Wave Functions of Pentadiagonal Matrices in the Weak Coupling Limit
aa r X i v : . [ phy s i c s . g e n - ph ] S e p WAVE FUNCTIONS FOR PENTADIAGONALMATRICES IN THE WEAK COUPLINGLIMIT
Larry ZamickDepartment of Physics and Astronomy,Rutgers University, Piscataway, New Jersey 08854September 15, 2020
Abstract
We consider a pentadiagonal matrix which will be described in thetext. We demonstrate practical methods for obtaining weak coupling ex-pressions for the lowest eigenvector in terms of the parameters in thematrix, v and w. It is found that the expressions simplify if the wavefunction coefficients are put in the denominator.
Matrix diagonalization is at the heart of the nuclear shell model . But morethan that it is of vital importance in many other branches , certainly atomic andmolecular physics as well as condensed matter. In nuclear physics we typicallyuse very complicated Hamiltonians which are often represented as a long stringof 2 body matrix elements. Despite this the output is in some cases tantaliz-ingly simple.In previous works [1-4] we studied simple matrices in part for theirown sake but also we tried to connect and gain insight with results of realisticcalculations. We here choose an example from personal experience. In ref [5], using a complicated Hamiltonian we found simple result that when a binningprocess was applied the magnetic dipole strength from the ground state appearsto decrease exponentially with excitation energy. In references [1] and [2] wefound a similar behavior when we used very simple tridiagonal matrix indicatingthere might be something general about this behavior.Note that although ref [5]will have been published after the “matrix” papers [1-4] it was written earlierand appeared in the archives earlier. It had a clear influence on these “matrix”papers.. 1
The Interaction
In the previous papers the matrices that were addressed included tridiagonal,pentadiagonal and heptadiagonal[1-4].However most of the analytic work wasperformed for the tridiagonal case. Here we consider the more complex penta-diagonal case and address the problem of the ground state wave functions inthe weak coupling limit..We start by showing in Table 1 an (11)x(11) pentadiagonal matrix whichis to also represent a nuclear Hamiltonian. We have dealt with such a matrixbefore[3] but onlt for the case w=v. On the diagonal we shall consider the casewhere E n =nE.Table 1. An (11) x(11) Pentadiagonal Matrix with 2 Parameters v and w.. { E0, v, w, 0, 0, 0, 0, 0, 0, 0, 0 } , { v , E1, v, w, 0, 0, 0, 0, 0, 0, 0 } , { w, v, E2, v, w, 0, 0, 0, 0, 0, 0 } , {
0, w, v, E3, v, w, 0, 0, 0, 0, 0 } , {
0, 0, w, v, E4, v, w, 0, 0, 0, 0 } , {
0, 0, 0, w, v, E5, v, w, 0, 0, 0 } , {
0, 0, 0, 0, w, v, E6, v, w, 0, 0 } , {
0, 0, 0, 0, 0, w, v, E7, v, w, 0 } , {
0, 0, 0, 0, 0, 0, w, v, E8, v, w } , {
0, 0, 0, 0, 0, 0, 0, w, v, E9, v } , {
0, 0, 0, 0, 0, 0, 0, 0, w, v, E10 }} .. 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. .. .Our objective is to obtain the lowest eigenvector of the pentadiagonal ma-trix shown above in the weak coupling limit i.e. when both v and w are muchsmaller than the energy separations on the diagonal . We use the word “ob-tain” rather than “derive”. While a derivation can be obtained it becomes verycomplicated. We here offer a practical method of obtaining the eigenvector com-ponents. Of course it is not diffiicult, with programs like Mathematica to obtainthese components for any strength of the interaction but we are here interestedin the analytical form.We associate the lowest eigenvector wiith a ground statewave function. 2ur method is to choose small values of v and w . We use Mathematicato obtain the eigenvectors, and fit the eigenvectors components to a plausibleformula. To make things competative we chose the following set of values:v = n *10 − and w = m *10 − with m and n small integers. Why the bigdifference between v and w? With w we can make a 2 state jump in one shotbut with v we have to make the jump in 2 steps.By examining the structure of the lowest eigenvalue we find that the com-ponents { a , a ,......,a , a } have the following structure: { − v , (c v + c w),( d v + d v w),( e v + e v w + e w ),...... } .When the results are present with the numbers in the numerator they arecomplicated looking , but in the denominator they are for the most part integers,and so the formulas look much simpler. Our previous works shed some light onthis, as will be seen in the next section...Table 2: Components of the ground state wave functions for pentadiagoal ma-trices in terms of v and w.. n a n / − w/