Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction

I. Angular Momentum Operator
Rotation R(θ): in polar coordinates the point r = (r, φ)
is transformed to r' = (r,φ + θ) by a rotation through the
angle θ around the z-axis
Define the rotated wave function as:
Polar coordinates:
Its value at r is determined by the value of φ at that point,
which is transformed to r by the rotation:
The shift in φ can also be expressed by a Taylor expansion
Rotation (in exponential representation):
where the operator J z for infinitesimal rotations is the
angular-momentum operator: