Angular Momentum Rotation Continuous Large Hilberty Space

ANGULAR MOMENTUM

L.D. LANDAU , E.M. LIFSHITZ , in Quantum Mechanics (Third Edition), 1977

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The chapter discusses the angular momentum. To derive the law of conservation of momentum, the homogeneity of space relative to a closed system of particles is used. Besides its homogeneity, space also has the property of isotropy: all directions in it are equivalent. Hence, the Hamiltonian of a closed system cannot change when the system rotates as a whole through an arbitrary angle about an arbitrary axis. It is sufficient to require the fulfillment of this condition for an infinitely small rotation. The quantity whose conservation for a closed system follows from the property of isotropy of space is the angular momentum of the system. The three components of the angular momentum cannot simultaneously have definite values except in the case where all three components simultaneously vanish. Angular momentum is fundamentally different from the linear momentum, whose three components are simultaneously measurable. The chapter also discusses the Eigenvalues of the angular momentum. To determine the eigenvalues of the component, in some direction, of the angular momentum of a particle, it is convenient to use the expression for its operator in spherical polar coordinates, taking the direction in question as the polar axis.

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Angular Momenta

Mario Reis , in Fundamentals of Magnetism, 2013

3.1 Angular momentum algebra

All of the angular momenta (spin, orbital, and total) obey the same algebra here described and then, instead of J (as will be used here), it is possible to substitute that by either L or S . Thus, let us first consider the angular momentum in Cartesian coordinates:

(3.1) $\overrightarrow{J}=J_x\hat{i}+J_y\hat{j}+J_z\hat{k}\text{.}$

These components obey the following commutation relation:

(3.2) $[J_x\text{,}{J}_y]=i\hslash J_z\text{,}$

(3.3) $[J_y\text{,}{J}_z]=i\hslash J_x\text{,}$

(3.4) $[J_z\text{,}{J}_x]=i\hslash J_y\text{.}$

The reason in which all of the angular momentum algebra is valid for all of the angular momenta came from the above set of equations, i.e., spin, orbital, and total angular momenta follow the same commutation relation.

Note these components do not commute each other and therefore these cannot be diagonalized simultaneously. Thus, we need to consider other quantity: the square of the total angular momentum operator:

(3.5) $J^2=J_x{J}_x+J_y{J}_y+J_z{J}_z\text{.}$

This new operator commutes with each of those components:

(3.6) $[J^2\text{,}{J}_u]=0(u=x\text{,}y\text{,}z)$

and therefore these can be diagonalized simultaneously. We can now define the basis $|j\text{,}{m}_j〉$ , where $J^2$ and $J_z$ are diagonals (this last was chosen by convenience; but we could choose either $J_x$ or $J_y$ ). Thus,

(3.7) $J^2|j\text{,}{m}_j〉=j(j+1){\hslash }^2|j\text{,}{m}_j〉\text{,}$

where j is the total angular momentum of the system, and

(3.8) $J_z|j\text{,}{m}_j〉=m_j\hslash |j\text{,}{m}_j〉\text{,}$

where $m_j=-j\text{,}-j+1\text{,}\dots \text{,}+j-1\text{,}+j$ , is the projection of the total angular momentum $\overrightarrow{J}$ on the z-axis.

We have thus a basis where both, $J^2$ and $J_z$ are diagonals. However, we do not know all of the total moment, since the components $J_x$ and $J_y$ remain to be determined. To go further we need to define two operators, named ladder operators:

(3.9) $J_{+}=J_x+{\mathit{iJ}}_y\text{and}{J}_{-}=J_x-{\mathit{iJ}}_y$

that commutes as

(3.10) $[J_{+}\text{,}{J}_{-}]=2\hslash J_z\text{,}[J_z\text{,}{J}_{\pm }]=\pm \hslash J_{\pm }\text{,}[J^2\text{,}{J}_{\pm }]=0\text{.}$

Thus, it is possible to determine:

(3.11) $J_{\pm }|j\text{,}{m}_j〉=\sqrt{(j\mp m_j)(j\pm m_j+1)}\hslash |j\text{,}{m}_j\pm 1〉\text{.}$

From Eq. (3.9), it is possible to write:

(3.12) $J_x=\frac{1}{2}(J_{+}+J_{-})\text{and}{J}_y=\frac{1}{2i}(J_{+}-J_{-})\text{and}$

and therefore determine the final operator $\overrightarrow{J}$ .

