On the Use of B-Splines as Ritz Variational Basis Functions to Solve the Schrodinger Equation (TISE) for a constrained free Quantum Particle

Anandaram Mandyam N

doi:10.12723/mjs.56.2

Authors

Anandaram Mandyam Professor of Physics (Retired), Bangalore University, Bangalore, Karnataka, India

DOI:

https://doi.org/10.12723/mjs.56.2

Keywords:

B-Splines, Variational basis, TISE, free particle, Gauss Legendre quadrature, Collocation, Eigen solution

Abstract

B-Splines as piecewise adaptation of Bernstein polynomials (aka, B-polys) are widely used as Ritz variational basis functions in solving many problems in the fields of quantum mechanics and atomic physics. In this paper they are used to solve the 1-D stationary Schrodinger equation (TISE) for a free quantum particle subject to a fixed domain length by using the Python software SPLIPY with different sets of computation parameters. In every case it was found that over 60 percent of energy levels had excellent accuracy thereby proving that the use of B-spline collocation is a preferred method.

References

H.Bachau et.al. “Applications of B-splines in atomic and molecular physics”, Rep. Prog. Phys. Vol. 64, pp. 1815-1942, 2001.

M.N. Anandaram, “Bernstein Polynomials: Properties and Applications to Bezier Curves, B-Splines and Solution of BVPs”, Submitted to MJOS

https://en.wikipedia.org/wiki/De_Boor's_algorithm

Carl deBoor, “B-Spline Basics”, https:// apps.dtic.mil/ dtic/tr/ fulltext/u2/a172773.pdf

C. deBoor and B. Swartz, “Collocation at Gaussian Points”, SIAM. Journal of Numerical Analysis, Vol.10, No.4, pp. 582-606, 1973. Download pdf copy from https:// www.researchgate.net/ publication/ 238747998_Collocation_at_Gaussian_Points

Access “SPLIPY 1.3.1” from https://github.com/sintefmath/Splipy/

Access “Bspline.py” from https://github.com/johntfoster/bspline

M. Garagiola, “Quantum Harmonic Oscillator in 1-D”, GitHub.com, 2016. https:// github.com/ marianogaragiola/ Schrodinger_ Equation_python