Perturbation Theory for Time-Dependent Quantum Systems Involving Complex Potentials

Abstract

We explore how to apply perturbation theory to complicated time-dependent Hamiltonian systems that involve complex potentials. To do this, we introduce a generalized time-dependent oscillator to which the complex potentials are connected through a weak coupling strength. We regard the complex potentials in the Hamiltonian as the perturbed terms. Quantum characteristics of the system, such as wave functions and expectation values of the Hamiltonian, are investigated on the basis of the perturbation theory. We apply our theory to particular systems with explicit choices of time-dependent parameters. Through such applications, the time behavior of the quantum wave packets and the spectrum of expectation values of the Hamiltonian are analyzed in detail. We confirm that the imaginary parts of expectation values of the Hamiltonian are not zero but very small, whereas the real parts deviate slightly from those of the unperturbed system.

Circuit Complexity in Interacting Quenched Quantum Field Theory

“…To compute the eigenvalues of the unperturbed quenched Hamiltonian in Equation ( 6), one can again use the invariant operator method. From the arguments given in [38], the energy eigenvalues for time-dependent harmonic oscillators can be evaluated by multiplying the time-dependent factor by the expression of the energy eigenvalues of time-independent harmonic oscillators. Hence, one can show that the energy eigenvalues for each of the N decoupled oscillators become:…”

“…To tackle this problem, the Lewis-Resenfeld invariant operator method has been developed, which allows one to determine the timedependent eigenstates of such systems [35]. Additionally, the method can be extended to consider the adiabatic evolution of time-dependent parameters [36,37], providing a means to compute time-independent perturbative corrections to the eigenstates [38]. The exact form of time-dependent parameters in these eigenstates can be found by solving the Ermakov-Milne-Pinney equation, which can be efficiently computed using the Mathematica software.…”

Strain-driven tunability of the optical, electronic, and mechanical properties of lead-free inorganic CsGeCl3 perovskites

“…The Monkhorst-Pack k-mesh grid of 12 × 12 × 1 is considered for proper Brillouin zone sampling [55]. During the optical properties calculation, the first-order time-dependent perturbation theory is embedded in the model for investigating the dynamical stability properties of the considered material configurations [56]. Moreover, a Gamma-centered 8 × 8 × 1 Monkhorst-Pack k-mesh grid is ensured for proper scrutinization of optical parameters such as optical absorption co-efficient and the complex dielectric function; usually broken into real e w 1 ( )and imaginary e w 2 ( ) parts while maintaining the complex relation, e w e w e w = + j .…”

Toward Large‐Scale Hydrogen Production from Water: What Have We Learned and What Are the Main Research Hurdles to Cross for Commercialization?

“…Computation based on time‐dependent (TD) ab initio methods (TD‐DFT, [ 91 ] perturbation theory–based, and other methods [ 92,93 ] ) for solid state studies are also needed so information related to interfacial charge transfer, charge carriers lifetime, and excited energy states can complement experimental results. [ 94,95 ] In particular, the problem of hole localization, which is linked to the surface structure and associated energy position, has been discussed and some attempts made to model the energy profile of both excited electron and hole TR signal using TD‐DFT.…”