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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):
In simpler terms, the total angular momentum operator characterizes how a quantum system is changed when it is rotated. The relationship between angular momentum operators and rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics, as discussed further below. The different types of rotation ...
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.
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,
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)
Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.
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.
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: