Angular momentum

Classical mechanics

$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
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This gyroscope remains upright while spinning due to its angular momentum. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Early Ideas on Motion The Greek philosophers, and Aristotle in particular were the first to propose that there are abstract principles governing nature Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. A gyroscope is a device for measuring or maintaining orientation, based on the principles of Angular momentum.

In physics, the 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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about In particular, if a point mass rotates about an axis, then the angular momentum with respect to a point on the axis is related to the mass of the object, the velocity and the distance of the mass to the axis. Point mass is an Idealistic term used to describe either Matter which is infinitely small or an object which can be thought of as infinitely small Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object

Angular momentum is important in physics because it is a conserved quantity: a system's angular momentum stays constant unless an external torque acts on it. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about Torque is the rate at which angular momentum is transferred in or out of the system. When a rigid body rotates, its resistance to a change in its rotational motion is measured by its moment of inertia. This article is about the moment of inertia of a rotating object. Angular momentum is an important concept in both physics and engineering, with numerous applications. For example, the kinetic energy stored in a massive rotating object such as a flywheel is proportional to the square of the angular momentum. The kinetic energy of an object is the extra Energy which it possesses due to its motion A flywheel is a mechanical device with significant Moment of inertia used as a storage device for Rotational energy. Conservation of angular momentum also explains many phenomena in sports and nature.

From a fundamental point of view, angular momentum is related to rotation of the system. Rotational symmetry of space is related to the conservation of angular momentum as an example of Noether's theorem. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has

1 Angular momentum in classical mechanics
2 Angular momentum in relativistic mechanics
3 Angular momentum in quantum mechanics
4 Angular momentum in electrodynamics
5 See also
6 External links
7 References

Angular momentum in classical mechanics

Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system

Definition

Angular momentum of a particle about a given origin is defined as:

$\mathbf{L}=\mathbf{r}\times\mathbf{p}$

where:

$\mathbf{L}$ is the angular momentum of the particle,

$\mathbf{r}$ is the position vector of the particle (evidently from the origin),

$\mathbf{p}$ is the linear momentum of the particle, and

$\times\,$ is the vector cross product. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

As seen from the definition, the derived SI units of angular momentum are newton metre seconds (N·m·s or kg·m²s^-1). SI derived units are part of the SI system of measurement units and are derived from the seven SI base units They are derived from SI basic units/defined The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units Because of the cross product, L is a pseudovector perpendicular to both the radial vector r and the momentum vector p and it is assigned a sign by the right-hand rule. In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an For the related yet different principle relating to electromagnetic coils see Right hand grip rule.

If a system consists of several particles, the total angular momentum about an origin can be obtained by adding (or integrating) all the angular momenta of the constituent particles. Angular momentum can also be calculated by multiplying the square of the displacement r, the mass of the particle and the angular velocity. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Do not confuse with Angular frequency The unit for angular velocity is rad/s

Orbital and spin angular momentum

It is very often convenient to consider the angular momentum of a collection of particles about their center of mass, since this simplifies the mathematics considerably. The angular momentum of a collection of particles is the sum of the angular momentum of each particle:

$\mathbf{L}=\sum_i \mathbf{R}_i\times m_i \mathbf{V}_i$

where $R i$ is the distance of particle i from the reference point, $m i$ is its mass, and $V i$ is its velocity. The center of mass is defined by:

$\mathbf{R}=\frac{1}{M}\sum_i m_i \mathbf{R}_i$

where the total mass of all particles is given by

$M=\sum_i m_i\,$

It follows that the velocity of the center of mass is

$\mathbf{V}=\frac{1}{M}\sum_i m_i \mathbf{V}_i\,$

If we define $\mathbf{r}_i$ as the displacement of particle i from the center of mass, and $\mathbf{v}_i$ as the velocity of particle i with respect to the center of mass, then we have

