantalizingly simple.In previous works [1][2][3][4] we studied simple matrices in part for their own sake but also we tried to connect and gain insight with results of realistic calculations. We here choose an example from personal experience.…”

“…In ref [5] , using a complicated Hamiltonian we found simple result that when a binning process was applied the magnetic dipole strength from the ground state appears to decrease exponentially with excitation energy. In references [1] and [2] we found a similar behavior when we used very simple tridiagonal matrix indicating there might be something general about this behavior.Note that although ref [5] will have been published after the "matrix" papers [1][2][3][4] it was written earlier and appeared in the archives earlier. It had a clear influence on these "matrix" papers.…”

Matrix Model: Emergence of a Quantum Number in the Strong Coupling Regime

“…In ref. [4] besides excitation energy plots we also consider top-down transitions. That is to say, we start with the highest energy state and calculate the cascade of transitions until the ground states is reached.…”