Quantization for brachistochrone problem

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 V=-mgy. 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.