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56) and for eikz: 00 eik2 = £ eikrcose= ( 2 / + ly'jtf^pfcos Θ) 1=0 ,Σ(21+1)Ιύη^-1πΙ2)Ρι(οο5θ) r-*co KF ikr e = Σ (2i +1) — 2 (—\)le-ikr kr — p '< cos V' (3-57> 2ikr Here the Pt(cos 0) are the Legendre functions, and j\(kr) the spherical Bessel functions1'; I is the orbital angular momentum quantum number and the complex number a{ is the partial wave amplitude. Putting Eqs. 57) in Eq. 54) yields Mr) * - i - Σ (2/ + 1){(1 + at)eikr - ( - 1)' ^ ' ^ ( c o s 0). , Blanpied, 1971. 58) 46 3 Quantum Mechanics and Scattering Each elkr term is an outgoing spherical wave, while each e~lkr term is an incoming one; thus Φ * tout + tin- (3-59) Now since conservation of probability requires that rateout = rate i n , we must have J|iAout|2^ = J | i A i n r ^ .

We can make finite transformations by repeated application of infinitesimal ones. Thus (T + isGf = T + insG + η(μ ~ ]) (ie)2G2 + · · ■. 23) Now if we put ε = λ/η and let n -> oo while keeping λ fixed, we get an exact expression for the finite transformation: ϊ + ΐ - G ) =T + ÜG + K-—- + ··· n J 2! = eaG. 24) This transformation is continuous, and G is called the generator of the transformation. Note that since G is Hermitian, U(X) is automatically unitary: U(XyU(X) = e-ikGeiW = T. There also exist discrete transformations which cannot be put into the form of Eq.

39) 2 Relativistic Wave Equations 28 This is a condition on the transformation matrix L. 14) is equally covariant; so φ'{χ) = φ{χ)1 (defining L) implies that LyaLL-l=aavyx. 40) Comparison with Eq. , φ\χ') = φ(χ)ΙΓ1. 40) have some immediate and important consequences. , φφ is a scalar quantity. 43) so this is a four-vector. Quantities of the form φθφ, with 0 some product of •y's, are known as bilinear covariants and play a central role when one introduces interaction. 1; the transformation properties are readily established.