# Angular Momentum as a Generator of Rotations in 3D

### From FSUPhysicsWiki

Let us consider an infinitesimal rotation described by a vector directed along the axis about which the rotation takes place and whose magnitude is the angle of the rotation. We then have

where is the change in the position vector of the particle due to such a rotation.

Let us now consider a function of position, Substituting in the rotated coordinate and expanding to first order in we obtain

**Figure 1: Illustration of the rotation considered here.**

The expression,

may therefore be interpreted as an operator that performs an infinitesimally small rotation of position coordinates. Noticing that

we may write this infinitesimal rotation operator as

Note that this expression only applies to infinitesimal rotations. We may construct a rotation operator for finite rotations, however, as follows. Let be a finite rotation. Let us imagine performing this rotation as a sequence of rotations by where is large. Each of these rotations may be treated as infinitesimal. The full rotation operator becomes

If we now let we obtain

In this form, we recognize that angular momentum is a generator of rotations, similarly to how linear momentum generates translations.

The transformation rule for an operator is thus

This expression is valid for any rotation. We see that, if the operator commutes with both position and momentum, then it will remain unchanged by a rotation.

We can also calculate the effect of the unitary operator on the wave function, as follows. We first determine the effect of the operator on a position eigenstate:

As expected, the effect is to produce another position eigenstate, this one at the image of the rotation. The effect on the wavefunction is therefore as follows.

This is just the wave function evaluated at the rotated point.