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Angular momentum operator

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	Topics Angular momentum operator Topic Home In quantum mechanicsQuantum mechanics Quantum mechanics is a first quantized quantum theory that supersedes classical mechanics at the atomic and subatomic levels... , the angular momentum operator is an operatorOperator In mathematics, an operator is a function, usually of a special kind depending on the topic.... analogous to classical angular momentumAngular momentum In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the... . The angular momentum operator plays a central role in the theory of atomic physicsAtomic physics Atomic physics is the field of physics that studies atoms as isolated systems comprised of electrons and an atomic nucleus.... and other quantum problems involving rotational symmetryFacts About Rotational symmetry Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space.... . In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion. Intuitive meaning Angular momentum quantifies the rotational aspect of motion. Like energy and linear momentum, angular momentum in an isolated system is conserved. The concept of an angular momentum operatorOperator Overview In mathematics, an operator is a function, usually of a special kind depending on the topic.... is necessary in quantum mechanics, as calculations of angular momentum must be made upon a wave function, rather than on a point or rigid body as classical calculations entail. This is because at the scale of quantum mechanics, the matterMatter In physics, matter is commonly defined as the substance of which physical objects are composed, not counting the contributio... analyzed is best described by a wave equation or probability amplitudeProbability amplitude In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity.... , rather than as a collection of fixed points or as a rigid body. Vector calculusVector calculus Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions.... is used in calculations of angular momentum, as angular momentum has compenents in each of the three spatial dimensions. Mathematical definition Angular momentum L is mathematically defined as the cross productCross product In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space.... of a wave function's position operatorPosition operator In quantum mechanics, the position operator corresponds to the position observable of a particle.... (r) and momentum operatorMomentum operator In quantum mechanics, momentum is defined as an operator on the wave function.... (p): In the special case of a single particle with no electric chargeElectric charge Summary Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic inte... and no spinSpin (physics) In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the mo... , the angular momentum operator can be written in the position basis as a single vector equation: where is the gradientGradient A generalization of these concepts is the gradient in vector calculus; and this article is mostly about this vector gradient... operator. This is a commonly encountered form of the angular momentum operator, though not the most general one. Further analysis of commutation properties The first commutation relation above is an example of what is generally known as a Lie algebraLie algebra In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a... . In this case, the Lie algebra is that of SU(2) or SO(3), the rotation groupRotation group In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space,... in three dimensions. The second commutation relation indicates that is a Casimir invariantCasimir invariant In mathematics, a Casimir invariant or Casimir operator, , of an -dimensional semisimple Lie algebra is defined as fo... . The third commutation relation states that the angular momentum is a constant of motionConstant of motion In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on... , and is a special case of Liouville's equationLiouville's theorem (Hamiltonian) In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statis... for quantum mechanicsQuantum mechanics Quantum mechanics is a first quantized quantum theory that supersedes classical mechanics at the atomic and subatomic levels... , or more precisely, of Ehrenfest's theorem. In classical physics It should be noted that the angular momentum in classical mechanicsClassical mechanics Classical mechanics is used to describe the motion of macroscopic objects, from projectiles to parts of machinery, as well a... obeys a similar commutation relation, where is the Poisson bracketPoisson bracket In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a cen... . Angular momentum computations in spherical coordinates This section includes mathematical equations involving partial differential equations and Dirac notation. Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. Then, the angular momentum in space representation is: When solving to find eigenstates of this operator, we obtain the following where are the spherical harmonicSpherical Harmonic Overview Spherical Harmonic is a fantasy novel by Catherine Asaro which tells the story of Dyhianna Selei, after the Radiance War fou... s. See also Runge-Lenz vector (used to describe the shape and orientation of bodies in orbit) Position operatorFacts About Position operator In quantum mechanics, the position operator corresponds to the position observable of a particle.... Momentum operatorMomentum operator In quantum mechanics, momentum is defined as an operator on the wave function.... Annihilation operator Creation operator Hamiltonian operator Ladder operator