$\mathbf{R}_i=\mathbf{R}+\mathbf{r}_i\,$ and $\mathbf{V}_i=\mathbf{V}+\mathbf{v}_i\,$

and also

$\sum_i m_i \mathbf{r}_i=0\,$ and $\sum_i m_i \mathbf{v}_i=0\,$

so that the total angular momentum is

$\mathbf{L}=\sum_i (\mathbf{R}+\mathbf{r}_i)\times m_i (\mathbf{V}+\mathbf{v}_i) = \left(\mathbf{R}\times M\mathbf{V}\right) + \left(\sum_i \mathbf{r}_i\times m_i \mathbf{v}_i\right)$

The first term is just the angular momentum of the center of mass. It is the same angular momentum one would obtain if there were just one particle of mass M moving at velocity V located at the center of mass. The second term is the angular momentum that is the result of the particles spinning about their center of mass. This second term can be even further simplified if the particles form a rigid body. In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected An analogous result is obtained for a continuous distribution of matter.

Fixed axis of rotation

For many applications where one is only concerned about rotation around one axis, it is sufficient to discard the pseudovector nature of angular momentum, and treat it like a scalar where it is positive when it corresponds to a counter-clockwise rotations, and negative clockwise. To do this, just take the definition of the cross product and discard the unit vector, so that angular momentum becomes:

$L = |\mathbf{r}||\mathbf{p}|\sin \theta_{r,p}$

where θ_r,p is the angle between r and p measured from r to p; an important distinction because without it, the sign of the cross product would be meaningless. From the above, it is possible to reformulate the definition to either of the following:

$L = \pm|\mathbf{p}||\mathbf{r}_{\perp}|$

where r_⊥ is called the lever arm distance to p.

The easiest way to conceptualize this is to consider the lever arm distance to be the distance from the origin to the line that p travels along. With this definition, it is necessary to consider the direction of p (pointed clockwise or counter-clockwise) to figure out the sign of L. Equivalently:

$L = \pm|\mathbf{r}||\mathbf{p}_{\perp}|$

where p_⊥ is the component of p that is perpendicular to r. As above, the sign is decided based on the sense of rotation.

For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector:

$\mathbf{L}= I \mathbf{\omega}$

where

$I\,$ is the moment of inertia of the object (in general, a tensor quantity)

$\mathbf{\omega}$ is the angular velocity. This article is about the moment of inertia of a rotating object. This article is about the moment of inertia of a rotating object. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually Do not confuse with Angular frequency The unit for angular velocity is rad/s

Conservation of angular momentum

The torque caused by the two opposing forces F_g and -F_g causes a change in the angular momentum L in the direction of that torque (since torque is the time derivative of angular momentum). A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about This causes the top to precess. Precession refers to a change in the direction of the axis of a rotating object

In a closed system angular momentum is constant. This conservation law mathematically follows from continuous directional symmetry of space (no direction in space is any different from any other direction). See Noether's theorem. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has

The time derivative of angular momentum is called torque:

$\tau = \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \mathbf{r} \times \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \mathbf{r} \times \mathbf{F}$

So requiring the system to be "closed" here is mathematically equivalent to zero external torque acting on the system:

$\mathbf{L}_{\mathrm{system}} = \mathrm{constant} \leftrightarrow \sum \tau_{\mathrm{ext}} = 0$

where $τ e x t$ is any torque applied to the system of particles. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about

In orbits, the angular momentum is distributed between the spin of the planet itself and the angular momentum of its orbit:

$\mathbf{L}_{\mathrm{total}} = \mathbf{L}_{\mathrm{spin}} + \mathbf{L}_{\mathrm{orbit}}$ ;

If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved.

The conservation of angular momentum is used extensively in analyzing what is called central force motion. If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant. Constant angular momentum is extremely useful when dealing with the orbits of planets and satellites, and also when analyzing the Bohr model of the atom. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is This article is about artificial satellites For natural satellites also known as moons see Natural satellite. In Atomic physics, the Bohr model created by Niels Bohr depicts the Atom as a small positively charged nucleus surrounded by Electrons History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny

The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. By bringing part of mass of her body closer to the axis she decreases her body's moment of inertia. Because angular momentum is constant in the absence of external torques, the angular velocity (rotational speed) of the skater has to increase.

