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Physical Review

Note on the General Theory of Scattering

M. L. Goldberger

Published Jun 1, 1951• 4 citations
Publisher

Abstract

The usual presentation of scattering theory proceeds from a discussion of the integral equations(1) ψa(±)=Φa+ 1/Ea±iε-H0 H1ψa(±).
DOI
10.1103/physrev.82.757

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We start the discussion by a formal theory of perturbation, which will be further elucidated by straightforward calculations. 2.1.1 The Golden Rule Assume that H0 is a time-independent Hamiltonian and all solutions of the eigenvalue problem H0|k〉 = Ek|k〉 are known; it is straightforward to construct the solution of the equation of motion i ∂|ψ(t)〉 ∂t = H0|ψ(t)〉 (2.1) which coincides at t = 0 with the given initial state |ψ(0)〉. Under these circumstances, the state vector |ψ(t)〉 = e−iH0t/|ψ(0) (2.2) represents the general solution of the equation of motion, Eq. 2.1. Since∑ |k〉〈k| = 1 we obtain |ψ(t)〉 = ∑ k e−iH0t/|k〉〈k|ψ(0)〉 = ∑ k e−iEkt/|k〉〈k|ψ(0)〉 (2.3) We now calculate transition amplitudes between the relevant unperturbed eigenstates due to the appearance of an external perturbation V at t = 0. We transform the equation of motion in the Schro¨dinger picture i ∂|ψ(t)〉 ∂t = (H0 + V )|ψ(t)〉 (2.4) to the interaction picture by a time-dependent unitary operator |ψˆ(t)〉 = exp (iH0t/)|ψ(t)〉 (2.5) so that in the interaction picture, the equation of motion for the state is i ∂|ψˆ(t)〉 ∂t = Vˆ (t)|ψˆ(t)〉 (2.6) where the new interaction operator is given by Vˆ (t) = exp (iH0t/)V exp (−iH0t/) (2.7) Next we introduce a time development operator Tˆ (t) such that |ψˆ(t)〉 = Tˆ (t)|ψˆ(0)〉 (2.8) The time development operator satisfies the integral equation Tˆ (t) = 1− i Vˆ (t)Tˆ (t′)dt′ (2.9) The relationship between the operator expressions in the two pictures is Tˆ (t) = exp (iH0t/)T (t) exp (−iH0t/) (2.10) The transition matrix elements of the time development operator between eigen- states of the unperturbed Hamiltonian are 〈q|Tˆ (t)|k〉 = δ(q − k)− i 〈q|Vˆ (t)Tˆ (t′)|k〉dt′ = δ(q − k)− i 〈q|Vˆ (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.11) By Eq. 2.10, and note H0|k〉 = Ek|k〉 and H0|q〉 = Eq|q〉, 〈q|Tˆ (t)|k〉 = δ(q − k)− i ei(Eq−Ek′ )t ′/〈q|V (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.12) The strategy of time-dependent perturbation theory is to assume that V is small and proceed with iterating Eq. 2.9 as a power series in terms of Vˆ Tˆ (t) = 1− i Vˆ ′dt′ − 1 2 Vˆ ′′dt′′ + · · · (2.13) and the transition amplitude from initial state |k〉 to final state |q〉 in the perturbation expansion becomes 〈q|Tˆ (t)|k〉 = δ(q − k)− i − 1 2 × ∫ t′ 0 ei(Ek′−Ek)t ′′/〈k′|V (t′′)|k〉dt′dt′′ + · · · (2.14) It is equivalent to obtain the above equation by iteratively replacing the term 〈k′|Tˆ (t′)|k〉 on the right side of Eq. 2.12 by the whole right-side expression of Eq. 2.12, i.e., ∑ k′ = ∫ t′ 0 + ∑ k′ =k ei(Eq−Ek′ )t ′′/〈q|V (t′′)|k′〉〈k′|Tˆ (t′′)|k〉dt′′ (2.15) By Eq. 2.12 〈k|Tˆ (t′)|k〉 = 1− i ei(Ek−Ek′′ )t ′′/〈k|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.16) The first term on the right side of the above equation results in the second term on the right side of Eq. 2.14. For k′ = k and by Eq. 2.12, the second term on the right side of Eq. 2.15 becomes 〈k′|Tˆ (t′)|k〉 = − i ei(Ek′−Ek′′ )t ′′/〈k′|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.17) We use Eq. 2.15 to separate the term k′′ = k from k′′ = k in the above equation, which becomes the third term on the right side of Eq. 2.14.
Up to this point in our study of quantum mechanics, we have ignored relativistic effects and taken spin as an extra observed property of particles with no specific model. In formulating an appropriate theory for the extension of quantum mechanics to include relativistic effects, it is logical to begin with the Hamiltonian based on our knowledge of relativity, where we have Hˆ = c[(pˆ− eA)2 +m2c2]1/2 + eφ , (11.1) where the first portion portion comes from W 2 = p2c2 + m2c4 using the canonical momentum pˆ = γmv + eA when there is a magnetic field present represented by the vector potential A, and then we have added the scalar electrostatic potential. This form for the Hamiltonian is very formidable, since it includes a square root with an operator inside (pˆ → (~/i)∇). By rearranging and squaring, Equation (11.1) can be put into a more symmetric form that avoids this difficulty, so that (Hˆ − eφ)2 − (pˆ− eA)2c2 = m2c4 . (11.2) One possible interpretation of this equation is that we should make the usual substitutions pˆ→ ~ i ∇ and Hˆ → −~ i ∂ ∂t , which leads to[( −~ i ∂ ∂t − eφ )2 − ( ~ i ∇− eA )2 c2 ] ψ = m2c4ψ . (11.3) This equation is called the Klein-Gordon equation, and represents one possible way to generalize the Schro¨dinger equation to include relativistic effects.