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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 trivial case of the angular momentum of a body in an orbit is given by = where is the mass of the orbiting object, is the orbit's frequency and is the orbit's radius.. The angular momentum of a uniform rigid sphere rotating around its axis, instead, is given by = where is the sphere's mass, is the frequency of rotation and is the sphere's radius.
The momentum operator can be described as a symmetric (i.e. Hermitian), unbounded operator acting on a dense subspace of the quantum state space. If the operator acts on a (normalizable) quantum state then the operator is self-adjoint. In physics the term Hermitian often refers to both symmetric and self-adjoint operators. [7] [8]
In physics, the Pauli–Lubanski pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang Pauli and Józef Lubański. [1] It describes the spin states of moving particles. [2]
Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the del operator, which is itself a vector (useful in momentum-related quantum operators, in the table below). An operator in n-dimensional space can be written:
A particular application of the ladder operator concept is found in the quantum-mechanical treatment of angular momentum. For a general angular momentum vector J with components J x, J y and J z one defines the two ladder operators [3] + = +, =, where i is the imaginary unit.
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, [,] =, [,] =, [,] =, where i is the purely imaginary number and the Planck constant ħ has been set equal to one. The Casimir operator
Angular momentum operators are self-adjoint operators j x, j y, and j z that satisfy the commutation relations [,] =,, {,,}, where ε klm is the Levi-Civita symbol. Together the three operators define a vector operator , a rank one Cartesian tensor operator , j = ( j x , j y , j z ) . {\displaystyle \mathbf {j} =(\mathrm {j_{x}} ,\mathrm {j_{y ...