Book contents
- Frontmatter
- Contents
- Preface
- Part I Discrete time concepts
- Part II Classical discrete time mechanics
- Part III Discrete time quantum mechanics
- 16 Discrete time quantum mechanics
- 17 The quantized discrete time oscillator
- 18 Path integrals
- 19 Quantum encoding
- Part IV Discrete time classical field theory
- Part V Discrete time quantum field theory
- Part VI Further developments
- Appendix A Coherent states
- Appendix B The time-dependent oscillator
- Appendix C Quaternions
- Appendix D Quantum registers
- References
- Index
17 - The quantized discrete time oscillator
from Part III - Discrete time quantum mechanics
Published online by Cambridge University Press: 05 May 2014
- Frontmatter
- Contents
- Preface
- Part I Discrete time concepts
- Part II Classical discrete time mechanics
- Part III Discrete time quantum mechanics
- 16 Discrete time quantum mechanics
- 17 The quantized discrete time oscillator
- 18 Path integrals
- 19 Quantum encoding
- Part IV Discrete time classical field theory
- Part V Discrete time quantum field theory
- Part VI Further developments
- Appendix A Coherent states
- Appendix B The time-dependent oscillator
- Appendix C Quaternions
- Appendix D Quantum registers
- References
- Index
Summary
Introduction
The quantized harmonic oscillator has proven to be perhaps the most important system in continuous time (CT) quantum mechanics (QM). It was the atomic oscillators assumed to line the containing walls of a black-body radiation field that Planck referred to when he postulated the quantization of energy in 1900 (Planck, 1900, 1901), thereby triggering the development of QM. Subsequently, the quantized harmonic oscillator resurfaced in many contexts, most notably in quantum optics and relativistic quantum field theory, including string theory.
There are two reasons for the importance of the quantized oscillator: (i) it can be solved completely; and (ii) it has ladder operators that generate Planck's quantized energy levels in particle QM and particle-like states in relativistic quantum field theory.
Temporal discretization does not destroy the properties of the oscillator. Rather, it enhances them. We shall show that the parabolic barrier encountered in the classical theory of the discrete time (DT) oscillator, which was discussed in the previous chapter, provides a natural upper bound for important physical quantities such as energy and momentum. In other words, the quantized DT oscillator has a natural cutoff. This may turn out to be of fundamental importance in the regularization (renormalization) of relativistic quantum field theories, where there exist immense problems related to the fact that the CT oscillator has no upper bound to its energy spectrum.
- Type
- Chapter
- Information
- Principles of Discrete Time Mechanics , pp. 192 - 208Publisher: Cambridge University PressPrint publication year: 2014