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 \({\mathcal{P}}^{(\pm )}=\pm \sqrt{2m({\mathcal{H}}-{\mathcal{V}}(q))}\), which generates the space evolution under the potential \({\mathcal{V}}(q)\) and Hamiltonian \({\mathcal{H}}\). No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian (\(-{\mathcal{H}}\)). While the classical dynamics do not change, the corresponding Quantum operator \({{{\hat{\mathcal P}}}}^{(\pm )}\) naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time 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 lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of \({{{\hat{\mathcal P}}}}^{(\pm )}\) and the kinetic energy \({\mathcal{K}}_{0}\) (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 \({\mathcal{K}}_{0}\ne 0\). No time evolution is present for \({\mathcal{K}}_{0}=0\), and the ground state with energy \({\hbar {\omega}} /2\) is stable. For \({\mathcal{K}}_{0}>0\), the ground state becomes coupled to the state with energy \(-{\hbar {\omega}} /2\), and this coupling allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of \({\mathcal{K}}_{0}=k{\hbar {\omega}}\) (\(k=1,2,\ldots\)). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case \({\mathcal{K}}_{0}=0\) leads to plane-waves-like solutions at the threshold. Above the threshold (\({\mathcal{K}}_{0}>0\)), we obtain a plane-wave-like solution, and for the bounded states (\({\mathcal{K}}_{0}<0\)), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.