ls3_unit_2 - THE INTERACTION OF RADIATION AND MATTER...

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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY II. C ANONICAL Q UANTIZATION OF E LECTRODYNAMICS : 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 B 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 B 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 B r r , t ( 29 = r ∇× r A r r , t ( 29 [ II-2a ] r E r r , t ( 29 = - t r A r r , t ( 29 - r ∇ϕ r r , t ( 29 [ II-2b ] so that r r r A r r , t ( 29 [ ] - ∇ 2 r A r r , t ( 29 + 1 c 2 2 t 2 r A 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.
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY 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 A r r , t ( 29 = 0 -- so that 2 r A r r , t ( 29 - 1 c 2 2 t 2 r A 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 A 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 transverse 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 H 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 field problem: 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 ]
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY We turn now explicitly to a treatment of the free electromagnetic field -- formally
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