Stieltjes electrostatic model interpretation for bound state problems

Abstract

In this paper, it is shown that Stieltjes electrostatic model and quantum Hamilton Jacobi formalism are analogous to each other. This analogy allows the bound state problem to mimic as n unit moving imaginary charges $i\hbar $ , which are placed in between the two fixed imaginary charges arising due to the classical turning points of the potential. The interaction potential between n unit moving imaginary charges $i\hbar $ is given by the logarithm of the wave function. For an exactly solvable potential, this system attains stable equilibrium position at the zeros of the orthogonal polynomials depending upon the interval of the classical turning points.