Chapter 6 Methods of Approximation - Particle Physics Group

CHAPTER 6. METHODS OF APPROXIMATION 116|A ± | 2 ω−ω fiFigure 6.4: The functions |A ± | 2 = sin2 (∆ωt/2)(∆ω/2) 2ω fi→ 2πtδ(∆ω) of height t 2 /2 and width 4π/t.Periodic perturbations. In practice we will often be interested in the response to periodicexternal forces of the formW(t) = θ(t)(W + e iωt + W − e −iωt ) with W † − = W + . (6.112)Then the time integration can be performed with the resultP (1)i→f = 1 ∣ ∣∣ A+ 〈ϕ 2 f |W + |ϕ i 〉 + A − 〈ϕ f |W + |ϕ i 〉 ∣ 2 (6.113)in terms of the integralsA ± =∫ t0dt ′ e i(ω fi±ω)t ′ = ei(ω fi±ω)t − 1i(ω fi ± ω)= e i 2 (ω fi±ω)t sin ( (ω fi ± ω)t/2 )(ω fi ± ω)/2. (6.114)Figure 6.4 shows that the functions A ± (ω) are well-localized about ω = ∓ω fi , respectively, andconverge to δ-functions( ) 2|A ± | 2 = sin ∆ωt/2→ 2πtδ(∆ω) with ∆ = ω ± ω∆ω/2fi (6.115)for late times t ≫ 1/ω fi , where the prefactor follows from the integral∫ ( ) 2 ∞ dξ sin(tξ)−∞ ξ = πt ⇒ lim1t→∞ t( ) 2 sin(tξ)ξ = πδ(ξ). (6.116)For t → ∞ the interference terms between A + W + and A − W − in (6.113) can hence be negleted,P i→f→ 2πt(δ(E f − E i − ω) ∣ ∣〈f|W − |i〉 ∣ ∣ 2 + δ(E f − E i + ω) ∣ ∣〈f|W + |i〉 ∣ ∣ 2) (6.117)For frequencies ω ≈ ±ω fi the transition probabilities become very large so that the contributionof A − W − is called resonant term (absorption of an energy quantum ω fi ) while A + W + is calledanti-resonant (emission of an energy quantum ω fi ). Since the probability becomes linear in tit is useful to introduce the transition rateFor discrete energy levels we thus obtain Fermi’s golden ruleΓ i→f = lim t→∞ ( 1 t P i→f). (6.118)Γ i→f = 2π |〈f|W ±|i〉| 2 δ(E f − E i ± ω) (6.119)which was derived by Pauli in 1928, and called golden rule by Fermi in his 1950 book NuclearPhysics. The δ-function infinity for discrete energy levels is of course an unphysical artefact of