Quantization for brachistochrone problem
Physics Department, Akdeniz University,
P.O. Box 510, 07058 Antalya, Turkey
Revised: 29 January 2003
Published online: 30 July 2003
The brachistochrone curve corresponds to the minimization of the time functional. In this paper we discuss the dynamics of a massive particle, which moves classically on the brachistochrone curve under the potential . We derive the Lagrangian and the Hamiltonian of the particle and show that this problem corresponds to the particle in an infinite wall with a harmonic oscillator potential and the solutions of Schrödinger's equation are confluent hypergeometric functions. We also discuss the periodic potential problem for the brachistochrones and obtain the band structure of Kronig-Penney model for the particle with positive energy in a certain limit.
PACS: 03.65.-w – Quantum mechanics / 03.65.Ge – Solutions of wave equations: bound states / 71.15.Ap – Basis sets (LCAO, plane-wave, APW, etc.) and related methodology (scattering methods, ASA, linearized methods, etc.)
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2003