1.2. SPIN ANGULAR MOMENTUM
1.2.2
9
Infinitesimal Rotations of Vector Fields
Let us consider a rotation about the z axis by a small angle δφ. Under such a rotation, the components of a vector r = (x, y, z) are transformed to x y z
= x + δφ y, = −δφ x + y, = z,
neglecting terms of second and higher order in δφ. The difference δψ(x, y, z) = ψ(x , y , z ) − ψ(x, y, z) between the values of a scalar function ψ evaluated in the rotated and unrotated coordinate systems is (to lowest order in δφ), ∂ ∂ −y δψ(x, y, z) = −δφ x ψ(x, y, z) = −iδφ Lz ψ(x, y, z). ∂y ∂x The operator Lz , in the sense of this equation, generates an infinitesimal rotation about the z axis. Similarly, Lx and Ly generate infinitesimal rotations about the x and y axes. Generally, an infinitesimal rotation about an axis in the direction n is generated by L · n. Now, let us consider how a vector function A(x, y, z) = [Ax (x, y, z), Ay (x, y, z), Az (x, y, z)] transforms under an infinitesimal rotation. The vector A is attached to a point in the coordinate system; it rotates with the coordinate axes on transforming from a coordinate system (x, y, z) to a coordinate system (x , y , z ). An infinitesimal rotation δφ about the z axis induces the following changes in the components of A:
δAy
= Ax (x , y , z ) − δφAy (x , y , z ) − Ax (x, y, z) = −iδφ [Lz Ax (x, y, z) − iAy (x, y, z)] , = Ay (x , y , z ) + δφAx (x , y , z ) − Ay (x, y, z)
δAz
= −iδφ [Lz Ay (x, y, z) + iAy (x, y, z)] , = Az (x , y , z ) − Az (x, y, z)
δAx
= −iδφ Lz Az (x, y, z) . Let us introduce the 3 × 3 matrix sz defined by 0 −i 0 sz = i 0 0 . 0 0 0 With the aid of this matrix, one can rewrite the equations for δA in the form δA(x, y, z) = −iδφ Jz A(x, y, z), where Jz = Lz + sz . If we define angular momentum to be the generator of infinitesimal rotations, then the z component