The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars (indeed, decreasing the size of object 10⁴ times results in increase of its angular velocity by the factor 10⁸). A white dwarf, also called a degenerate dwarf, is a small Star composed mostly of Electron-degenerate matter. A neutron star is a type of remnant that can result from the Gravitational collapse of a massive Star during a Type II, Type Ib or Type A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e

The conservation of angular momentum in Earth-Moon system results in the transfer of angular momentum from Earth to Moon (due to tidal torque the Moon exerts on the Earth). This in turn results in the slowing down of the rotation rate of Earth (at about 42 nsec/day), and in gradual increase of the radius of Moon's orbit (at ~4. 5 cm/year rate).

Angular momentum in relativistic mechanics

In modern (late 20th century) theoretical physics, angular momentum is described using a different formalism. The twentieth century of the Common Era began on Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant). In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has For a system of point particles without any intrinsic angular momentum, it turns out to be

$\sum_i \bold{r}_i\wedge \bold{p}_i$

(Here, the wedge product is used. ).

Angular momentum in quantum mechanics

In quantum mechanics, angular momentum is quantized -- that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. In Physics, a quantum leap or quantum jump is a change of an Electron from one energy state to another within an Atom. The angular momentum of a subatomic particle, due to its motion through space, is always a whole-number multiple of $\hbar$ ("h-bar," known as Dirac's constant), defined as Planck's constant divided by 2π. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Furthermore, experiments show that most subatomic particles have a permanent, built-in angular momentum, which is not due to their motion through space. This spin angular momentum comes in units of $\hbar/2$ . In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin For example, an electron standing at rest has an angular momentum of $\hbar/2$ .

Basic definition

The classical definition of angular momentum as $\mathbf{L}=\mathbf{r}\times\mathbf{p}$ depends on six numbers: $r x$ , $r y$ , $r z$ , $p x$ , $p y$ , and $p z$ . Translating this into quantum-mechanical terms, the Heisenberg uncertainty principle tells us that it is not possible for all six of these numbers to be measured simultaneously with arbitrary precision. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.

Mathematically, angular momentum in quantum mechanics is defined like momentum - not as a quantity but as an operator on the wave function:

$\mathbf{L}=\mathbf{r}\times\mathbf{p}$

where r and p are the position and momentum operators respectively. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In physics an operator is a function acting on the space of Physical states As a resultof its application on a physical state another physical state is obtained A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system 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

$\mathbf{L}=-i\hbar(\mathbf{r}\times\nabla)$

where $\nabla$ is the vector differential operator "Del" (also called "Nabla"). Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Quantum mechanics, the angular momentum operator is an Operator analogous to classical Angular momentum. &nablaDel Nabla is the Symbol \nabla The name comes from the Greek word for a Hebrew Harp, which had a similar shape This orbital angular momentum operator is the most commonly encountered form of the angular momentum operator, though not the only one. It satisfies the following canonical commutation relations:

$[L_l, L_m ] = i \hbar \sum_{n=1}^3 \epsilon_{lmn} L_n$ ,

where ε_lmn is the (antisymmetric) Levi-Civita symbol. In Physics, the canonical commutation relation is the relation between Canonical conjugate quantities (quantities which are related by definition such that one is The Levi-Civita symbol, also called the Permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in Tensor From this follows

$\left[L_i, L^2 \right] = 0$

Since,

$L_x = -i\hbar (y {\partial\over \partial z} - z {\partial\over \partial y})$

$L_y = -i\hbar (z {\partial\over \partial x} - x {\partial\over \partial z})$

$L_z = -i\hbar (x {\partial\over \partial y} - y {\partial\over \partial x})$

it follows, for example,

$\begin{align} \left[L_x,L_y\right] & = -\hbar^2 \left( (y {\partial \over \partial z} - z {\partial\over \partial y})(z {\partial\over \partial x} - x {\partial\over \partial z}) - (z {\partial\over \partial x} - x {\partial\over \partial z})(y {\partial \over \partial z} - z {\partial\over \partial y})\right) \\ & = -\hbar^2 \left( y {\partial\over \partial x} - x {\partial\over \partial y}\right) = i \hbar L_z. \\ \end{align}$

Addition of quantized angular momenta

For more details on this topic, see Clebsch-Gordan coefficients. In Physics, the Clebsch-Gordan coefficients are sets of numbers that arise in Angular momentum coupling under the laws of Quantum mechanics.

