Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.4.3.
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The operators for an infinitesimal rotation in 3-d are
If we have a finite (larger than infinitesimal) rotation about one of the coordinate axes, we can create the operator by dividing up the finite rotation angle into small increments and take the limit as , just as we did with finite translations. For example, for a finite rotation about the axis, we have
What if we have a finite rotation about some arbitrarily directed axis? Suppose we have a vector as shown in the figure:
The vector makes an angle with the axis, and we wish to rotate about the axis by an angle . Note that this argument is completely general, since if the axis of rotation is not the axis, we can rotate the entire coordinate system so that the axis of rotation is the axis. The generality enters through the fact that we’re keeping the angle arbitrary.
The rotation by shifts the tip of along the circle shown by a distance in a counterclockwise direction (looking down the axis). This shift is in a direction that is perpendicular to both and , so the little vector representing the shift in is
Thus under the rotation , a vector transforms as
Just as with translations, if we rotate the coordinate system by an amount , this is equivalent to rotating the wave function by the same angle, but in the opposite direction, so we require
A first order Taylor expansion of the quantity on the RHS gives
The operator generating this rotation will have the form (in analogy with the forms for the coordinate axes above):
where is an angular momentum operator to be determined.
Writing out the RHS of 8, we have
Comparing this with 9, we see that
where is the unit vector along the axis of rotation. Since all rotations about the same axis commute, we can use the same procedure as above to generate a finite rotation about an arbitrary axis and get