Few words about the vectorial space in which these vectors are: the Hilbert space. Mathematically speaking, it is a generalization of the Euclidian space. For a total angular momentum J, there are $2j+1$ different states ( $m_j$ possibilities). Thus, the matrices obtained from above have dimension $(2j+1)\times (2j+1)$ . We leave as an exercise to write the matrix operators $J_x\text{,}{J}_y$ , and $J_z$ , based on the above description.

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CONSERVATION OF ANGULAR MOMENTUM

George B. Arfken , ... Joseph Priest , in International Edition University Physics, 1984

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Angular momentum is the rotational analog of linear momentum and is also associated with a conservation principle. Kepler's second law of planetary motion is an example of angular momentum conservation. An ice skater's rotational motion also demonstrates the conservation of angular momentum. When unstable nuclei decay with the emission of particles, angular momentum is conserved. The conservation of angular momentum is a universal principle. This chapter introduces the law of conservation of angular momentum by considering the criterion for its validity and illustrates its scope with varied examples. It defines the angular momentum for a particle and then presents the extension of that definition to a system of particles. The chapter presents the angular momentum of a system of particles as a sum of two types of angular momenta, spin and orbital, by using the center-of-mass concept. That separation of angular momentum into two types simplifies drawing conclusions about the rotational motion of the system as a whole.

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Angular Momentum and Vortices in Optics

Gerard Nienhuis , in Structured Light and Its Applications, 2008

2.10 CONCLUSION

Angular momentum (AM) is an important and delicate quantity of light. It arises both from circulating phase gradients and from rotating vector properties of the field. This suggests the possible separation of optical AM into different types, with the flavor of orbital and spin AM. We discuss the possibility and limitations of such a separation, both for a classical and a quantum description of light. For an arbitrary Maxwell field, such a separation is possible. The separate parts have a well-defined significance and generate useful transformations. However, they do not have all the properties of AM. There is a number of special modes of light where the presence of AM is rather obvious. These are also the fields that contain vortices in the form of phase or polarization singularities. Examples of exact solutions of Maxwell's equations are multipole fields with a spherical or cylindrical structure. For these fields, the polarization and phase contributions to AM are inseparable. For both types of multipole fields, the z-component of the AM has a well-defined value per photon, and for the spherical multipole fields the same statement holds for the total AM. For paraxial beams, AM in the propagation direction is separated in a natural way in an orbital and a polarization part. This is not only true for the AM integrated over space, but also for the densities of spin and orbital AM. This remains true for nonmonochromatic beams. We consider the appearance of AM both for stationary beams and for beams in which the light pattern is rotating.

We point out that there is a one-to-one correspondence between monochromatic paraxial beams and solutions of the two-dimensional isotropic harmonic oscillator. This analogy is applied to the description of the dynamics of vortices in paraxial beams in a number of simple examples. We wish to stress, however, that orbital AM in light beams does not require the presence of vortices. A counterexample is a light beam with general astigmatism, where the elliptical wave front has a different orientation than the elliptical intensity pattern in each transverse plane [38]. In this case, the rotation of the ellipses during propagation gives rise to AM [39,40].

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Fundamentals of Free-Space Optical Communications Systems, Optical Channels, Characterization, and Network/Access Technology

Arun K. Majumdar , in Optical Wireless Communications for Broadband Global Internet Connectivity, 2019

4.13.4 Intersatellite Links

The propagation path for intersatellite links such as GEO-GEO, LEO-GEO, and LEO-LEO optical links do not suffer the atmospheric absorption, scattering, turbulence effects, and cloud outages. Since the two satellites move with different relative velocity, the acquisition and tracking can be one of the major challenges to maintain the two satellites for LOS links. Other challenges for establishing optical link connectivity are discussed [19]: point-ahead-angle, doppler shift, satellite vibration and tracking, and background noise sources. The transmission and detection schemes need to be power efficient because of the large propagation distance, and coherent detection is more efficient for intersatellite links for optimum SNR. Gbps data rates reported recently are data rates of 5.6 Gbit/s between LEO and LEO [6,10] and 1.8   Gbit/s between GEO and LEO [20].

