Book contents
- Frontmatter
- Contents
- Preface
- 1 The Quantum World and Its Description
- 2 Matter and Field
- 3 Quantum Mechanics in the Heisenberg (Matrix) Picture
- 4 Quantum Mechanics in the Schrödinger (Wave) Picture. Dirac‘s Notation
- 5 One-Dimensional Potential Steps, Barriers, and Wells
- 6 The WKB Approximation. Electronic Properties of Solids
- 7 The Free Particle. The Time-Dependent Schrödinger Equation
- 8 Dynamics of Quantum Systems
- 9 The One-Dimensional Harmonic Oscillator
- 10 The Physics Underlying Quantum Phenomena
- 11 Angular Momentum Theory
- 12 Central Potentials. The Hydrogen Atom
- 13 Stationary Perturbation Theory and the Variational Method
- 14 The Electron Spin. Entangled States
- 15 Identical-Particle Systems: Quantum Statistics. The Density Matrix
- 16 Atoms and Molecules
- 17 Time-Dependent Perturbations. Field Quantization and Second Quantization
- 18 Elastic Scattering Theory
- 19 Relativistic Equations. An Introduction
- Appendix A Mathematical Tools
- References
- Index
13 - Stationary Perturbation Theory and the Variational Method
Published online by Cambridge University Press: 27 December 2025
Book contents
- Frontmatter
- Contents
- Preface
- 1 The Quantum World and Its Description
- 2 Matter and Field
- 3 Quantum Mechanics in the Heisenberg (Matrix) Picture
- 4 Quantum Mechanics in the Schrödinger (Wave) Picture. Dirac‘s Notation
- 5 One-Dimensional Potential Steps, Barriers, and Wells
- 6 The WKB Approximation. Electronic Properties of Solids
- 7 The Free Particle. The Time-Dependent Schrödinger Equation
- 8 Dynamics of Quantum Systems
- 9 The One-Dimensional Harmonic Oscillator
- 10 The Physics Underlying Quantum Phenomena
- 11 Angular Momentum Theory
- 12 Central Potentials. The Hydrogen Atom
- 13 Stationary Perturbation Theory and the Variational Method
- 14 The Electron Spin. Entangled States
- 15 Identical-Particle Systems: Quantum Statistics. The Density Matrix
- 16 Atoms and Molecules
- 17 Time-Dependent Perturbations. Field Quantization and Second Quantization
- 18 Elastic Scattering Theory
- 19 Relativistic Equations. An Introduction
- Appendix A Mathematical Tools
- References
- Index
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Summary
Chapter 13 is devoted to two important approximation methods used to solve time-independent problems, namely stationary perturbation theory and the variational method. Perturbation theory is developed for both non-degenerate states and degenerate states, with illustrative examples. The Stark effect is considered separately. The dielectric constant of a transparent medium as a function of frequency is shown to result from the modification of the wave function due to the perturbation of the radiation field. Two complementary topics are presented: the canonical transformation method and the Feynman-Hellman method. Finally, the possibilities of the variational method are illustrated by using it first to determine an upper bound for the energy of the stationary states of a quantum system, and then to study more complicated cases, including the partially shielded Coulomb potential produced by a neutral atom.
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- Quantum MechanicsA Physical Approach, pp. 331 - 362Publisher: Cambridge University PressPrint publication year: 2025
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