Book contents
- Frontmatter
- Contents
- Preface
- Part I Discrete time concepts
- Part II Classical discrete time mechanics
- Part III Discrete time quantum mechanics
- Part IV Discrete time classical field theory
- 20 Discrete time classical field equations
- 21 The discrete time Schrödinger equation
- 22 The discrete time Klein–Gordon equation
- 23 The discrete time Dirac equation
- 24 Discrete time Maxwell equations
- 25 The discrete time Skyrme model
- 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
22 - The discrete time Klein–Gordon equation
from Part IV - Discrete time classical field theory
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
- Part IV Discrete time classical field theory
- 20 Discrete time classical field equations
- 21 The discrete time Schrödinger equation
- 22 The discrete time Klein–Gordon equation
- 23 The discrete time Dirac equation
- 24 Discrete time Maxwell equations
- 25 The discrete time Skyrme model
- 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
When, in 1926, Schrödinger came to publish his non-relativistic wave equation (Schrödinger, 1926), namely the equation we discretized in the previous chapter, he had previously considered a special-relativistic wave equation, but discarded it on account of some of its properties, which he believed were unphysical. In particular, that relativistic equation has a conserved current density that cannot be interpreted as a classical probability current density because it can take on negative values, something that a true probability density would not do. If Schrödinger had been aware of antiparticles, which were discovered several years later, it is conceivable that he would have persisted with that relativistic wave equation. The equation he discarded is sometimes referred to as the Schrödinger–Fock–Klein–Gordon equation, but more commonly is known as just the Klein–Gordon (K–G) equation.
The significance of Schrödinger's rejection of the K–G equation lies in his decision to forgo Lorentz covariance in favour of an intuitive, albeit non-relativistic, interpretation of the wave equation that now bears his name. Schrödinger's nonrelativistic wave equation was extraordinarily successful when applied to atomic physics, which greatly contributed to the rise of quantum physics.
Within two years, however, the situation was dramatically restored in favour of special relativity. In 1928 Dirac published his famous Lorentz covariant wave equation for the electron (Dirac, 1928).
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- Principles of Discrete Time Mechanics , pp. 246 - 252Publisher: Cambridge University PressPrint publication year: 2014