An Introduction to Elementary Particles, Edition: 2nd by W.S.C. WILLIAMS (Eds.)


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82). The sum is performed over all the different pure spin states χ that occur in the incident beam. 86) 60 II. 88) A as for the pure state. The important case that we shall treat is that of s = i . This occurs in the analysis of the scattering of spin-0 particles by a target of spin-4 particles (protons), which can be polarized. Density matrices are required to describe the initial polarization of protons in the target and the polari­ zation of protons recoiling at a particular angle. p p , and Μ are 2 X 2 matrices, and it follows that we can write them as linear sums of the 2 X 2 unit matrix 1 and the Pauli matrices σ\ for example, it Λ = U + M, · Σ 2 (η = 1,2).

84) χ \ Λι which follows from the definition given by Eq. 82) and where χ is the matrix representation of | %i>. The matrix representing the spin state of the scattered beam (subscript 2) is given by operating on χ with a matrix M, and we normalize so that the differential scattering cross-section is given by λ Χ The density matrix p for the scattered wave spin states is 2 Pi = Xili = Μχα^Μ^ = Mp M^. 85) r which follows from Eq. 82). The sum is performed over all the different pure spin states χ that occur in the incident beam.

8): z z T-+ V = DTD~ . 12 Tensor Operators 51 D T(k, k ) D~ = Σ &'\'ζ' l 2 k ^ ( · °) 2 7 V by analogy with Eq. 63). If we make the rotation through infinitesimal angle a about the axis Eq. 70) becomes (1 + iaJjfi) T(k, k ) (1 - iaJJH) z = Σ <*·**' Ii + 1 i a ^ l> k J k z > τ &> **'); V hence, Γ(£, k ) + ™L {J, T(k, k ) - T(k, k ) J,} η z z = Σ z { z T(k,k ') z z -iZ k/ \Jx\k, k y T(k, k/) . Jx T(k, k ) - T(k, k ) J = Σ z z k z x Substituting for the matrix elements on the right from Eqs.

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