Bound States In A One-Dimensional Coulomb Potential

Live Script computing bound state wave functions and energies of a particle in 1D Coulomb-like potential wells.

This Live Script explores numerically the problem of a quantum particle bound by a one-dimensional Coulomb-like potential. The one-dimensional potential V(x) = -k/|x| is singular at the origin and the spectrum of special interest. A numerical calculation, like a realistic system, has an effective regularization mechanism, and we explore too hard, soft, and Haines-Roberts cutoffs near x = 0. The bound state energy spectrum is shown to be close to that for hydrogen in the Bohr model.

This script may interest students and instructors in physics and is appropriate for a first course in modern physics. 'Try this' suggestions, coding 'Challenges,' hyperlinks, and references are included for further exploration. Additional educational Live Scripts by the author are here. Some related Live Scripts in quantum mechanics by the author are listed in the References.