"Generator of rotations"...rotations of WHAT?

[AxiomOfChoice]

The angular momentum operator is defined as the "generator of rotations." Fine. But rotations of WHAT? What's being rotated? The wave function (doesn't make sense; isn't the wave function a scalar), perhaps? An arbitrary vector in the coordinate system under investigation?

[CompuChip]

It is the latter. Equivalently, you are rotating the basis of the three-dimensional space (a passive rotation). Or, yet differently formulated, you are rotating the space itself while keeping the basis fixed (active rotation), which also causes the coordinates of vectors to change.

[DrFaustus]

AxiomOfChoice -> Rotations of yourself... or of your coordinate system, if you prefer. To which correspond transformations of observables/states. And the wave function is nothing but an explicit representation of a vector in some specific base (the "coordinate representation").

[AxiomOfChoice]

Thanks guys! That's very helpful.

Right now, I'm studying a triatomic molecule confined to move in two dimensions. Suppose the central nucleus is fixed at the origin, and let \theta be the angle between the radius vector of the first (free to move) nucleus and the second (free to move) nucleus. How do I show that L = -i\hbar \dfrac{\partial}{\partial \theta} in this case?

[xepma]

Use the definition of L:
L = r x p (outer product)
By plugging in the appropriate components you will get your answer.

Also: you can view the generator of a symmetry transformations in two ways. The active and the passive way. In the passive way, you change the way you "view" the system - this in some sense just a coordinate transformation, like boosting yourself to a moving reference frame. The physical state of the system stays the same.

In the active point of view you physically change the system. That is, the symmetry transformation actively transforms the physical state, meaning we map one state in your Hilbert space, to another. If both states carry the same "physical information" (for a lack of a better description) then you are dealing with a symmetry of the system. An example is the Ising model: flipping all the spins in the opposite direction actively changes the state, but leaves you with the same energy etc.

To complete the discussion: in a gauge "symmetry" you also activily transform the state. But the physical state itself doesnt change - you simply have redundant way of describing the same physical state. A gauge redundancy would be a better way of putting it, because it is not a "true" symmetry of the system (i.e. we dont map between different physical states when performing a gauge transformation).