Bound state solutions of the Schrödinger equation for potentials with dirac delta functions

Abstract

A general method for the bound state solutions of the SchrÄodinger equation for analytically solvable potentials with any ¯nite number of Dirac delta functions is intro- duced for n-dimensional systems. Then, the potentials with Dirac delta functions are used to model some physical systems. The eigenvalue equations for harmonic and linear potentials with a ¯nite number of Dirac delta functions located randomly are derived for one dimensional systems. For the latter potential, the behavior of the eigenvalues of the ground and the ¯rst excited states for various strengths and locations of Dirac delta functions is investigated. The eigenvalues and the number of bound states for a PT -symmetric system with two Dirac delta functions are studied. In case of a contact interaction, to get the changes from a liner potential, the changes in the masses of s states for charmonium is presented. It is also shown that the Fermi energy of a trian- gular well changes if there is an impurity in the well. By describing a dimple potential with a Dirac delta function, it is shown that tight and deep dimple potentials can in- crease the condensate fraction and critical temperature of a Bose-Einstein condensate. We conclude that addition of the point interactions which can be modelled by Dirac delta functions changes the properties of the physical systems considerably.