Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 Basic Concepts in Quantum Mechanics
- Chapter 2 One-Dimensional Potential Problems
- Chapter 3 Three-Dimensional Problems
- Chapter 4 Approximation Methods in Quantum Mechanics
- Chapter 5 Equilibrium Statistical Mechanics
- Chapter 6 Nonequilibrium statistical Mechanics
- Chapter 7 Multielectron Systems and Crystalline Symmetries
- Chapter 8 Motion of Electrons in a Periodic Potential
- Chapter 9 Phonons and Scattering Mechanisms in Solids
- Chapter 10 Generation and Recombination Processes In Semiconductors
- Chapter 11 Junctions
- Chapter 12 Semiconductor Photonic Detectors
- Chapter 13 Optoelectronic Emitters
- Chapter 14 Field-Effect Devices
- References
- Index
Chapter 4 - Approximation Methods in Quantum Mechanics
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Chapter 1 Basic Concepts in Quantum Mechanics
- Chapter 2 One-Dimensional Potential Problems
- Chapter 3 Three-Dimensional Problems
- Chapter 4 Approximation Methods in Quantum Mechanics
- Chapter 5 Equilibrium Statistical Mechanics
- Chapter 6 Nonequilibrium statistical Mechanics
- Chapter 7 Multielectron Systems and Crystalline Symmetries
- Chapter 8 Motion of Electrons in a Periodic Potential
- Chapter 9 Phonons and Scattering Mechanisms in Solids
- Chapter 10 Generation and Recombination Processes In Semiconductors
- Chapter 11 Junctions
- Chapter 12 Semiconductor Photonic Detectors
- Chapter 13 Optoelectronic Emitters
- Chapter 14 Field-Effect Devices
- References
- Index
Summary
In the previous chapters, we have solved a series of problems that all have an exact solution. However, there are only a select few problems in quantum mechanics that provide exact analytical solutions. Those problems that can be solved exactly are the free particle, the one-dimensional barrier potential, the finite and the infinite square wells, the infinite triangular well, the harmonic oscillator, and the hydrogen atom, including both its nonrelativistic and relativistic solutions. However, there are many problems different from the above-mentioned few as well as problems that cannot be approximated by these solutions. What happens then if we want to solve a different problem, such as the helium atom or the hydrogen atom in an applied electric field? To solve these problems it is necessary to use approximation techniques. In this chapter, we consider some of these techniques, the most important being the time-independent and the time-dependent perturbation theories. We begin with a discussion of time-independent disturbances in a nondegenerate system.
Time-independent Perturbation Theory: The Nondegenerate Case
We seek to determine the solution to a problem in which a small disturbance is applied to an otherwise determinable system. Perturbation theory is used most often in cases for which the solution to the undisturbed (unperturbed) system is known and the perturbation is small. For example, perturbation theory is useful in solving for the energy levels in a hydrogen atom in the presence of a small external electric field.
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- The Physics of SemiconductorsWith Applications to Optoelectronic Devices, pp. 210 - 248Publisher: Cambridge University PressPrint publication year: 1999