Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.2.3.
The angluar momentum operator is the generator of rotations in the plane. We did the derivation for infinitesimal rotations, but we can generalize this to finite rotations in a similar manner to that used for translations. The unitary transformation for an infinitesimal rotation is
For rotation through a finite angle , we divide up the angle into small angles, so . Rotation through the full angle is then given by
The limit follows because the only non-trivial operator involved is , so no commutation problems arise.
In rectangular coordinates, has the relatively non-obvious form
so it’s not immediately clear that 2 does in fact lead to the desired rotation. Trying to calculate the exponential with expressed this way is not easy, given that the two terms and don’t commute.
It turns out that has a much simpler form in polar coordinates, and there are two ways of converting it to polar form. First, we recall the transformation equations.
From the chain rule, we can convert the derivatives:
Using similar methods, we get for the other derivative
Plugging these into 4 we have
Another method of converting to polar coordinates is to consider the effect of for an infinitesimal rotation on a state vector expressed in polar coordinates . Shankar states that
If you don’t believe this, it can be shown using a method similar to that for the one-dimensional translation. In this case, we’re dealing with position eigenkets in polar coordinates, so we have
Applying this, we get
where in the last line, we used the substitution . (The substitution is used just to give the radial variable a different name in the integrand.) We can use the same limits of integration for and , since we just need to ensure that the integral covers the total range of angles. It then follows that
Combining this with 1 we have
Expanding the RHS to order we have
from which 17 follows again.
Once we have in this form, the exponential form of a finite rotation is easier to interpret, for we have, from 2
Applying this to a state function , we see that we get the Taylor series for , so the exponential does indeed represent a rotation through a finite angle.