Who is Afraid of Special Functions and Orthogonal Polynomials in Quantum Mechanics

Abhijit Sen, N. Gurappa, Rajneesh Atre and Prasanta K. Panigrahi

Physics Education

We explicate a novel procedure to solve general linear dierential equations, which routinely manifest in Quantum Mechanics. These equations include Hermite, Laguerre, Jocobi, Bessel and hypergeometric etc. The present approach connects the desired solutions to monomials x, of an appropriate degree . In the process the underlying symmetry of the equations under study, as well as that of the solutions are made transparent. We demonstrate the ecacy of the method by showing the common structure of the solution space of a wide variety of dierential equations. We also illustrate the use of the procedure to develop approximate solutions through the examples of sextic and anharmonic oscillators. We further use the method in nding solutions of many particle interacting systems through the examples of Calogero-Sutherland and Sutherland type of models.