Calculating bound states resonances and scattering amplitudes for arbitrary 1D potentials with piecewise parabolas

Abstract

Even for one-dimensional (1D) potentials, the calculation of highly excited bound states and resonances (in particular broad and overlapping ones) often requires heavy numerical tools. The method presented here is based on representing an arbitrary 1D potential as a set of piecewise parabolas, where the solutions of the Schrödinger equation within each parabolic region are analytic. Outgoing, incoming, or zero-valued boundary conditions are imposed to solve for bound and resonance states, while asymmetric boundary conditions are enforced to calculate scattering amplitudes. The developed method is also applicable for complex potentials, as for example found in non-Hermitian symmetric systems (such as optical waveguides, where the spectrum varies from real to complex as a function of gain and loss). We demonstrate for several model Hamiltonians that calculations of energies, decay rates, scattering amplitudes, and wave functions are obtained in high accuracy and with minimal computational effort in comparison to other widely used methods. We believe our approach will be useful in many areas of physics and chemistry, and for example can be used to calculate various types of resonances, to solve scattering problems in cold molecular collisions, and to create analytical basis functions for multidimensional potentials.

SUSY designed broken PT-symmetric optical filters

“…As noted in [40], a 1D PTsymmetric potential can be viewed from the two sides differently, one side being a gain side while the other one a loss side and consequently a generalized energy conservation relation was therein derived. Moreover, the presence of a resonance (corresponding to a real bound state of the Schrödinger equation) can be detected as a unitary transmission peak at the corresponding eigenenergy [41]. In accordance, in Fig( 2) the PT-symmetric potential with δ = 1 exhibits the maximal transmission (the peak expected in the energy space becomes very wide in direct λ space) at the expected location λ = 2π/k s 1/|e| ≈ 89nm (for e = −0.5 in units of k 2 s ).…”

“…On the other hand, if light is sent perpendicularly to a waveguide, transversal resonances are excited and can be identified by transmission measurements i.e. total transmission is expected at the exact location of the resonances [41] (on the contrary, for illumination along the waveguide main axis, transversal resonances turn to be optical modes propagating along the waveguide). In the case of broken PT-symmetric potentials the net effect of the potential (from one side only) on the field can lead to gain effects that result in transmission larger than one.…”

On a solution method for the bound energy states of a particle in a one-dimensional symmetric finite square well potential

“…However, under these circumstance, the second argument of the two auxiliary functions (that was k¢ for the symmetric potential) will not be the same. Analogs in spirit to the procedure outlined in this work may be applied to even more complex problems such as computing the strong field Stark and Zeeman resonances (energies and widths) in atomic and molecular systems including DC and AC strong field effects, atomic, molecular and nuclear multiphoton resonances, different resonant effects in cooperative laser-electron nuclear processes and resonant effects in the interaction of strong electromagnetic pulses with solids, to mention a few [6][7][8][9][10][11][12].…”

“…Studies of one-dimensional (1D), two-dimensional, threedimensional dynamic quantum wells, quantum dots, quantum localized lattices as well as nanosystems are also of great interest [1,2]. The hydrogen atom, quantum oscillators, problems involving a quantum oscillator under the action of a periodic external potential, motion of a charged particle with spin in a constant or uniform periodic magnetic field and many others are some of the widely known problems that can be solved either analytically or numerically using various quantum theory methods [3][4][5][6][7][8][9][10][11][12].…”

“…We write all the formulas throughout the paper without simplifications. With other words, we do not use atomic units which are another familiar choice in the field [12].…”