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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):
The rotation operator gates (), and () are the analog rotation matrices in three Cartesian axes of SO(3), [c] along the x, y or z-axes of the Bloch sphere projection. As Pauli matrices are related to the generator of rotations, these rotation operators can be written as matrix exponentials with Pauli matrices in the argument.
That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3) § A note on Lie algebras. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group ...
Rotation operator may refer to: An operator that specifies a rotation (mathematics) Three-dimensional rotation operator; Rot (operator) aka Curl, a differential operator in mathematics; Rotation operator (quantum mechanics)
The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator. Physical pure states in quantum mechanics are represented as unit-norm vectors (probabilities are normalized to one) in a special complex Hilbert space. Time evolution in this vector space is given by the application of the evolution operator.
In quantum mechanics, these three operators are the components of a vector operator known as angular momentum. Examples are the angular momentum of an electron in an atom, electronic spin, and the angular momentum of a rigid rotor. In all cases, the three operators satisfy the following commutation relations,
Hence a brief treatment on this is given here. A more complete description in a quantum mechanical context can be found here. Consider a family of unitary operators representing a rotation about some axis. Since the rotation has one degree of freedom, the operator acts on a field of scalars such that:
In quantum mechanics, the Casimir invariant is the "angular-momentum-squared" operator; integer values of spin j characterize bosonic representations, while half-integer values fermionic representations.