Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Part I From classical to wave mechanics
- 1 Experimental foundations of quantum theory
- 2 Classical dynamics
- 3 Wave equations
- 4 Wave mechanics
- 5 Applications of wave mechanics
- 6 Introduction to spin
- 7 Perturbation theory
- 8 Scattering theory
- Part II Weyl quantization and algebraic methods
- Part III Selected topics
- References
- Index
7 - Perturbation theory
Published online by Cambridge University Press: 14 January 2010
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Part I From classical to wave mechanics
- 1 Experimental foundations of quantum theory
- 2 Classical dynamics
- 3 Wave equations
- 4 Wave mechanics
- 5 Applications of wave mechanics
- 6 Introduction to spin
- 7 Perturbation theory
- 8 Scattering theory
- Part II Weyl quantization and algebraic methods
- Part III Selected topics
- References
- Index
Summary
The subject of perturbation theory in non-relativistic quantum mechanics is introduced. First, perturbation theory for stationary states in the absence of degeneracy is studied. The case of nearby levels, perturbations of a one-dimensional harmonic oscillator, the occurrence of degeneracy, Stark and Zeeman effects are then described in detail. After a brief outline of the variational method, the Dyson series for time-dependent perturbation theory is derived, with application to harmonic perturbations. The Fermi golden rule is also presented. The chapter ends with an assessment of four branches of the subject: regular perturbation theory, asymptotic perturbation theory, and summability methods, spectral concentration and singular perturbations.
Approximate methods for stationary states
It is frequently the case, in physical problems, that the full Hamiltonian operator H consists of an operator H0 whose eigenvalues and eigenfunctions are known exactly, and a second term V resulting from a variety of sources (e.g. the interaction with an electric field, or a magnetic field, or the relativistic corrections to the kinetic energy, or the effects of spin). The problems and aims of perturbation theory for stationary states can be therefore summarized as follows.
(i) Once the domain of (essential) self-adjointness of the ‘unperturbed’ Hamiltonian H0 is determined, find on which domain the ‘sum’ H = H0 + V represents an (essentially) self-adjoint operator (the reader should remember from appendix 4. A that in general, once some operators A and B are given, the intersection of their domains, on which their sum is defined, might be the empty set).
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- Information
- From Classical to Quantum MechanicsAn Introduction to the Formalism, Foundations and Applications, pp. 244 - 296Publisher: Cambridge University PressPrint publication year: 2004