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Concept
Rotation operator (quantum mechanics)
Summary
This article concerns the rotation operator, as it appears in quantum mechanics.
With every physical rotation , we postulate a quantum mechanical rotation operator which rotates quantum mechanical states.
In terms of the generators of rotation,
where is rotation axis, is angular momentum, and is the reduced Planck constant.
Translation operator (quantum mechanics)
The rotation operator , with the first argument indicating the rotation axis and the second the rotation angle, can operate through the translation operator for infinitesimal rotations as explained below. This is why, it is first shown how the translation operator is acting on a particle at position x (the particle is then in the state according to Quantum Mechanics).
Translation of the particle at position to position :
Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the identity operator, which does nothing):
Taylor development gives:
with
From that follows:
This is a differential equation with the solution
Additionally, suppose a Hamiltonian is independent of the position. Because the translation operator can be written in terms of , and , we know that This result means that linear momentum for the system is conserved.
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 about the -axis to leaving 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 expansion we get:
with the -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 -axis, implies . This result means that angular momentum is conserved.
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