Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Chapter 12, Exercise 12.2.2.

We can also derive the generator of rotations by considering passive transformations of the position and momentum operators, in a way similar to that used for deriving the generator of translations. In a passive transformation, the operators are modified while the state vectors remain the same. For an infinitesimal rotation about the axis in two dimensions, the unitary operator has the form

For a finite rotation by the transformations are given by

For the infinitesimal transformation, and these equations reduce to

In the passive transformation scheme, we move the transformation to the operators to get

Substituting 1 into these equations gives us the commutation relations satisfied by . For example, in the first equation we have

Equating the last two lines, we get

Similarly, for the other three equations we get

We can use these commutation relations to derive the form of by using the commutation relations for coordinates and momenta:

with all other commutators involving and being zero. Starting with 17, we see that

We can therefore deduce that

where is some unknown function. We must include since the commutators of with and are all zero, so adding on still satisfies 17. (You can think of it as similar to adding on the constant in an indefinite integral.)

Now from 18, we have

so combining this with 23 we have

The undetermined function is now a function only of and , since the dependence of on and has been determined uniquely by the commutators 17 and 18.

From 19 we have

We can see that this is satisfied already by 25, except that we now know that the function cannot depend on , since then . Thus we have narrowed down to

Finally, from 20 we have

This is satisfied by 27 if we take (well, technically, we could take to be some constant, but we might as well take the constant to be zero), giving us the final form for :