ls3_unit_4

ls3_unit_4 - THE INTERACTION OF RADIATION AND MATTER:...

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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 34 R. Victor Jones, April 17, 2000 34 IV. INTERACTION HAMILTONIAN -- COUPLING FIELDS AND CHARGES: 23 To build a complete quantum picture of the interaction of matter and radiation our first and most critical task is to construct a reliable Lagrangian-Hamiltonian formulation of the problem. In this treatment, we will confine ourselves to a nonrelativistic view which, fortunately, is adequate for most circumstances. We start with a representation of the Lorentz force for a single charged particle -- viz. r f = q r E + r v × r B { } [ IV-1a ] or in terms of the electromagnetic potentials r = q r ϕ− r A t + r × r × r ( ) Χ Ψ Ω Ϋ ά έ [ IV-1b ] Let us now write the total time derivative of a component of the vector potential d A α dt = A α r β d r β + A α t = r v r ( ) A α + A α t [ IV-2 ] so that r v × r × r A ( ) = r r v r A ( ) r v r ( ) r A = r r v r A ( ) d r A + r A t [ IV-3 ] Therefore, we may write the Lorentz force as f α = q r α v β A β [ ] d A α Χ Ψ Ω Ϋ ά έ [ IV-4 ] 23 Much of what follows draws heavily on material in Chapter 5 of Rodney Loudon's The Quantum Theory of Light (second edition), Oxford (1983) . Marlan O. Scully and M. Suhail Zubairy in Section 5.1 of Quantum Optics, Cambridge (1997) have a slightly different, but quite insightful treatment of the subject of the atom- field interaction Hamiltonian.
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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 35 R. Victor Jones, April 17, 2000 35 We may also write A α = v α ϕ − v β A β [ ] so that Equation [ IV-4] becomes f α = r α q ϕ− v β A β [ ] { } + d dt v α q v β A β [ ] { } = m d v α = m d v α 1 2 v β v β Ρ Σ Τ Φ = d v α 1 2 m v 2 Ρ Σ Τ Φ [ IV-5 ] which may, in turn, be written d v α 1 2 m v 2 q v β A β ( ) Ρ Σ Τ Φ Χ Ψ Ω Ϋ ά έ r α 1 2 m v 2 q v β A β ( ) Ρ Σ Τ Φ = 0 . [ IV-6 ] This last equation may now be compared to the Lagrangian equation of motion -- i.e. d v α L r r , r v ( ) Χ Ψ Ω Ϋ ά έ r α L r r , r ( ) = 0 [ IV-7a ] where, in general, L r r , r ( ) = Kinetic Energy [ ] Potential Energy [ ] = T r r , r ( ) U r r , r ( ) . [ IV-7b ] Therefore, we identify L r r , r ( ) = 1 2 m v 2 q v r A ( ) Ρ Σ Τ Φ [ IV-8 ] as the Lagrangian for a single charged particle. We may write d L r r , r ( ) = r
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ls3_unit_4 - THE INTERACTION OF RADIATION AND MATTER:...

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