ls3_unit_2

# ls3_unit_2 - THE INTERACTION OF RADIATION AND MATTER...

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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 ( ) = t r B r r , t ( ) [ II-1a ] r × r B r r , t ( ) = μ 0 r J r r , t ( ) + ε 0 μ 0 t r E r r , t ( ) [ II-1b ] r r E r r , t ( ) = ρ r r , t ( ) ε 0 [ II-1c ] r r B r r , t ( ) = 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 B r r , t ( ) = r × r A r r , t ( ) [ II-2a ] r E r r , t ( ) = t r A r r , t ( ) r ϕ r r , t ( ) [ II-2b ] so that r r r A r r , t ( ) [ ] − ∇ 2 r A r r , t ( ) + 1 c 2 2 t 2 r A r r , t ( ) + 1 c 2 t ϕ r r , t ( ) = μ 0 r J r r , t ( ) [ 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 ( ) -- which have an infinite number of dequees of freedom.

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−ε 0 r t r A r r , t ( ) −ε 0 2 ϕ r r , t ( ) = ρ r r , t ( ) w [ II-3b ] In QED (Quantum Electrodynamics) it is convenient and traditional to make use of the Coulomb gauge -- i.e. r r A r r , t ( ) = 0 -- so that 2 r A r r , t ( ) 1 c 2 2 t 2 r A r r , t ( ) = μ 0 r J T r r , t ( ) [ II-4a ] 2 ϕ r r , t ( ) = −ρ r r , t ( ) ε 0 [ II-4b ] where r J T r r , t ( ) = r J r r , t ( ) r J L r r , t ( ) = r J r r , t ( ) −ε 0 t ϕ r r , t ( ) is the so called transverse current density. Since r A r r , t ( ) is completely determined by the transverse current density in the Coulomb gauge, electromagnetic problems become in a sense separable -- i.e. The transverse field problem: r r E T r r , t ( ) = 0 r × r E T r r , t ( ) = μ 0 t r H r r , t ( ) r × r H T r r , t ( ) = r J T r r , t ( ) + 1 c 2 t r E T r r , t ( ) [ II-5a ] The longitudinal field problem: r r E L r r , t ( ) = ρ
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ls3_unit_2 - THE INTERACTION OF RADIATION AND MATTER...

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