Given a quantized total angular momentum $\overrightarrow{j}$ which is the sum of two individual quantized angular momenta $\overrightarrow{l_1}$ and $\overrightarrow{l_2}$ ,

$\overrightarrow{j} = \overrightarrow{l_1} + \overrightarrow{l_2}$

the quantum number $j$ associated with its magnitude can range from $| l 1 - l 2 |$ to $l 1 + l 2$ in integer steps where $l 1$ and $l 2$ are quantum numbers corresponding to the magnitudes of the individual angular momenta. Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. ^^

Angular momentum as a generator of rotations

If $φ$ is the angle around a specific axis, for example the azimuthal angle around the z axis, then the angular momentum along this axis is the generator of rotations around this axis:

$L_z = -i\hbar {\partial\over \partial \phi}.$

The eigenfunctions of L_z are therefore $e^{i m_l \phi}$ , and since $φ$ has a period of $2π$ , m_l must be an integer. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Mathematics, an eigenfunction of a Linear operator, A, defined on some Function space is any non-zero function f in

For a particle with a spin S, this takes into account only the angular dependence of the location of the particle, for example its orbit in an atom. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin It is therefore known as orbital angular momentum. The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number However, when one rotates the system, one also changes the spin. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin Therefore the total angular momentum, which is the full generator of rotations, is $J i = L i + S i$ Being an angular momentum, J satisfies the same commutation relations as L, as will explained below. See also Azimuthal quantum number#Addition of quantized angular momenta In Quantum mechanics, the total angular quantum momentum numbers parameterize the total Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number namely

$[J_l, J_m ] = i \hbar \sum_n \epsilon_{lmn} J_n$

from which follows

$\left[J_l, J^2 \right] = 0.$

Acting with J on the wavefunction $ψ$ of a particle generates a rotation: $e^{i \phi J_z} \psi$ is the wavefunction $ψ$ rotated around the z axis by an angle $φ$ . A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system For an infinitesmal rotation by an angle $d φ$ , the rotated wavefunction is $ψ + i d φ J z ψ$ . A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system This is similarly true for rotations around any axis.

In a charged particle the momentum gets a contribution from the electromagnetic field, and the angular momenta L and J change accordingly. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product

If the Hamiltonian is invariant under rotations, as in spherically symmetric problems, then according to Noether's theorem, it commutes with the total angular momentum. In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. See also Azimuthal quantum number#Addition of quantized angular momenta In Quantum mechanics, the total angular quantum momentum numbers parameterize the total So the total angular momentum is a conserved quantity

$\left[J_l, H \right] = 0$

Since angular momentum is the generator of rotations, its commutation relations follow the commutation relations of the generators of the three-dimensional rotation group SO(3). In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves This article is about rotations in three-dimensional Euclidean space In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n This is why J always satisfies these commutation relations. In d dimensions, the angular momentum will satisfy the same commutation relations as the generators of the d-dimensional rotation group SO(d). In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n

SO(3) has the same Lie algebra (i. This article is about rotations in three-dimensional Euclidean space 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 e. the same commutation relations) as SU(2). Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n Generators of SU(2) can have half-integer eigenvalues, and so can m $j$ . Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes Indeed for fermions the spin S and total angular momentum J are half-integer. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin See also Azimuthal quantum number#Addition of quantized angular momenta In Quantum mechanics, the total angular quantum momentum numbers parameterize the total In fact this is the most general case: j and m $j$ are either integers or half-integers.

Technically, this is because the universal cover of SO(3) is isomorphic SU(2), and the representations of the latter are fully known. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n In the study of the Representation theory of Lie groups, the study of representations of SU(2 is fundamental to the study of representations of Semisimple Lie J_i span the Lie algebra and J² is the Casimir invariant, and it can be shown that if the eigenvalues of J_z and J² are m_j and j(j+1) then m_j and j are both integer multiples of one-half. 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 In Mathematics, a Casimir invariant or Casimir operator is a distinguished element of the centre of the Universal enveloping algebra of a Lie algebra In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes j is non-negative and m_j takes values between -j and j.

