2 i c p t a 0 d given the state f t f e i c

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Unformatted text preview: l' l' l' z z { l' q c = ÅÅÅÅÅ—Å ‚Ø ÅÅÅÅÅpÅÅ ‚ 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 Å 4 pw l,l' HW12.nb ØØ 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 2 2 —w q c 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 2 2 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 l,l' p Ø Ø*Ø Ø Ø Ø Ø Ø 3 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 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 = ‚Ø ÅÅÅÅÅÅÅÅpÅÅ ‚ Ial I pM al † I pM + al † I pM al I pMM Å 2 = ‚Ø — w p Ial p p ,l † IØ M a p l Ø Ø* Ø Ø Ø 1 l I pM + ÅÅÅÅ M 2 We find that the Hamiltonian is nothing but an infinite collection of harmonic oscillators of definite momentum and helicity. (c) The only piece of Hamitonian we need is H =...
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