Eur. Phys. J. D 25, 123-128 (2003)
https://doi.org/10.1140/epjd/e2003-00237-y

Quantization for brachistochrone problem

Physics Department, Akdeniz University, P.O. Box 510, 07058 Antalya, Turkey

Corresponding authors: ^a myucel@pascal.sci.akdeniz.edu.tr - ^b nunal@pascal.sci.akdeniz.edu.tr

Received: 18 July 2002
Revised: 29 January 2003
Published online: 30 July 2003

Abstract

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.