Quantum Dynamics of Brachistochrone Problem
In this study we discuss the quantum dynamics of a particle, which moves classically on the brachistochrone curve corresponding to the minimization of the time functional, in a linear gravity potential. We derive the Lagrangian and the Hamiltonian of the particle, which moves also on the brachistochrone curve by the minimization of the action functional. The solutions of the Schrödinger’s equation for this Hamiltonian give the energy spectrum, and the confluent hypergeometric functions as the wave functions. The problem combines the infinitewell and harmonic oscillator potentials. We also discuss the solutions of the Schrödinger’s equation for the particle in the periodic extension of the original brachistochrone problem. We show that the band structure arised from Floquet theory and the problem is equivalent to the periodic δ-potential problem for the particle with positive energy in the limit of infinite potential.
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