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Physical Review Letters

Time Evolution of Simple Quantum-Mechanical Systems. II. Two-State System in Intense Fields

W. R. Salzman

Published Feb 1, 1971• 28 citations
Publisher

Abstract

The Schr"odinger equation for a two-state quantum-mechanical system with sinusoidal perturbation is numerically integrated with respect to time. From these results a general formula for the induced transition probability (as a function of time, perturbation frequency, and perturbation strength) is extracted.
DOI
10.1103/physrevlett.26.220

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The interaction of a quantum system with an oscillating field is studied in a formalism which replaces the semiclassical time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The formalism is developed as a mathematical equivalent to the semiclassical treatment, and interpreted as a classical approximation to the quantum treatment of the field. Combined with a perturbation theory for two nearly degenerate states, the formalism provides a convenient method for determining resonance transition probabilities including frequency shifts and multiple quantum transitions. The theory is illustrated by a detailed study of the simple case of a two-state system excited by a strong oscillating field.
Some phenomena cannot be solved using the exact solution of the Schrödinger equation. However, using approximated methods such as perturbation theory we are able to obtain remarkable results considering terms up to second order. The physical phenomena studied here are the absorption and stimulated emission by the classical feature. In addition, we are going to discuss the physical meaning of perturbation theory when we treat the behaviour how initial state ket evolves as time goes on. We will evaluate the transition of probability between states, the rate and the energy involved in the atomic transitions. It will be introduced fundamental concepts of quantum mechanics, such as interaction picture, and a general perturbation theory using constant and sinusoidal time-dependent perturbation term. Concerning to these phenomena, we considered the perturbed term of the Hamiltonian of the interaction between the vector potential and electron momentum.
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It is argued that the Schrödinger equation does not yield a correct description of the quantum-mechanical time evolution of states of isolated (open) systems featuring events. A precise general law for the time evolution of states replacing the Schrödinger equation is formulated within the so-called ETH-Approach to Quantum Mechanics. This law eliminates the infamous "measurement problem." Our general results are illustrated by an analysis of simple models describing a very heavy atom coupled to the quantized radiation field in a limit where the speed of light tends to infinity. The discussion of these models is the main subject of our paper.
In this paper, the general optical Bloch equation describing interaction between intensive fields and two level system is obtained without using rotating wave approximation. The first-order solution of this equation is obtained by iterative method and analytic formulas of transition probability as a function of time, perturbation frequency and strength are presented. These results are in agreement with those obtained by an iterative step-wise power series and numerical integration solution for the time dependent Schr(?)dinger equation. And these simplified and clear mathmatical formulas facilitate the application of physics.