Position as an Independent Variable and the Emergence of the 1/2-Time Fractional Derivative in Quantum Mechanics

Using the position as an independent variable, and time as the dependent variable, we derive the function P ( ± ) = ± 2 m ( H - V ( q ) ) , which generates the space evolution under the potential V ( q ) and Hamiltonian H . No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian ( - H ). While the classical dynamics do not change, the corresponding Quantum operator P ^ ( ± ) 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 P ^ ( ± ) and the kinetic energy 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 ± ħ ω / 2 , and the corresponding pair of states are coupled for K 0 ≠ 0 . No time evolution is present for K 0 = 0 , and the ground state with energy ħ ω / 2 is stable. For K 0 > 0 , the ground state becomes coupled to the state with energy - ħ ω / 2 , and this coupling allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of K 0 = k ħ ω ( k = 1 , 2 , … ). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case K 0 = 0 leads to plane-waves-like solutions at the threshold. Above the threshold ( K 0 > 0 ), we obtain a plane-wave-like solution, and for the bounded states ( K 0 < 0 ), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.