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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.

We consider infinitesimal times, in which case 5 = 1 and it follows that iHAt/n, [D, Η] = 0 . 2) it follows that [J , Η] = 0 . 2 Using the operator equation ift(dQ/dt) = [Q, A f ] , we see that the eigenvalues of J and J (or of L and L in the absence of spin) do not change with time; that is, they are conserved. , of the Hamiltonian) under rotations leads to conservation laws. 6 as applied to angular mometum. 4 Spin and Total Angular Momentum We have already explicitly anticipated the existence of spin as a part of total angular momentum.

The eigenstates involved are orthonormal, so that if we multiply Eq. 39) by < j , j , j , j ' \ we find f a Σ 0"> Jz I Ja , jaz , jb , JbzXJa , jaz , jb , jbz I j , j z} = a z b $ j J' P az b z J J' a bz bz · ( 2 A l ) J'Jz Similarly Σ =8jj'dj j .. 42) z z JazJbz Now Ο » Λ I ja * Jaz » Λ > Λ ζ ) Ο α » jaz f jb » 7ί>ζ I j> j z ) * > so that Eqs. 42) confirm that the matrix of the coefficients is unitary, since this property requires = β for the matrix elements. Since the Clebsch-Gordan coefficients are zero unless j = j + j , the double sums in Eqs.

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