ls3_unit_2 - THE INTERACTION OF RADIATION AND MATTER...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY II. CANONICAL QUANTIZATION OF ELECTRODYNAMICS: With the foregoing preparation, we are now in a position to apply the classical analogy or canonical quantization program to achieve the second quantization of the electromagnetic field. 5 As our starting point and for reference, we, once again, set forth the vacuum or microscopic Maxwell's equations in the time domain: r ∇× r E r r , t ( 29 = - t r B r r , t ( 29 [ II-1a ] r r r r , t ( 29 = μ 0 r J r r , t ( 29 0 μ 0 t r E r r , t ( 29 [ II-1b ] r ∇⋅ r E r r , t ( 29 = ρ r r , t ( 29 ε 0 [ II-1c ] r r r r , t ( 29 = 0 [ II-1d ] The canonical formulation of classical electrodynamics ( Jeans' Theorem ) is most conveniently achieved in terms of the (magnetic) vector potential in the time domain -- viz. r r r , t ( 29 = r r A r r , t ( 29 [ II-2a ] r E r r , t ( 29 = - t r r r , t ( 29 - r ∇ϕ r r , t ( 29 [ II-2b ] so that r r r r r , t ( 29 [ ] - ∇ 2 r r r , t ( 29 + 1 c 2 2 t 2 r r r , t ( 29 + 1 c 2 t ϕ r r , t ( 29 = μ 0 r J r r , t ( 29 [ II-3a ] 5 In common usage, the process of treating the cordinates q i and p i as quantized variables is called first quantization . Second quantization is the process of quantizing fields -- say, r A r r , t ( 29 -- which have an infinite number of dequees of freedom.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY 0 r t r A r r , t ( 29 0 2 ϕ r r , t ( 29 = ρ r r , t ( 29 w [ II-3b ] In QED (Quantum Electrodynamics) it is convenient and traditional to make use of the Coulomb gauge -- i.e. r r r r , t ( 29 = 0 -- so that 2 r r r , t ( 29 - 1 c 2 2 t 2 r r r , t ( 29 =-μ 0 r J T r r , t ( 29 [ II-4a ] 2 ϕ r r , t ( 29 = -ρ r r , t ( 29 ε 0 [ II-4b ] where r J T r r , t ( 29 = r J r r , t ( 29 - r J L r r , t ( 29 = r J r r , t ( 29 0 t ϕ r r , t ( 29 is the so called transverse current density. Since r r r , t ( 29 is completely determined by the transverse current density in the Coulomb gauge, electromagnetic problems become in a sense separable -- i.e. The field problem: r r E T r r , t ( 29 = 0 r ∇× r E T r r , t ( 29 = -μ 0 t r H r r , t ( 29 r r T r r , t ( 29 = r J T r r , t ( 29 + 1 c 2 t r E T r r , t ( 29 [ II-5a ] The longitudinal r r E L r r , t ( 29 = ρ r r , t ( 29 ε 0 r J L r r , t ( 29 = -ε 0 t r E L r r , t ( 29 [ II-5b ]
Background image of page 2
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY We turn now explicitly to a treatment of the free electromagnetic field -- formally the case of r J T r r , t ( 29 0 wherein 2 r A r r , t ( 29 - 1 c 2 2 t 2 r r r , t ( 29 = 0
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/31/2011 for the course PHYSICS 108 taught by Professor Staff during the Winter '08 term at UC Davis.

Page1 / 10

ls3_unit_2 - THE INTERACTION OF RADIATION AND MATTER...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online