# Viiia 8 which is of course the generalized form of

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[ VIIIA-8 ] which is, of course, the generalized form of the Fermi golden rule . The s econd order approximation ( for 1 τ ): Using the third term in the iteration set forth in Equation [ I-27c ] of IRM:ST, we can write P i f 2 ( 29 t , t 0 ( 29 = lim ε→ 0 1 h ψ f H int ψ i exp ε t - i ϖ i t ( 29 ϖ i f + i ε ( 29 + ψ f T 0 t ( 29 - i h 2 d t exp ε t ( 29 H int I t ( 29 -∞ t × d ′′ t exp ε ′′ t ( 29 H 1int I ′′ t ( 29 -∞ t T 0 - 1 0 ( 29 ψ i 2 [ VIIIA-9a ] or in the Schrödinger picture

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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P i f 2 ( 29 t , t 0 ( 29 = lim ε→ 0 1 h ψ f H int ψ i exp ε t - i ϖ i t ( 29 ϖ i f + i ε ( 29 + ψ f T 0 t ( 29 - i h 2 d t T 0 - 1 t ( 29 exp ε t ( 29 H int T 0 t ( 29 -∞ t × d ′′ t T 0 - 1 ′′ t ( 29 exp ε ′′ t ( 29 H int T 0 ′′ t ( 29 -∞ t T 0 - 1 0 ( 29 ψ i 2 [ VIIIA-9b ] Using the closure theorem we see that P i f 2 ( 29 t , t 0 ( 29 = lim ε→ 0 1 h ψ f H int ψ i exp ε t - i ϖ i t ( 29 ϖ i f + i ε ( 29 + - i h 2 exp - i ϖ f t ( 29 ψ f H int ψ m ψ m H int ψ i m × d t d ′′ t exp i ϖ f t t - i ϖ m t - ′′ t ( 29 ′′ t - i ϖ i ′′ t [ ] -∞ t -∞ t 2 [ VIIIA-10 ] Integrating and writing the matrix elements in more concise notation, we see that P i f 2 ( 29 t , t 0 ( 29 = lim ε→ 0 exp ε t - i ϖ i t ( 29 ϖ i f + i ε ( 29 1 h f H int i + 1 h 2 f H int m m H int i ϖ i m + i ε 2 m 2 [ VIIIA-11 ] Therefore, following the arguments presented above, we see that the second order approximation for 1 τ is given by
T HE I
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