We now write down the Cartesian components of the angular momentum operator in spherical coordinates. We will make use of this result later in determining the eigenfunctions of the angular momentum squared and of one of its components.
The Cartesian coordinates
and
can be written in terms of the spherical coordinates
and
as follows:
Let us start with the
component of the angular momentum,
In Cartesian coordinates, this is
If we make use of the chain rule, then we obtain
Similarly, the
and
components may be found to be
and
Problem
(Richard L. Liboff, Introductory Quantum Mechanics, 2nd Edition, pp. 377-379)
Show, using the above results, that the operator,

when applied to a function
of the azimuthal angle
rotates the angle
to
That is, show that
Solution