Quantum Theory - Particle Physics Group

CHAPTER 6. METHODS OF APPROXIMATION 122Our next step is to transform the Schrödinger equation (6.103) of the interaction pictureinto an integral equation by integrating it over the interval from t 0 = 0 to t,|ψ i (t)〉 I = |ϕ i 〉 + 1i∫ t0dt ′ W I (t ′ )|ψ i (t ′ )〉 I , (6.104)where we used the boundary condition (6.97). For small W I (t) we can solve this equation byiteration, i.e. we insert |ψ i 〉 I = |ϕ i 〉+O(W) on the r.h.s. and continue by inserting the resultinghigher order corrections of |ψ i 〉 I . We thus obtain the Neumann series 7|ψ i (t)〉 = |ϕ i 〉}{{}initial stateThe transition amplitude now becomes+A i→f = 〈ϕ f |ψ i (t)〉 = δ if − i ∫ t+ 1 dt ′ W I (t ′ )|ϕ i 〉 +i 0} {{ }first order correction∫1 t ∫ t ′dt ′ dt ′′ W(i) 2 I (t ′ )W I (t ′′ )|ϕ i 〉 +... (6.107)0 0} {{ }second order correction∫ t0dt ′ 〈ϕ f | W I (t ′ ) |ϕ i 〉 + O(W 2 I ). (6.108)In the remainder of this section we focus on the leading contribution to the transition from aninitial state |ϕ i 〉 to a final state |ϕ f 〉 with f ≠ i,A (1)i→f= − i = − i ∫ t0∫ t0dt ′ 〈ϕ f | e i H 0t ′ W(t ′ )e − i H 0t ′ |ϕ i 〉 (6.109)dt ′ e i (E f −E i )t ′ 〈ϕ f |W(t ′ ) |ϕ i 〉 (6.110)where we used H 0 |ϕ i 〉 = E i |ϕ i 〉 and 〈ϕ f |H 0 = 〈ϕ f |E f to evaluate the time evolution operators.For first order transitions we thus obtain the probabilityi→f = 1 ∣∫ ∣∣∣ tdt ′ e iω fit ′ 〈ϕ 2 f |W(t ′ )|ϕ i 〉∣P (1)with the Bohr angular frequency7 Introducing the time ordering operator T byTA(t 1 )B(t 2 ) = θ(t 1 − t 2 )A(t 1 )B(t 2 ) + θ(t 2 − t 1 )B(t 2 )A(t 1 ) =02(6.111)ω fi = E f − E i. (6.112){A(t 1 )B(t 2 ) if t 1 > t 2B(t 2 )A(t 1 ) if t 1 < t 2(6.105)the Neumann series can be subsumed in terms of a formal expression for the time evolution operator|ψ i (t)〉 I = U I (t)|ϕ i 〉, U I (t) = Te − i R t0 dt′ W I(t ′ )(6.106)as is easily checked by expansion of the exponential.