2023/07/25 by Marcus W. Beims, Beims, Marcus W, Arlans J. S. de Lara +1
Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #FOS: Physical sciences #Mechanical and Optical Resonators #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2307.15081
openalex publication_date 2023/07/25 · openalex created_date 2023/08/01 · openalex updated_date 2026/08/03
Using the position as an independent variable, and time as the dependent variable, we derive the function \cal P(±), which generates the space evolution under the potential \cal V(q) and Hamiltonian \cal 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 \cal P(±) and the kinetic energy \cal 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 \cal K0≠ 0. No time evolution is present for \cal K0=0, and the ground state with energy ℏω/2 is stable. For \cal K0>0, the ground state becomes coupled to the state with energy -ℏω/2, and this coupling allows to describe higher excited states. Energy quantization of the HO leads to quantization of \cal K0=kℏω (k=1,2,…). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case \cal 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.