Classical and quantum particles in the brachistochrone upper half-space

The brachistochrone upper half-space geometry is introduced and investigated. The classical geodesics on this curved space are found to be planar cycloids. We show that the center and radius of the cycloid depend on the initial conditions. Furthermore, we solve the Schrodinger equation for a free particle in this curved space. Due to the geometrical structure of the curved space, the free particle-free means no external nongeometric potential is applied-experiences a purely geometric potential. We utilize the point canonical transformation to bring the Schrodinger equation into its standard form where the geometric potential takes the form of a harmonic oscillator.