Angular Momentum and Rotations

Problems involving rotation occur frequently in the quantum theory. This section discusses two models which look rather artificial, but which serve to introduce some important ideas. They have a direct application to the rotations of molecules, but are especially useful in the context of atomic structure, discussed in Chapters 4 and 5. 

We first consider a particle moving at constant speed in a circle in the (x,y) plane. It is convenient to use the polar coordinates illustrated in Fig. 3.10. In terms of the cartesian coordinates (x,y) we have 

We are interested in the case where r is constant, and 0 changes at a constant rate. Let 

An important quantity in rotating systems is the angular momentum. For a particle moving in the (x,y) plane its definition is 

If the particle is not confined to the (jr,y) plane we need to use the spherical polar coordinates (see Fig. 3.11), defined by 

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There are two primary reasons for looking at rotations in NMR of liquid crystals. First, rotational motion of the spin-bearing molecules determines, in part, relaxation behavior of the spin system. Second, one or more r.f. pulse(s) in NMR experiments has the effect of rotating the spin angular momentum of the spin system. Therefore, it is necessary to deal with spatial rotations of the spin system and with spin rotations. The connection between rotations and angular momentum (j) is expressed by a rotation operator... 

Table 1.1. Cross-sections of rotational energy and angular momentum...

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

The physical meaning of and f L.., is obvious they govern the relaxation of rotational energy and angular momentum, respectively. The former is also an operator of the spectral exchange between the components of the isotropic Raman Q-branch. So, equality (7.94a) holds, as the probability conservation law. In contrast, the second one, Eq. (7.94b), is wrong, because, after substitution into the definition of the angular momentum correlation time... 

Gillen K. T., Douglas D. S., Malmberg M. S., Maryott A. A. NMR relaxation study of liquid CCI3F. Reorientational and angular momentum correlation times and rotational diffusion, J. Chem. Phys. 57, 5170-9 (1972). 

This chapter presents a physical description of the interaction of flames with fluids in rotating vessels. It covers the interplay of the flame with viscous boundary layers, secondary flows, vorticity, and angular momentum and focuses on the changes in the flame speed and quenching. There is also a short discussion of issues requiring further studies, in particular Coriolis acceleration effects, which remain a totally unknown territory on the map of flame studies. 

Great care has to be given to the physics of rotation and to the treatment of its interaction with mass loss. For differentially rotating stars, the structure equations need to be written differently [9] than for solid body rotation. For the transport of the chemical elements and angular momentum, we consider the effects of shear mixing, meridional circulation, horizontal turbulence and in the advanced stages the dynamical shear is also included. Caution has to be given that advection and diffusion are not the same physical effect. 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

As a consequence of the transformation, the equation of motion depends on three extra coordinates which describe the orientation in space of the rotating local system. Furthermore, there are additional terms in the Hamiltonian which represent uncoupled momenta of the nuclear and electronic motion and moment of inertia of the molecule. In general, the Hamiltonian has a structure which allows for separation of electronic and vibrational motions. The separation of rotations however is not obvious. Following the standard scheme of the various contributions to the energy, one may assume that the momentum and angular momentum of internal motions vanish. Thus, the Hamiltonian is simplified to the following form. 

The total vibrational energy is a sum of energies of 3N-6 distinct harmonic oscillators. Indeed, 3N-6 is the final number of coordinates in Equation 8. Namely, the number of 3N+3 coordinates of the initial equation has been reduced by three through the elimination of internal rotations. Furthermore, the equation of nuclear motion (mainly its potential) has to be invariant under rotations and translations of a molecule as a whole (which is equivalent to the momentum and angular momentum preservation laws). The latter requirement leads to a further reduction of the number of coordinates by six (five in the case of linear molecules for which there are only two possible rotations). 

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

In this chapter we discuss only the scalar aspect of rotational excitation, i.e., the forces which promote rotational excitation and how they show up in the final state distributions. The simple model of a triatomic molecule with total angular momentum J = 0, outlined in Section 3.2, is adequate for this purpose without concealing the main dynamical effects with too many indices and angular momentum coupling elements. The vector properties and some more involved topics will be discussed in Chapter 11. 

The (inertial) I-frame is an element of ordinary 3D (real) space it is origin localizable and can be rotated with respect to a second frame (e.g., Euler angles). Abstract quantum states are necessarily invariable to translations and rotations of such frames. Projected quantum states must be invariant to origin translations and I-frame rotations linear and angular momentum... 

Moment of Inertia. We next discuss momentum, rotation, torque, moment of inertia, and angular momentum. A body with velocity v and mass m is defined to have (linear) momentum p ... 

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