4.13.4.1 Orbital Angular Momentum for Atmospheric Free-Space Optical Communication System: Potential Scheme for Gigabit Range Data Rate

Optical angular momentum (OAM) is associated with the helicity photon wavefront and related to spatial distribution. Application of OAM beams in high data rate FSO communication has been explored recently [21,22]. In order to fully utilize the terabit capacity of the optical carrier for potential FSO communication systems, research has been devoted to OAM-multiplexed free-space laser communication links. Increased capacity and spectral efficiency for high data rate communication are being studied using information carrying OAM beams, specifically for long distance, deep space, or near Earth optical communication for future potential technology. OAM for FSO applications is discussed in Ref.[19]. As described in Ref.[19], the OAM transmitted beam can be expressed as follows:

(4.65) $U\left(r,\theta ,t\right)=S\left(t\right)·A\left(r\right)·\exp \left(il\theta \right)$

where A(r) is the amplitude of the waist of the Gaussian beam, exp(ilθ) is the azimuthal phase term, r is the radial distance from the center axis of the beam, and S(t) is the signal containing the data to be transmitted. This is for a single OAM beam, and when multiplexing N such information carrying OAM beams, the total multiplex OAM beam can be written as

(4.66) $\text{Transmitted}\text{OAM}\text{beam}:U_{mux}\left(r,\theta ,t\right)=\sum _{m=1}^N{S}_m\left(t\right)A_m\left(r\right)\exp \left(i{l}_m\theta \right)$

Received demultiplexed can then be obtained by multiplying an inverse azimuth phase term as follows.

Received demultiplexed OAM beam:

(4.67) $U_{Rx}\left(r,\theta ,t\right)=\exp \left[(-i{l}_n\theta )\right]\sum _{m=1}^N{S}_m\left(t\right)A_m\left(r\right)\exp \left(i{l}_m\theta \right)$

The data rates will depend on the number of light beams and the modes of the resultant OAM beam and the types of modulation used. Terabit free-space data transmission employing OAM multiplexing for short distances in the presence of turbulence have been reported [23,24], which show the future potential of achieving a very high data rate at least in the multigigabit ranges between satellites and satellite-to-ground links for near-Earth FSO communications even in the presence of some level of turbulence to make high data rate Internet connectivity possible.

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CONSERVATION OF ANGULAR MOMENTUM

George B. Arfken , ... Joseph Priest , in University Physics, 1984

Summary

The angular momentum of a particle is defined by

(11.1) $\text{L}=\text{r}\times \text{p}$

where r is the position vector and p is the linear momentum vector of the particle. The magnitude of L can be written

(11.4) $\text{L}=rp\text{sin}\theta =r{p}_{\perp }=r_{\perp }p$

where r _⊥ is the projection of r on a line perpendicular to p, and p _⊥ is the projection of p on a line perpendicular to r.

The rate of change of L with time is governed by

(11.8) ${\tau }_{ex}=\frac{d\text{L}}{dt}$

where τ_ex is the net torque acting on the particle. When τ_ex = 0, then the angular momentum of the particle is conserved. If

$\begin{array}{l}{\tau }_{ex}=0\hfill \\ \text{then}\hfill \\ \Delta \text{L}\text{=}\text{0}\text{L}\text{=}\text{constant}\hfill \end{array}$

The total angular momentum of a system of particles is defined by

(11.13) $\text{L}={\displaystyle \sum r_i\times p_i}$

The net internal torque due to the interparticle forces sums to zero, and the net external torque determines the time rate of change of angular momentum for the system of particles. Thus Eq. 11.8 is also satisfied for a system of particles where τ_ex is the net external torque, and L is total angular momentum of the system. Furthermore, if τ_ex = 0, then the angular momentum of the system is conserved. The total angular momentum of a system can be expressed as a sum of spin and orbital angular momentum. Thus

(11.17) $\text{L}={\text{L}}_s+{\text{L}}_o$

where the spin angular momentum

(11.26) ${\text{L}}_s={\displaystyle \sum {\text{r}}_i\text{'}}\times {\text{p}}_i\text{'}$

is the angular momentum relative to the center of mass, and the orbital angular momentum

(11.25) ${\text{L}}_o=M{\text{r}}_{\text{c}\text{m}}\times {\text{v}}_{\text{c}\text{m}}$

is the angular momentum of the total mass of the system located at the center of mass and moving with the center-of-mass velocity. Here r _cm and v _cm are the center-of-mass position and velocity vectors, respectively, and primes denote quantities relative to the center of mass.

If the angular momentum is conserved, ΔL = 0, and

(11.27) $\Delta {\text{L}}_s=-\Delta {\text{L}}_o$

Any change in the spin angular momentum of a system whose total angular momentum is conserved is matched by an equal and opposite change in the orbital angular momentum of the system.

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Introductory Article: Classical Mechanics

G. Gallavotti , in Encyclopedia of Mathematical Physics, 2006

Rigid Body

Another fundamental integrable system is the rigid body in the absence of gravity and with a fixed point O. It can be naturally described in terms of the Euler angles θ ₀, φ ₀, ψ ₀ (see Figure 3 ) and their derivatives ${\stackrel{.}{\theta }}_0,{\stackrel{.}{\phi }}_0,{\stackrel{.}{\psi }}_0$ .

Figure 3. The Euler angles of the comoving frame i ₁, i ₂, i ₃ with respect to a fixed frame x , y , z . The direction n is the "node line, intersection between the planes x , y and i ₁, i ₂.

Let I ₁, I ₂, I ₃ be the three principal inertia moments of the body along the three principal axes with unit vectors i ₁, i ₂, i ₃. The inertia moments and the principal axes are the eigenvalues and the associated unit eigenvectors of the 3 × 3 inertia matrix $\mathcal{I}$ , which is defined by ${\mathcal{I}}_{hk}={\sum }_{i=1}^n{m}_i({\mathit{x}}_i{)}_h({\mathit{x}}_i{)}_k$ , where h, k = 1, 2, 3 and x _i is the position of the ith particle in a reference frame with origin at O and in which all particles are at rest: this comoving frame exists as a consequence of the rigidity constraint. The principal axes form a coordinate system which is comoving as well: that is, in the frame (O; i ₁, i ₂, i ₃) as well, the particles are at rest.

The Lagrangian is simply the kinetic energy: we imagine the rigidity constraint to be ideal (e.g., as realized by internal central forces in the limit of infinite rigidity, as mentioned in the section "Lagrange and Hamilton forms of equations of motion"). The angular velocity of the rigid motion is defined by

[43] $\mathit{\omega }={\stackrel{.}{\theta }}_0\mathit{n}+{\stackrel{.}{\phi }}_{0}z+{\stackrel{.}{\psi }}_0{\mathit{i}}_3$

expressing that a generic infinitesimal motion must consist of a variation of the three Euler angles and, therefore, it has to be a rotation of speeds ${\stackrel{.}{\theta }}_0,{\stackrel{.}{\phi }}_0,{\stackrel{.}{\psi }}_0$ around the axes n , z , i ₃ as shown in Figure 3 .

Let (ω ₁, ω ₂, ω ₃) be the components of ω along the principal axes i ₁, i ₂, i ₃: for brevity, the latter axes will often be called 1, 2, 3 . Then the angular momentum M , with respect to the pivot point O, and the kinetic energy K can be checked to be

[44] $\begin{array}{l}\mathit{M}=I_1{\omega }_1{\mathit{i}}_1+I_2{\omega }_2{\mathit{i}}_2+I_3{\omega }_3{\mathit{i}}_3\\ K=\frac{1}{2}(I_1{\omega }_1^2+I_2{\omega }_2^2+I_2{\omega }_3^2)\end{array}$

and are constants of motion. From Figure 3 it follows that ${\omega }_1={\stackrel{.}{\theta }}_{0}cos{\psi }_0+{\stackrel{.}{\phi }}_{0}sin{\theta }_{0}sin{\psi }_0,{\omega }_2=-{\stackrel{.}{\theta }}_{0}sin{\psi }_0+{\stackrel{.}{\phi }}_{0}sin{\theta }_{0}cos{\psi }_0$ and ${\omega }_3={\stackrel{.}{\phi }}_{0}cos{\theta }_0+{\stackrel{.}{\psi }}_0$ , so that the Lagrangian, uninspiring at first, is

[45] $\begin{array}{l}\mathcal{L}\stackrel{\text{def}}{=}\frac{1}{2}{I}_1({\stackrel{.}{\theta }}_{0}cos{\psi }_0+{\stackrel{.}{\phi }}_{0}sin{\theta }_{0}sin{\psi }_0{)}^2\\ +\frac{1}{2}{I}_2(-{\stackrel{.}{\theta }}_{0}sin{\psi }_0+{\stackrel{.}{\phi }}_{0}sin{\theta }_{0}cos{\psi }_0{)}^2\\ +\frac{1}{2}{I}_3({\stackrel{.}{\phi }}_{0}cos{\theta }_0+{\stackrel{.}{\psi }}_0{)}^2\end{array}$

Angular momentum conservation does not imply that the components ω _j are constants because i ₁, i ₂, i ₃ also change with time according to

$\frac{\text{d}}{\text{d}t}{\mathit{i}}_j=\mathit{\omega }\wedge {\mathit{i}}_j,j=1,2,3$

Hence, $\stackrel{.}{\mathit{M}}=0$ becomes, by the first of [44] and denoting I ω = (I ₁ ω ₁, I ₂ ω ₂, I ₃ ω ₃), the Euler equations $I\stackrel{.}{\mathit{\omega }}+\mathit{\omega }\wedge I\mathit{\omega }=0$ , or

[46] $\begin{array}{l}{I}_1{\stackrel{.}{\omega }}_1=(I_2-I_3){\omega }_2{\omega }_3\\ I_2{\stackrel{.}{\omega }}_2=(I_3-I_1){\omega }_3{\omega }_1\\ I_3{\stackrel{.}{\omega }}_3=(I_1-I_2){\omega }_1{\omega }_2\end{array}$

which can be considered together with the conserved quantities [44].

Since angular momentum is conserved, it is convenient to introduce the laboratory frame (O; x ₀, y ₀, z ₀) with fixed axes x ₀, y ₀, z ₀ and (see Figure 4 ):

1.

(O; x , y , z ), the momentum frame with fixed axes, but with z -axis oriented as M , and x -axis coinciding with the node (i.e., the intersection) of the x ₀ – y ₀ plane and the x – y plane (orthogonal to M ). Therefore, x , y , z is determined by the two Euler angles ζ, γ of (O; x , y , z ) in (O; x ₀, y ₀, z ₀);

2.

(O; 1,2,3), the comoving frame, that is, the frame fixed with the body, and with unit vectors i ₁, i ₂, i ₃ parallel to the principal axes of the body. The frame is determined by three Euler angles θ ₀, φ ₀, ψ ₀;

3.

the Euler angles of (O; 1, 2, 3) with respect to (O; x , y , z ), which are denoted θ, φ, ψ;

4.

G, the total angular momentum: $G^2={\sum }_j{I}_j^2{\omega }_j^2$ ;

5.

M ₃, the angular momentum along the z ₀ axis; M ₃ = G cos ζ; and

6.

L, the projection of M on the axis 3, L = G cos θ.

Figure 4. The laboratory frame, the angular momentum frame, and the comoving frame (and the Deprit angles).

The quantities G, M ₃, L, φ, γ, ψ determine θ ₀, φ ₀, ψ ₀ and ${\stackrel{.}{\theta }}_0,{\stackrel{.}{\phi }}_0,{\stackrel{.}{\psi }}_0$ , or the $p_{{\theta }_0},p_{{\phi }_0},p_{{\psi }_0}$ variables conjugated to θ ₀, φ ₀, ψ ₀ as shown by the following comment.

Considering Figure 4 , the angles ζ, γ determine location, in the fixed frame (O; x ₀, y ₀, z ₀) of the direction of M and the node line m , which are, respectively, the z -axis and the x -axis of the fixed frame associated with the angular momentum; the angles θ, φ, ψ then determine the position of the comoving frame with respect to the fixed frame (O; x , y , z ), hence its position with respect to (O; x ₀, y ₀, z ₀), that is, (θ ₀, φ ₀, ψ ₀). From this and G, it is possible to determine ω because

[47] $\begin{array}{l}cos\theta =\frac{I_3{\omega }_3}{G},tan\psi =\frac{I_2{\omega }_2}{I_1{\omega }_1}\\ {\omega }_2^2=I_2^{-2}(G^2-I_1^2{\omega }_1^2-I_3^2{\omega }_3^2)\end{array}$

and, from [43], ${\stackrel{.}{\theta }}_0,{\stackrel{.}{\phi }}_0,{\stackrel{.}{\psi }}_0$ are determined.

