Fourier Spectra of Quantum Systems Eхcited by Laser Radiation and the Exact Solution of their Dynamics Equations without Integration

Abstract

Simulation of the coherent excitation of molecules by laser radiation is carried out. It is based on simple models, i.e., quantum systems with N + 1 energy level. The exact solution of differential equations describing the process in terms of the simplest semi-classical Rabi model is obtained without integration of differential equations but using discrete mathematics with Fourier transform and discrete orthogonal polynomials. The Fourier transform realizes the transition from continuum t-space with time-dependent probability amplitudes an(t) of a quantum system to discrete Fourier space where Fourier spectra Fn(ω) are spectral images of an(t). The spectra are shown to be described by some discrete orthogonal polynomial sequence corresponding to the quantum system. The example shows how using specially constructed polynomials one can calculate the Fourier spectra and find the probability amplitudes an(t) describing the excitation of a quantum system. The established one-to-one correspondence between the polynomial characteristics and the coefficients of differential equations allows us to calculate all the characteristics of quantum systems whose excitation is described by the solution. Thus, the transition from functions an(t) to their spectral space allows one to solve some dynamic equations without integration reducing the problem to calculating the finite sum from 0 to N.