Abstract
There are three sets of equations that are relativistic invariant, equations for vector and scalar potentials, Klein-Gordon equation and Dirac equation, and they could be derived from a single equation. Solutions are analyzed and with special emphases on applying various Green functions. Particular emphases is devoted to analyzing Dirac equation, being representative of relativistic quantum dynamics for spin half particles.
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Notes
- 1.
Analogous way is to retain the integration variable e real but shifting the integration path into the complex e plane. The result is the same.
- 2.
The meaning of this charge density is not very physical but it is instructive for understanding how its sudden change affects the electromagnetic field.
- 3.
The product of two eigenfunctions results from applying perturbation method in analysis of the effect on a charge by external force. In that case one of the eigenfunction, \(f_{1}(\overrightarrow{u})\), corresponds to the unperturbed solution (“incoming”) and \(f_{2}(\overrightarrow{u})\) is from the set in which perturbed solution is expanded (“outgoing”).
- 4.
For a very narrow probability amplitude in momentum space (a very wide in the coordinate space) this assumption is nearly exact.
- 5.
It should be noted that from the continuity equation the current is not uniquely determined, as discussed in Chap. 3.
- 6.
The same is valid for \(\widehat{f}_{0}(\overrightarrow{k})\) but the signs in superscript is left for convenience and does not imply positive energy component.
- 7.
\(\Theta \left( t\right) \) and \(\partial _{t}\) can interchange their order because \(\delta \left( t\right) \) that results from this gives zero contribution in f.
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© 2016 Springer-Verlag Berlin Heidelberg
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Bosanac, S.D. (2016). Relativistic Wave Equations. In: Electromagnetic Interactions. Springer Series on Atomic, Optical, and Plasma Physics, vol 94. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52878-5_2
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DOI: https://doi.org/10.1007/978-3-662-52878-5_2
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