The Lagrangian [45] gives immediately (after expressing ω , i.e., n , z , i ₃ , in terms of the Euler angles θ ₀, φ ₀, ψ ₀) an expression for the variables $p_{{\theta }_0},p_{{\phi }_0},p_{{\psi }_0}$ conjugated to θ ₀, φ ₀, ψ ₀:

[48] $p_{{\theta }_0}=\mathit{M}\cdot {\mathit{n}}_0,p_{{\phi }_0}=\mathit{M}\cdot z_0,p_{{\psi }_0}=\mathit{M}\cdot {\mathit{i}}_3$

and, in principle, we could proceed to compute the Hamiltonian.

However, the computation can be avoided because of the very remarkable property (Deprit), which can be checked with some patience, making use of [48] and of elementary spherical trigonometry identities,

[49] $\begin{array}{l}{M}_3\text{d}\text{γ}+G\text{d}\phi +L\text{d}\psi \\ =p_{{\phi }_0}\text{d}{\phi }_0+p_{{\psi }_0}\text{d}{\psi }_0+p_{{\theta }_0}\text{d}{\theta }_0\end{array}$

which means that the map $((M_3,\text{γ}),(L,\psi ),(G,\phi ))\leftrightarrow ((p_{{\theta }_0},{\theta }_0)(p_{{\phi }_0},{\phi }_0),(p_{{\psi }_0},{\psi }_0))$ is a canonical map. And in the new coordinates, the kinetic energy, hence the Hamiltonian, takes the form

[50] $K=\frac{1}{2}\left[\frac{L^2}{I_3}+(G^2-L^2)\left(\frac{{sin}^2\psi }{I_1}+\frac{{cos}^2\psi }{I_2}\right)\right]$

This again shows that G, M ₃ are constants of motion, and the L, ψ variables are determined by a quadrature, because the Hamilton equation for ψ combined with the energy conservation yields

[51] $\begin{array}{ll}\stackrel{.}{\psi }=& \pm \left(\frac{1}{I_3}-\frac{{sin}^2\psi }{I_1}-\frac{{cos}^2\psi }{I_2}\right)\\ & \times \sqrt{\frac{2E-G^2\left(\frac{{sin}^2\psi }{I_1}+\frac{{cos}^2\psi }{I_2}\right)}{\frac{1}{I_3}-\frac{{sin}^2\psi }{I_1}-\frac{{cos}^2\psi }{I_2}}}\end{array}$

In the integrability region, this motion is periodic with some period T _L (E, G). Once ψ(t) is determined, the Hamilton equation for φ leads to the further quadrature

[52] $\stackrel{.}{\phi }=\left(\frac{{sin}^2\psi (t)}{I_1}+\frac{{cos}^2\psi (t)}{I_2}\right)G$

which determines a second periodic motion with period T _G (E, G). The γ, M ₃ are constants and, therefore, the motion takes place on three-dimensional invariant tori ${\mathcal{T}}_{E,G,M_3}$ in phase space, each of which is "always" foliated into two-dimensional invariant tori parametrized by the angle γ which is constant (by [50], because K is M ₃-independent): the latter are in turn foliated by one-dimensional invariant tori, that is, by periodic orbits, with E, G such that the value of T _L (E, G)/T _G (E, G) is rational.

Note that if I ₁ = I ₂ = I, the above analysis is extremely simplified. Furthermore, if gravity g acts on the system the Hamiltonian will simply change by the addition of a potential −mgz if z is the height of the center of mass. Then (see Figure 4 ), if the center of mass of the body is on the axis i ₃ and z = h cos θ ₀, and h is the distance of the center of mass from O, since cos θ ₀ = cos θ cos ζ − sin θ sin ζ cos φ, the Hamiltonian will become $\mathcal{H}=K-\text{<mi mathvariant="italic" is="true">mgh</mi>}cos{\theta }_0$ or

[53] $\begin{array}{ll}\mathcal{H}& =\frac{G^2}{2I_3}+\frac{G^2-L^2}{2I}-\text{<mi mathvariant="italic" is="true">mgh</mi>}(\frac{M_{3}L}{G^2}-{\left(1-\frac{M_3^2}{G^2}\right)}^{1/2}\\ & \times {\left(1-\frac{L^2}{G^2}\right)}^{1/2}cos\phi )\end{array}$

so that, again, the system is integrable by quadratures (with the roles of ψ and φ "interchanged" with respect to the previous case) in suitable regions of phase space. This is called the Lagrange's gyroscope.

A less elementary integrable case is when the inertia moments are related as I ₁ = I ₂ = 2I ₃ and the center of mass is in the i ₁– i ₂ plane (rather than on the i ₃-axis) and only gravity acts, besides the constraint force on the pivot point O; this is called Kowalevskaia's gyroscope.