Relation to spherical harmonics

Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of This article is about rotations in three-dimensional Euclidean space In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial Then, the angular momentum in space representation is:

$L^2 = -\frac{\hbar^2}{\sin\theta}\frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial}{\partial \theta}\right) - \frac{\hbar^2}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2}$

When solving to find eigenstates of this operator, we obtain the following

$L^2 | l, m \rang = {\hbar}^2 l(l+1) | l, m \rang$

$L_z | l, m \rang = \hbar m | l, m \rang$

where

$\lang \theta , \phi | l, m \rang = Y_{l,m}(\theta,\phi)$

are the spherical harmonics. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of

Angular momentum in electrodynamics

When describing the motion of a charged particle in the presence of an electromagnetic field, the "kinetic momentum" p is not gauge invariant. The electromagnetic field is a physical field produced by electrically charged objects. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations As a consequence, the canonical angular momentum $\mathbf{L} = \mathbf{r} \times \mathbf{p}$ is not gauge invariant either. Instead, the momentum that is physical, the so-called canonical momentum, is

$\mathbf{p} -\frac {e \mathbf{A} }{c}$

where $e$ is the electric charge, c the speed of light and A the vector potential. In Mathematics and Classical mechanics, canonical coordinates are particular sets of coordinates on the Phase space, or equivalently on the Cotangent Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Vector calculus, a vector potential is a Vector field whose curl is a given vector field Thus, for example, the Hamiltonian of a charged particle of mass m in an electromagnetic field is then

$H =\frac{1}{2m} \left( \mathbf{p} -\frac {e \mathbf{A} }{c}\right)^2 + e\phi$

where $φ$ is the scalar potential. In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system A scalar Potential is a fundamental concept in Vector analysis and Physics (the adjective 'scalar' is frequently omitted if there is no danger of confusion This is the Hamiltonian that gives the Lorentz force law. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation The gauge-invariant angular momentum, or "kinetic angular momentum" is given by

$K= \mathbf{r} \times \left( \mathbf{p} -\frac {e \mathbf{A} }{c}\right)$

The interplay with quantum mechanics is discussed further in the article on canonical commutation relations. In Physics, the canonical commutation relation is the relation between Canonical conjugate quantities (quantities which are related by definition such that one is

External links

Conservation of Angular Momentum - a chapter from an online textbook

Angular Momentum in a Collision Process - derivation of the three dimensional case

References

Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck, "Quantum Mechanics" (1977). This article is about the moment of inertia of a rotating object. In Quantum mechanics, the procedure of constructing Eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling Areal Velocity is the rate at which Area is swept out by a particle as it moves along a Curve. A control moment gyroscope (CMG is an Attitude control device generally used in Satellite attitude control systems The rotational energy or angular kinetic energy is the Kinetic energy due to the rotation of an object and is part of its total kinetic energy. The rigid rotor is a mechanical model that is used to explain rotating systems Yrast (Swedish pronunciation, English) is a technical term in Nuclear physics that refers to a state of a nucleus with a minimum of Energy Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Quantum mechanics, spatial quantization is the Quantization of Angular momentum in three-dimensional space John Wiley & Sons.
E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1935) Cambridge at the University Press, ISBN 0-521-09209-4 See chapter 3.
Edmonds, A. R. , Angular Momentum in Quantum Mechanics, (1957) Princeton University Press, ISBN 0-691-07912-9.
Jackson, John David, "Classical Electrodynamics". Second Ed. , 1975. Third Ed. , 1998. John Wiley & Sons.
Serway, Raymond A. ; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed. ). Brooks/Cole. ISBN 0-534-40842-7.
Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed. ). W. H. Freeman. ISBN 0-7167-0809-4.

Dictionary

-noun

(physics) The vector product that describes the rotary inertia of a system about an axis and is conserved in a closed system. For an isolated rigid body, it is a measure of the extent to which an object will continue to rotate in the absence of an applied torque.

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