A Spinor Approach To The SU(2) Clebsch-Gordan Coefficients

It is well know that the finite dimensional(irreducible)representation of an angular momentum vector operator can be constructed by employing a pair of spinors.The Clebsch-Gordan(CG)coefficients couple the individual eigenstates of two independent angular momentum operators J1 and J2 to form the eigenstates of the total angular momentum operator J=J1+J2.So,if we use two independent pairs of spinors to construct the individual states of J1 and J2,and take care of the coupling of these two pairs of spinors,it is possible to work out the CG coefficients.Our goal is to derive all CG coefficients using this spinor approach.It seems that this approach is simpler than the conventional approaches for calculating the CG coefficients.In this paper,we construct the set of general simultaneous eigenstates of J12,J22,J1z,J2z,?j1j2,m1m2,by introducing two independent pairs of spinors.After that,we use the antisymmetric tensor ??? to construct a spin-zero(invariant)quantity(?)(???????)j1+j2+j.Based on this invariant quantity,we are able to construct the simultaneous eigenstates of J2,Jz,J12,J21.?j1j2,jj,in terms of ?j1j2,m1m2 with very little calculations.Here j satisfies the familiar conditions |j1-j2|?j?j1+j2.In the most traditional approach,it is not easy to construct ?j1j2,jj when j?j1+j2;Usually,a length calculation is unavoidable.Based on ?j1j2,jj,it is easy to work out the general simultaneous eigenstates of J2,Jz,J12,J22 ?j1,j2,jm in terms of ?j1j2,m1m2.One then can read off all possible non-vanishing CG coefficients.We are able to derive the explicit expression of the CG coefficients of SU(2)group with this spinor approach.This expression is exactly the same as the expression derived by the previous physicists.Finally,we present some simple examples to illustrate the spinor approach to the CG coefficients.