The total hamiltonian is w h p 2 w p ial p p l

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Unformatted text preview: j z " ####### — Ø l' wq q k { q c = ÅÅÅÅÅ—Å ‚Ø ÅÅÅÅÅpÅÅ ‚ Å 4 w I e l I pM ÿ e l' I pM al I pM al' † I pM + e l I pM ÿ e l' I pM al † I pM al' I pM Ø 2 2 Ø *Ø Ø Ø Ø*Ø Ø Ø Ø Ø Ø + e l I pM ÿ e l' I- pM al I pM al' I- pM + e l I pM ÿ e l' I- pM al † I pM al' † I- pMM —w c2 q2 Because wq = c q ê —, ÅÅÅÅÅ—Å ÅÅÅÅÅqÅÅ = ÅÅÅÅÅÅÅÅÅqÅÅ , the sum of two terms cancel pieces with two annihilation operators or two creation Åw 4 4 l,l' p Ø Ø Ø Ø Ø operators. The total Hamiltonian is —w H = ‚Ø ÅÅÅÅÅÅÅÅpÅÅ ‚ Å 2 Ø Ø Ø Ø* I e l I pM ÿ e l' I pM p l,l' Ø Ø*Ø Ø* Ø Ø al I pM al' † I pM + e l I pM ÿ e l' I pM al † I pM al' I pMM Ø Ø Ø Ø* Ø Ø Ø Ø Ø Using the orthonormality of the polarization vectors, e l I pM ÿ e l' I pM = dl,l' , it further simplifies to Ø Ø Ø Ø —w H = ‚ ÅÅÅÅÅÅÅÅpÅÅ ‚ Ia I pM a † I pM + a † I pM a I pMM Å Ø Ø* Ø Ø j„Ø j q k " ####### w q ‚ —2 l'...
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This note was uploaded on 01/17/2014 for the course PHYSICS 221b taught by Professor Staff during the Winter '08 term at University of California, Berkeley.

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