angular momentum

angular momentum

angular momentum: see momentum.

Property that describes the rotary inertia of a system in motion about an axis. It is a vector quantity, having both magnitude and direction. The magnitude of the angular momentum of an object is the product of its linear momentum (mass math.m × velocity math.v) and the perpendicular distance math.r from the centre of rotation, or math.mmath.vmath.r. The direction is that of the axis of rotation. The angular momentum of an isolated system is constant. This means that a rigid spinning object continues to spin at a constant rate unless acted upon by an external torque. A spinning gyroscope in an airplane remains fixed in its orientation, independent of the airplane's motion, because of the conservation of direction as well as magnitude.

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In physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector. The angular momentum of a system of particles is the sum of that of the particles within it.

Angular momentum is an important concept in both physics and engineering, with numerous applications. 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. Rotational symmetry of space is related to the conservation of angular momentum as an example of Noether's theorem. The conservation of angular momentum explains many phenomena in nature.

Angular momentum in classical mechanics

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 relative to the origin,
mathbf{p} is the linear momentum of the particle, and
times, is the vector cross product.

As seen from the definition, the derived SI units of angular momentum are newton metre seconds (N·m·s or kg·m2s-1 or joule seconds). 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.

Angular momentum of a collection of particles

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.

Angular momentum in the center of mass frame

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}_itimes 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 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 moving relative to their center of mass. This second term can be even further simplified if the particles form a rigid body, in which case a spin appears. 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 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 mathbf{r}_{perp} 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 mathbf{p}_{perp} 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.

As the kinetic energy K of a massive rotating body is given by

mathbf{K}= I mathbf{omega^2}/2

it is proportional to the square of the angular momentum.

Conservation of angular momentum

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.

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 tau_{ext} 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 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.

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).

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

sum_i bold{r}_iwedge 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. 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.

Basic definition

The classical definition of angular momentum as mathbf{L}=mathbf{r}timesmathbf{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. 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}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

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:

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

where εlmn is the (antisymmetric) Levi-Civita symbol. From this follows

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

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.

Angular momentum as a generator of rotations

If phi 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 = -ihbar {partialover partial phi}.

The eigenfunctions of Lz are therefore e^{i m_l phi}, and since phi has a period of 2pi, ml must be an integer.

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 J_i = L_i + S_i Being an angular momentum, J satisfies the same commutation relations as L, as will explained below. namely

[J_ell, J_m ] = i hbar sum_n varepsilon_{lmn} J_n
from which follows
left[J_ell, J^2 right] = 0.

Acting with J on the wavefunction psi of a particle generates a rotation: e^{i phi J_z} psi is the wavefunction psi rotated around the z axis by an angle phi. For an infinitesmal rotation by an angle dphi, the rotated wavefunction is psi+ i dphi J_z psi. 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.

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

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).

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.

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.

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:
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
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.

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

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

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

where phi 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

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.

See also

External links

References

  • Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck, "Quantum Mechanics" (1977). 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.

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