For more details, see Gallavotti (1983).

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DISTRIBUTIONS

YVONNE CHOQUET-BRUHAT , CÉCILE DEWITT-MORETTE , in Analysis, Manifolds and Physics, 2000

Answer 3

For angular momentum l the above functional is replaced by

$\begin{array}{l}\frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}\text{d}r\left\{{\left(\text{d}u\left(r\right)/\text{d}r\right)}^2+\left[l\left(l+1\right)r^2\right]u^2\left(r\right)\right\}}}{{\left({\displaystyle \underset{0}{\overset{\infty }{\int }}\text{d}r{u}^6\left(r\right)/r^4}\right)}^{1/3}}\\ \frac{{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\text{dz}\left\{{\left(\text{d}\phi \left(z\right)/\text{dz}\right)}^2+{\left(l+\frac{1}{2}\right)}^2{\phi }^2\left(z\right)\right\}}}{{\left({\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\text{dz}{\phi }^6\left(z\right)}\right)}^{1/3}}=G_l\left(\phi \right).\end{array}$

Since G _l (ϕ) differs from G ₀(ϕ) only through the replacement ϕ²/4 → ϕ²(2l + 1)²/4, we may get back G ₀(ϕ) from G _l (ϕ) by a scale transformation: $\varphi (z)=\tilde{\varphi }$ this leads to

$G_l\left(\varphi \right)={\left(2l+1\right)}^{4/3}{G}_0\left(\varphi \right).$

Since the infimum of G ₀(ϕ) is known, we obtain finally

$v_1\le {\tilde{S}}_3\frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}\text{d}r{r}^2}{\left|V\right|}_{-}^{3/2}}{{(2l+1}^2}$

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C-H bend motion of acetylene

GUOZHEN WU , in Nonlinearity and Chaos in Molecular Vibrations, 2005

12.8 Geometric interpretation of vibrational angular momentum

The vibrational angular momentum l caused by the two C—H bends is l ₊′− l ₋′ or (n ₄₊′ + n ₅₊′)-(n _4- + n _5-′). It is the difference between the '+' and '-' rotations.Fig.12.8 shows the case of L4 with N _b = 6, l = 2. The '+' motion possesses larger rotational momentum or angular momentum since for the '-' motion, the phase angle between the two C—H bends can range between -π and π, i.e., with no definite phase angle, while for the '+' motion, the phase angle is centred around π (or -π), i.e., in trans configuration with nonzero l. This is depicted in Fig.12.9.

Fig.12.8. The phase relation for the case with nonzero vibrational angular momentum

Fig.12.9. For the '-' motion (a), the phase angle between the two C—H bends can range between −π and π, i.e., with no definite phase angle, while for the '+' motion (b), the phase angle is centred around π (or -π).

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Angular Momentum Methods for Atoms

Valerio Magnasco , in Elementary Methods of Molecular Quantum Mechanics, 2007

9.4.1 Clebsch–Gordan Coefficients and Wigner 3-j and 9-j Symbols

Two angular momentum vectors, | l ₁ m ₁〉 and |l ₂ m2〉, can be coupled to a resultant |LM〉 by the relation:

(63) $|LM〉=\sum _{m_1{m}_2}|l_1{m}_1〉|l_2{m}_2〉〈l_1{m}_1{l}_2{m}_2||LM〉,$

where 〈l ₁ m ₁ l ₂ m ₂ |LM〉 is a real number called vector coupling or Clebsch-Gordan coefficient, and we used Dirac notation. The transformation (63) is unitary, the inverse being:

(64) $|l_1{m}_1〉|l_2{m}_2〉=\sum _{LM}|LM〉〈LM||l_1{m}_1{l}_2{m}_2〉,$

where the coefficient 〈LM|l ₁ m ₁ l ₂ m ₂〉 is the complex conjugate of the corresponding coefficient 〈l ₁ m ₁ l ₂ m ₂ |LM〉.

