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Encyclopedia > Angular momentum
Classical mechanics
$vec{F} = frac{mathrm{d}}{mathrm{d}t}(m vec{v})$
Newton's Second Law
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This gyroscope remains upright while spinning due to its angular momentum.

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. 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. 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. Conservation of angular momentum also explains many phenomena in sports and nature. In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... For other senses of this word, see torque (disambiguation). ... Moment of inertia, also called mass moment of inertia or the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... Spoked flywheel Flywheel from stationary engine. ...

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. Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ...

## Angular momentum in classical mechanics GA_googleFillSlot("encyclopedia_square");

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

Image File history File links Torque_animation. ... For other senses of this word, see torque (disambiguation). ...

### Definition

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

$mathbf{L}=mathbf{r}timesmathbf{p}$

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.

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. For other uses, see Mass (disambiguation). ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ...

### 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: In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

$mathbf{L}=sum_i mathbf{R}_itimes m_i mathbf{V}_i$

where Ri is the distance of particle i from the reference point, mi is its mass, and Vi 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

$mathbf{L}=sum_i (mathbf{R}+mathbf{r}_i)times m_i (mathbf{V}+mathbf{v}_i) = left(mathbf{R}times Mmathbf{V}right) + left(sum_i mathbf{r}_itimes m_i mathbf{v}_iright)$

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. An analogous result is obtained for a continuous distribution of matter. In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ...

### 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 θr,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. Leverage redirects here. ...

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: Moment of inertia, also called mass moment of inertia or the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ...

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

where

$I,$ is the moment of inertia of the object (in general, a tensor quantity)
$mathbf{omega}$ is the angular velocity.

Moment of inertia, also called mass moment of inertia or the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ...

### Conservation of angular momentum

The torque caused by the two opposing forces Fg and -Fg causes a change in the angular momentum L in the direction of that torque (since torque is the time derivative of angular momentum). This causes the top to precess.

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... For other senses of this word, see torque (disambiguation). ... Precession (also called gyroscopic precession) is the phenomenon by which the axis of a spinning object (e. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ...

The time derivative of angular momentum is called torque: For other senses of this word, see torque (disambiguation). ...

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

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 τext is any torque applied to the system of particles.

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 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). This article or section does not adequately cite its references or sources. ... For the story by Larry Niven, see Neutron Star (story). ... For other uses, see Black hole (disambiguation). ...

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. 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). For a system of point particles without any intrinsic angular momentum, it turns out to be (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In physics, a Noether charge is a physical quantity conserved as an effect of a continuous symmetry of the underlying system. ...

$sum_i bold{r}_iwedge bold{p}_i$

(Here, the wedge product is used.). In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...

## 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. 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. 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 $hbar/2$. For example, an electron standing at rest has an angular momentum of $hbar/2$. For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... Quantum Leap is a science fiction television series that ran for 97 episodes from March 1989 to May 1993 on NBC. It follows the adventures of Dr. Samuel Beckett (played by Scott Bakula), a brilliant scientist who after researching time-travel, and doing experiments in something he calls The Imaging... Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... In physics, spin refers to 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. ...

### Basic definition

The classical definition of angular momentum as $mathbf{L}=mathbf{r}timesmathbf{p}$ depends on six numbers: rx, ry, rz, px, py, and pz. 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. 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. In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ...

Mathematically, angular momentum in quantum mechanics is defined like momentum - not as a quantity but as an operator on the wave function: This article is about momentum in physics. ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ...

$mathbf{L}=mathbf{r}timesmathbf{p}$

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 This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... In physics, spin refers to 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 quantum mechanics, 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. ...

$mathbf{L}=-ihbar(mathbf{r}timesnabla)$

where $nabla$ is the vector differential operator "Del" (also called "Nabla"). 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: In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ... Nabla is a symbol, shown as . ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant . ...

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

where εlmn is the (antisymmetric) Levi-Civita symbol. From this follows The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ...

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

Since,

$L_x = -ihbar (y {partialover partial z} - z {partialover partial y})$
$L_y = -ihbar (z {partialover partial x} - x {partialover partial z})$
$L_z = -ihbar (x {partialover partial y} - y {partialover partial x})$

it follows, for example,

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

### Addition of quantized angular momenta

For more details on this topic, see Clebsch-Gordan coefficients.

Given a quantized total angular momentum $overrightarrow{j}$ which is the sum of two individual quantized angular momenta $overrightarrow{l_1}$ and $overrightarrow{l_2}$, This article may be too technical for most readers to understand. ...

