Using the position as an independent variable, and time as the dependent variable, we derive the function ${\cal P}^{(\pm)}=\pm\sqrt{2m({\cal H}-{\cal V}(q))}$, which generates the space evolution under the potential ${\cal V}(q)$ and Hamiltonian ${\cal H}$. No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian ($-{\cal H}$). While the classical dynamics do not change, the corresponding quantum operator ${\cal \hat P}^{(\pm)}$ naturally leads to a $1/2-$fractional time evolution, consistent with a recent proposed spacetime symmetric formalism of the quantum mechanics. Using Dirac's procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the $1/2$-fractional derivative, the two-coupled time-independent Dirac equations (TIDE) lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the ($\pm$) solutions of ${\cal \hat P}^{(\pm)}$ and the kinetic energy ${\cal K}_0$ (separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are $\pm\hbar\omega/2$, and the corresponding pair of states are coupled for ${\cal K}_0\ne 0$. No time evolution is present for ${\cal K}_0=0$, and the ground state with energy $\hbar\omega/2$ is stable. For ${\cal K}_0>0$, the ground state becomes coupled to the state with energy $-\hbar\omega/2$, and \textit{this coupling} allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of ${\cal K}_0=k\hbar\omega$ ($k=1,2,\ldots$). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case ${\cal K}_0=0$ leads to plane-waves-like solutions at the threshold. Above the threshold (${\cal K}_0>0$), we obtain a plane-wave-like solution, and for the bounded states (${\cal K}_0<0$), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.