Abstract
Using the position as an independent variable, and time as the dependent variable, we derive the function P^(±), which generates the space evolution under the potential V(q) and Hamiltonian H. Canonically conjugated variables are the time and minus the Hamiltonian. While the classical dynamics do not change, the corresponding quantum operator naturally leads to a 1/2-fractional time evolution, consistent with a recently proposed spacetime symmetric formalism of quantum mechanics. Using Dirac's procedure, separation of variables is possible, and while the coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the coupled time-independent Dirac equations (TIDE) lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of P^(±) and the kinetic energy K₀ is the coupling strength. We obtain a pair of coupled states for systems with finite forces. The potential shifts for the harmonic oscillator (HO) are ±ℏω/2, and the corresponding pair of states are coupled for K₀≠ 0. No time evolution is present for K₀=0, and the ground state with energy ℏω/2 is stable. For K₀>0, the ground state becomes coupled to the state with energy -ℏω/2, and \textit{this coupling} allows to describe higher excited states. Energy quantization of the HO leads to quantization of K₀=kℏω (k=1,2,…). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case K₀=0 leads to plane-waves-like solutions at the threshold. Above the threshold, we obtain a plane-wave-like solution, and for the bounded states the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.