Information about Angular Momentum Operator
In quantum mechanics, the angular momentum operator is an operator that is the quantum analog of the classical angular momentum. It plays a central role in the theory of atomic physics and other quantum problems with rotational symmetry.
where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the position basis as
where
is the gradient operator. This is a commonly encountered form of the angular momentum operator, though not the most general one. It has the following properties
where
denotes the Levi-Civita symbol and, even more importantly,
it commutes with the Hamiltonian of such a chargeless and spinless particle
The first commutation relation is an example of what is generally known as a Lie algebra. In this case, the Lie algebra is that of SU(2) or SO(3), the rotation group in three dimensions. The second commutation relation indicates that
is a Casimir invariant. The third commutation relation states that the angular momentum is a constant of motion, and is a special case of Liouville's equation for quantum mechanics, or more precisely, of Ehrenfest's theorem.
where
is the Poisson bracket.
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.
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Definition
In quantum mechanics, the angular momentum is defined like momentum - not as a quantity but as an operator on the wave function:
where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the position basis as
where
){kind=small}
is the gradient operator. This is a commonly encountered form of the angular momentum operator, though not the most general one. It has the following properties
where
denotes the Levi-Civita symbol and, even more importantly,
it commutes with the Hamiltonian of such a chargeless and spinless particle
.
The first commutation relation is an example of what is generally known as a Lie algebra. In this case, the Lie algebra is that of SU(2) or SO(3), the rotation group in three dimensions. The second commutation relation indicates that
is a Casimir invariant. The third commutation relation states that the angular momentum is a constant of motion, and is a special case of Liouville's equation for quantum mechanics, or more precisely, of Ehrenfest's theorem.
In spherical coordinates
Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. Then, the angular momentum in space representation is:- :
- :
- :
- :
In classical physics
It should be noted that the angular momentum in classical mechanics obeys a similar commutation relation,
where
is the Poisson bracket.
See also
quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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operator is a function, that operates on (or modifies) another function. Often, an "operator" is a function that acts on functions to produce other functions (the sense in which Oliver Heaviside used the term); or it may be a generalization of such a function, as in linear algebra,
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angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque.
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Atomic physics (or atom physics) is the field of physics that studies atoms as isolated systems comprised of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and the processes by which these arrangements change.
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rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag (see opposite) has
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operator is a function, that operates on (or modifies) another function. Often, an "operator" is a function that acts on functions to produce other functions (the sense in which Oliver Heaviside used the term); or it may be a generalization of such a function, as in linear algebra,
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A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. It is a function from a space that consists of the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e.
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Flavour in particle physics
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spin is the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point.
In classical mechanics, the spin angular momentum
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In classical mechanics, the spin angular momentum
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gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
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The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. It is named after the Italian mathematician and physicist Tullio Levi-Civita.
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commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
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In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system.
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In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations.
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In mechanics and geometry, the rotation group is the group of all rotations about the origin of 3-dimensional Euclidean space R3 under the operation of composition.
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In mechanics and geometry, the rotation group is the group of all rotations about the origin of 3-dimensional Euclidean space R3 under the operation of composition.
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In mathematics, a Casimir invariant or Casimir operator is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir invariant of the three-dimensional
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In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical
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For the theorem in complex analysis, see .
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.
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quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system.
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In mechanics and geometry, the rotation group is the group of all rotations about the origin of 3-dimensional Euclidean space R3 under the operation of composition.
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spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle from the positive x-axis.
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Spherical Harmonic is a fantasy novel from the Saga of the Skolian empire series of books by Catherine Asaro which tells the story of Pharaoh Dyhianna Selei (Dehya), ruler of the Skolian Imperialate, after the Radiance War fought by the Imperialate and their enemy Eubian Concord.
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Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies.
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In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation.
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Laplace–Runge–Lenz vector (or simply the LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a planet revolving around a sun.
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