hw212a - Physics 512 Homework Set #12 Solutions Winter 2003...

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Physics 512 Winter 2003 Homework Set #12 – Solutions 1. The diferential cross section For the ejection oF an electron with momentum ~ k f by an incident photon oF momentum ~ k ( ω = c | ~ k | ) and polarization ˆ ± (the photoelectric efect) may be written as d = α | ~ k f | 2 πm ¯ L 3 |h f | e i ~ k · ~r ~p · ˆ ± | i i| 2 where the matrix element reFers to the initial (bound) and ±nal (Free plane wave) electron states. Derive this expression using the quantum theory oF radiation (instead oF the classical treatment shown in class). We begin by computing the transition rate for this process to occur. For the rate, Fermi’s golden rule for harmonic perturbations gives w = 2 π ¯ h |h f ;0 | V | i ; { n ( α ) ~ k =1 }i| 2 ρ f ( E f ) Here we have used a notation where i and f denote the initial and ±nal electron state, while the remaining pieces are a shorthand label for the photons in the Fock space. For absorption of a photon, we have V = e mc ~ A (+) · ~p = e mc 2 π ¯ hc 2 1 L 3 / 2 X ~ k,α 1 ω a α ( ~ k ) ~p · ˆ ± ( α ) e i ~ k · Evaluating the matrix element in the photon Fock space, we see that the lowering operator annihilates the initial photon, leaving the Fock vacuum (which is normalized so that h 0 | 0 i ). This selects out only one particular term in the sum (corresponding to the single initial state photon). As a result h f | V | i ; { n ( α ) ~ k }i = e mc r 2 π ¯ hc 2 ωL 3 h f | e i ~ k · ~p · ˆ ± | i i Squaring this and using Fermi’s golden rule yields w = (2 π ) 2 e 2 m 2 3 |h f | e i ~ k · ~p · ˆ ± | i i| 2 = (2 π ) 2 e 2 m 2 3 m | ~ k f | ¯ h 2 ± L 2 π ² 3 |h f | e i ~ k · ~p · ˆ ± | i i| 2 ρ f ( E f ) = αc | ~ k f | 2 ¯ |h f | e i ~ k · ~p · ˆ ± | i i| 2 d where, for the ±nal state, we have used the free electron density of states ρ ( E f )= m | ~ k f | ¯ h 2 ± L 2 π ² 3 d 1
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Finally, this may be converted to a diferential cross section by dividing by the incident (single photon) flux c/L 3 . This gives the desired result d = α | ~ k f | 2 πm ¯ L 3 |h f | e i ~ k · ~r ~p · ˆ ± | i i| 2 Note that this matrix element is only ±or the electron states. Moreover, the ²nal ejected electron wave±unction must be normalized as ψ 1 /L 3 / 2 . Taking this into account,
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hw212a - Physics 512 Homework Set #12 Solutions Winter 2003...

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