Irreducible representations of the rotation group in terms of Euler's theorem

Abstract

In all of the previous discussions of the irreducible representations of the rotation group, the matrix elements corresponding to finite rotations are given in terms of the Euler angles of the rotation. In the present paper we describe rotations by means of a vector θ where the direction of θ is the direction of the axis of rotation and |θ| is the angle of rotation. The matrix elements corresponding to the finite rotation given by θ, are matrix elements of exp [i(θ·J)] where the matrixesJi(i=1, 2, 3) are the infinitesimal generators of the representation of the rotation group. Our representation thus takes the form given by Stone's theorem for one-dimensional groups. We express the matrix elements corresponding to the finite rotations in closed form in terms of Jacobi polynomials. Aside from the mathematical elegance of the result, our representation is useful in showing how wave functions transform in a rotated frame of reference when the axis of rotation is prescribed. Although in principle one could obtain the transformation properties of the wave function in terms of the Euler angle parametrization, the expressions are extremely cumbersome for a general axis of rotation. In the present representation, on the other hand, the expressions are simple. As part of the proof we also give the transformation from the basis in whichJ3 is diagonal to that in which θ·J is diagonal. The matrix elements for finite rotation for the cases of spin 1/2, 1 and 3/2 have been computed in the new representation and are given in the Appendix.