Abstract
To have a consistent theory of emission and absorption of electromagnetic ra- diation by quantum systems, not only should the particle system be quantized, but the electro-magnetic field as well. Hence we have to quantize Maxwell’s equations. We do this by analogy with the quantization of Classical Mechan- ics as we have done this in Section 4.1. This analogy suggests we have to put the classical electro-magnetic field theory, which is originally given in terms of Maxwell’s PDEs, into Hamiltonian form. As before we do this in two steps: by introducing the action principle, and performing a Legendre transform. Then we define the quantization map by associating with canonically conjugate classical fields the corresponding operators, and quantizing observables cor- respondingly. Since Maxwell’s equations are wave equations for vector fields with constraints, to provide the reader with a simpler guide, we first quantize the scalar wave equation (and the slightly more general Klein-Gordon equa- tion). The reader familiar with the quantization of the Klein-Gordon equation can proceed directly to the next section on quantization of the Maxwell equations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gustafson, S.J., Sigal, I.M. (2012). Quantum Electro-Magnetic Field - Photons. In: Mathematical Concepts of Quantum Mechanics. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21866-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-21866-8_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21865-1
Online ISBN: 978-3-642-21866-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)