In recent Literature, the Clebsch–Gordan coefficient is usually expressed in terms of the more symmetric Wigner 3-j symbol through the relation (Brink and Satchler, 1993):

(65) $\begin{array}{l}〈l_1{m}_1{l}_2{m}_2|LM〉=(-1)l_1-l_2+M\sqrt{2L+1}\left(\begin{array}{c}\begin{array}{ccc}{l}_1& l_2& L\\ m_1& m_2& -M\end{array}\\ 3-j\end{array}\right).\\ \text{Clebsch}-\text{Gordan}\end{array}$

The Wigner 3- j symbol $\left(\begin{array}{ccc}{l}_1& l_2& l_3\\ m_1& m_2& m_3\end{array}\right)$ is non-zero provided:

(66) $\begin{array}{l}{m}_1+m_2+m_3=0\\ |l_{\mu }-l_v|\le l_{\gamma }\le (l_{\mu }+l_v),\end{array}$

and has the following general expression (Rose, 1957; Brink and Satchler, 1993):

(67) $\begin{array}{l}\left(\begin{array}{ccc}{l}_1& l_2& l_3\\ m_1& m_2& m_3\end{array}\right)={\Delta }_{m_1+m_2+m_3,0}\times (-1)l_1+m_2-m_3\\ \times \sqrt{\frac{(l_1+l_2-l_3)!(l_1-l_2+l_3)!(-l_1+l_2+l_3)!(l_3+m_3)!(l_3-m_3)!}{(l_1+l_2+l_3+1)!(l_1+m_1)!(l_1-m_1)!(l_2+m_2)!(l_2-m_2)!}}\\ \times \sum _{\lambda }(-1)\lambda \frac{(l_1-m_1+\lambda )!(l_2+l_3+m_1-\lambda )!}{\lambda !(-l_1+l_2+l_3-\lambda )!(l_1-l_2+m_3+\lambda )!(l_3-m_3-\lambda )!},\end{array}$

where:

(68) $\text{max}(0,l_2-l_1-m_3)\le \lambda \le \text{min}(l_3-m_3,l_2-l_1+l_3).$

The summation over λ is limited to all integers giving non-negative factorials. The Wigner 3-j symbols are today available as standards on the Mathematica software (Wolfram, 1996), where they are evaluated in a sophisticated way in terms of hypergeometric functions (Abramowitz and Stegun, 1965) for integer or half-integer (spin) values of (l, m).

The vector coupling of three and four angular momenta implies use of Wigner 6-j or Wigner 9-j symbols, respectively.

A Wigner 9-j symbol is given as:

(69) $\left\{\begin{array}{ccc}{l}_1& l_2& L\\ {l^{\prime }}_1& {l^{\prime }}_2& L^{\prime }\\ L_1& L_2& \lambda \end{array}\right\}.$

It is invariant under interchange of rows and columns, and enjoys the property:

(70) $\begin{array}{c}\left(\begin{array}{ccc}\lambda & L& L^{\prime }\\ 0& 0& 0\end{array}\right)\left\{\begin{array}{ccc}\lambda & L& L^{\prime }\\ L_1& l_1& {l^{\prime }}_1\\ L_2& l_2& {l^{\prime }}_2\end{array}\right\}\\ =\sum _{M_1{M}_2}\left(\begin{array}{ccc}{L}_1& L_2& L\\ M_1& M_2& 0\end{array}\right)\\ \times \sum _{m_1{m}_2}\sum _{{m^{\prime }}_1{m^{\prime }}_2}\left(\begin{array}{ccc}{l}_1& l_2& L\\ m_1& m_2& 0\end{array}\right)\left(\begin{array}{ccc}{l^{\prime }}_1& {l^{\prime }}_2& L^{\prime }\\ {m^{\prime }}_1& {m^{\prime }}_2& 0\end{array}\right)\left(\begin{array}{ccc}{l}_1& {l^{\prime }}_1& L_1\\ m_1& {m^{\prime }}_1& M_1\end{array}\right)\\ \times \left(\begin{array}{ccc}{l}_2& {l^{\prime }}_2& L_2\\ m_2& {m^{\prime }}_2& M_2\end{array}\right),\end{array}$

that reduces to summations over Wigner 3-j symbols, more easily calculable on Mathematica. Used backwards, relation (70) is said to express the contraction of summations of 3-j symbols to 9-j (Brink and Satchler, 1993). Expression (70) is met in the spherical tensor expansion of the product $\frac{1}{r^{12}}\cdot \frac{1}{r_{1^{\prime }}{2}^{\prime }}$ , which occurs in the second-order theory of long-range intermolecular forces (Ottonelli, 1998; Magnasco and Ottonelli, 1999).

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