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Oxford University Press eBooks

Transitions

Peter Joshua Atkins, Julio de Paula, Ronald Friedman

Published Dec 12, 2013• 0 citations
Publisher

Abstract

Contents Time-dependent perturbation theory 140 The general procedure 140 Example 16.1: Using time-dependent perturbation theory for a slowly switched perturbation 140 Oscillating perturbations and spectroscopic transitions 142 Example 16.2: Analysing spectroscopic transitions of a harmonic oscillator 142 The energy of time-varying...
DOI
10.1093/hesc/9780199609819.003.0023

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If the perturbation is turned on for a brief period of time and then turned off, the particle ends up in the original potential. But it may have been induced to transition from one eigenstate to another during the perturbation. The calculation of the probability of that transition is the topic of time-dependent perturbation theory (TDPT). Again, formula are presented rather than derived, and two examples are provided: a linear perturbation, and an oscillating perturbation. These examples, respectively, show how selection rules arise, and how light of a given frequency excites energy transitions of the same frequency. This latter phenomenon is called resonance .
Abstract Quantum perturbation is an important tool for developing approximate solutions to quantum problems that cannot be solved exactly. This chapter provides a brief, systematic development of both time-independent and time-dependent quantum perturbation theory and the physical interpretation of its results, including both first- and second-order corrections. A quartic perturbation of the simple harmonic oscillator is used to illustrate time-independent perturbation theory. A spin-1/2 system subject to an oscillating field is used in time-dependent perturbation theory to describe the absorption and emission of energy from the quantum system.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
Consider a two-level atom represented by the spin operators σ ^ ± , σ ^ Z , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn19_1.tif"/> interacting with a one-dimensional harmonic oscillator, represented by the creation and annihilation operators â † and â . The Hamiltonian of the system is given by ( 19.1 ) H ^ = 1 2 ℏ ω 0 σ ^ Z + ℏ w 0 ( a ^ † a ^ + 1 2 )− 1 2 i ℏ g + ( σ ^ + a ^ − σ ^ − a ^ † ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/eqn19_1.tif"/>