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 lz = x py - y px 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 .
Read more about this topic: Rotation Operator (quantum Mechanics)
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