Modeling quantum harmonic oscillator in complex domain

Flux continuity and probability conservation in complexified Bohmian mechanics

“…The resultant "complex-valued Bohmian trajectories" offer certain advantages; for instance, they are known not to be fixed-points, in general, for nondegenerate stationary states, so that it is possible to achieve nontrivial trajectory dynamics in this context. Although complexvalued Bohmian mechanics may still be in its infancy, interest has grown tremendously in the last few years [1,9,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. The field appears to have started in the 1980's with a paper by Leacock and Padgett [15] and a less well known (and very brief) article by Tourenne [16].…”

“…The field appears to have started in the 1980's with a paper by Leacock and Padgett [15] and a less well known (and very brief) article by Tourenne [16]. More recent authors have explored the complex Bohmian approach both for time-independent (stationary) and time-dependent (wavepacket propagation) problems, in both the analytical and synthetic contexts [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].…”

Strong chaos in one-dimensional quantum system

“…According to quantum Hamilton mechanics [4][5][6][7][8], the quantum Hamiltonian can be written as (1) where is what we call quantum potential :…”

“…However, if we take into account its multi-path behavior, we will see below that chaos does occur in 1D harmonic oscillator. The eigenfunction of the Schrödinger equation with the applied potential V(x)=kx 2 /2 is given by [8] ( 4) where , , and is the nth-order Hermite polynomial. The equation of motion in the eigenstate Ψ n is found by substituting Eq.…”

“…Indeed, quantum trajectories solved from Eq. (5) show that quantum motions in pure eigenstates are periodic with quantized periods of oscillation [8]. Therefore, the superposition of at least two eigenstates is a necessary condition to exhibit chaos in a 1D quantum system.…”