Position as an independent variable and the emergence of the $1/2$-time fractional derivative in quantum mechanics

Abstract

Using the position as an independent variable, and time as the dependent variable, we derive the function ${\mathcal{P}}^{(\pm )}$, which generates the space evolution under the potential $\mathcal{V}(q)$ and Hamiltonian $\mathcal{H}$. 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 ($\pm$) solutions of ${\hat{\mathcal{P}}}^{(\pm )}$ and the kinetic energy ${\mathcal{K}}_0$ 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 $\pm \hslash \omega /2$, and the corresponding pair of states are coupled for ${\mathcal{K}}_0\ne 0$. No time evolution is present for ${\mathcal{K}}_0=0$, and the ground state with energy $\hslash \omega /2$ is stable. For ${\mathcal{K}}_0>0$, the ground state becomes coupled to the state with energy $-\hslash \omega /2$, and \textit{this coupling} allows to describe higher excited states. Energy quantization of the HO leads to quantization of ${\mathcal{K}}_0=k\hslash \omega$ ($k=1,2,\dots$). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case ${\mathcal{K}}_0=0$ 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.

11 pages, 6 figures