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

the quantum number j associated with its magnitude can range from | l1l2 | to l1 + l2 in integer steps where l1 and l2 are quantum numbers corresponding to the magnitudes of the individual angular momenta. ^^ Quantum numbers describe values of conserved quantity 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: Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ...

$L_z = -ihbar {partialover partial phi}.$

The eigenfunctions of Lz are therefore $e^{i m_l phi}$, and since φ has a period of , ml must be an integer. In mathematics, an eigenfunction of a linear operator A defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. ...

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. It is therefore known as orbital angular momentum. However, when one rotates the system, one also changes the spin. Therefore the total angular momentum, which is the full generator of rotations, is Ji = Li + Si Being an angular momentum, J satisfies the same commutation relations as L, as will explained below. namely In physics, spin refers to 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. ... The Azimuthal quantum number (or orbital angular momentum quantum number) l is a quantum number for an atomic orbital which determines its orbital angular momentum. ... In physics, spin refers to 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 atomic physics, the total angular quantum momentum numbers parameterize the total angular momentum of a given electron, by combining its orbital angular momentum and its intrinsic angular momentum (i. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ... The Azimuthal quantum number (or orbital angular momentum quantum number) symbolized as l (lower-case L) is a quantum number for an atomic orbital which determines its orbital angular momentum. ...

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

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. For an infinitesmal rotation by an angle dφ, the rotated wavefunction is ψ + idφJzψ. This is similarly true for rotations around any axis. This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

In a charged particle the momentum gets a contribution from the electromagnetic field, and the angular momenta L and J change accordingly. This article is about momentum in physics. ...

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. So the total angular momentum is a conserved quantity The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... In atomic physics, the total angular quantum momentum numbers parameterize the total angular momentum of a given electron, by combining its orbital angular momentum and its intrinsic angular momentum (i. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

$left[J_l, H right] = 0$

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). 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 mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

SO(3) has the same Lie algebra (i.e. the same commutation relations) as SU(2). Generators of SU(2) can have half-integer eigenvalues, and so can mj. Indeed for fermions the spin S and total angular momentum J are half-integer. In fact this is the most general case: j and mj are either integers or half-integers. In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ... In physics, spin refers to 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 atomic physics, the total angular quantum momentum numbers parameterize the total angular momentum of a given electron, by combining its orbital angular momentum and its intrinsic angular momentum (i. ...

Technically, this is because the universal cover of SO(3) is isomorphic SU(2), and the representations of the latter are fully known. Ji span the Lie algebra and J2 is the Casimir invariant, and it can be shown that if the eigenvalues of Jz and J2 are mj and j(j+1) then mj and j are both integer multiples of one-half. j is non-negative and mj takes values between -j and j. In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... 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 groups. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Casimir invariant of a Lie algebra is a member of the center of the universal enveloping algebra of the Lie algebra. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...

### Relation to spherical harmonics

Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. Then, the angular momentum in space representation is: In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

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

When solving to find eigenstates of this operator, we obtain the following In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

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

where

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

are the spherical harmonics. Spherical Harmonic is a fantasy novel by Catherine Asaro which tells the story of Pharaoh Dyhianna (Dehya) Selei, ruler of the Skolian Imperialate, after the Radiance War fought by the Imperialate and their enemy Eubian Concord. ...

## 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. 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 The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...

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

where e is the electric charge, c the speed of light and A the vector potential. Thus, for example, the Hamiltonian of a charged particle of mass m in an electromagnetic field is then This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... A line showing the speed of light on a scale model of Earth and the Moon, taking about 1â…“ seconds to traverse that distance. ... In vector calculus, a vector potential is a vector field whose curl is a given vector field. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...

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

where φ is the scalar potential. This is the Hamiltonian that gives the Lorentz force law. The gauge-invariant angular momentum, or "kinetic angular momentum" is given by It has been suggested that this article or section be merged with Potential. ... In physics, the Lorentz force is the force exerted on a charged particle in an electromagnetic field. ...

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

The interplay with quantum mechanics is discussed further in the article on canonical commutation relations. In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant . ...

Moment of inertia, also called mass moment of inertia or the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ... In quantum mechanics, the orbital and spin angular momentum of bodies can interact in angular momentum coupling. ... Areal velocity is the rate at which area is swept by the position vector of a point which moves along a curve. ... Control Moment Gyro(scope) 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 is a technical term in nuclear physics that refers to a state of a nucleus with more angular momentum than all the states of lower energy. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ...

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