Rotation Operator (quantum Mechanics) - in Relation To The Orbital Angular Momentum

In Relation To The Orbital Angular Momentum

Classically we have for the angular momentum . This is the same in quantum mechanics considering and as operators. Classically, an infinitesimal rotation of the vector r=(x,y,z) about the z-axis to r'=(x',y',z) leaving z unchanged can be expressed by the following infinitesimal translations (using Taylor approximation):

From that follows for states:

And consequently:

Using from above with and Taylor development we get:

with l_z = x p_y - y p_x the z-component of the angular momentum according to the classical cross product.

To get a rotation for the angle, we construct the following differential equation using the condition :

Similar to the translation operator, if we are given a Hamiltonian which rotationally symmetric about the z axis, implies . This result means that angular momentum is conserved.

For the spin angular momentum about the y-axis we just replace with and we get the spin rotation operator .