Oscillator isotropic

The simplest case of an oscillating isotropic plate includes an infinite set of waves known as Lamb waves. Two sets of waves can be distinguished symmetric waves (So, Si, etc.) with particle displacements symmetric about the neutral plane and antisymmetric waves (Ao, Ai, etc.), whose displacements have odd symmetry. For sufficiently thin plates only two waves occur, Aq and Sq. Thinner membranes exhibit... 

Meath W J and Kumar A 1990 Reliable isotropic and anisotropic dipole dispersion energies, evaluated using constrained dipole oscillator strength techniques, with application to interactions involving H2, N2 and the rare gases Int. J. Quantum Chem. Symp. 24 501... 

A related measure of the intensity often used for electronic spectroscopy is the oscillator strengdi,/ This is a dimensionless ratio of the transition intensity to tliat expected for an electron bound by Hooke s law forces so as to be an isotropic hanuonic oscillator. It can be related either to the experimental integrated intensity or to the theoretical transition moment integral ... 

Explicit forms of the coefficients Tt and A depend on the coordinate system employed, the level of approximation applied, and so on. They can be chosen, for example, such that a part of the coupling with other degrees of freedom (typically stretching vibrations) is accounted for. In the space-fixed coordinate system at the infinitesimal bending vibrations, Tt + 7 reduces to the kinetic energy operator of a two-dimensional (2D) isotropic haiinonic oscillator. 

Now, consider the general case of a V2 multiply excited degenerate vibrational level where V2 > 2, which is dealt with by solving the Schrddinger equation for the isotropic 2D harmonic oscillator with the Hamiltonian assuming the fonn [95]... 

The classical scheme for dichroism measurements implies measuring absorbances (optical densities) for light electric vector parallel and perpendicular to the orientation of director of a planarly oriented nematic or smectic sample. This approach requires high quality polarizers and planarly oriented samples. The alternative technique [50, 53] utilizes a comparison of the absorbance in the isotropic phase (Dj) with that of a homeotropically oriented smectic phase (Dh). In this case, the apparent order parameter for each vibrational oscillator of interest S (related to a certain molecular fragment) may be calculated as S = l-(Dh/Di) (l/f), where / is the thermal correction factor. The angles of orientation of vibrational oscillators (0) with respect to the normal to the smectic layers may be determined according to the equation... 

An isotropic oscillator is one for which the restoring force is independent of the direction of the displacement and depends only on its magnitude. For such an oscillator, the directional force constants are equal to one another... 

Derive the result that the degeneracy of the energy level E for an isotropic three-dimensional harmonic oscillator is (n + l)(n + 2)/2. 

The energy of the isotropic harmonic oscillator in three dimensions can be written as... 

The Drude oscillators are typically treated as isotropic on the atomic level. However, it is possible to extend the model to include atom-based anisotropic polarizability. When anisotropy is included, the harmonic self-energy of the Drude oscillators becomes... 

Spinning a crystal during measurement of WAXS patterns is an old method that turns any scattering pattern into a fiber pattern. The rotational axis becomes the principal axis. Thereafter isotropization of the scattering data is simplified because the mathematical treatment can resort to fiber symmetry of the measured data. In the literature the method is addressed as the rotating-crystal method or oscillating-crystal method. 

Figure 8.9. The scattering curve of an isotropic ideal two-phase system after multiplication by i 4 (cf. Eq. (8.43)). The Porod region in which the oscillations are almost faded away is generally beginning after the 2nd order of the long period reflection... 

For an isotropic lattice, we take advantage of the transform given in Eqs. (A1.73) and (A1.74) to derive, from Eq. (A1.95), three equations in frequencies of local and quasilocal vibrations of an impurity oscillator ... 

Figure 2.6 The potential V(r) that corresponds to the dynamical symmetry (I). The potential is nonrigid because [cf. Eq. (2.113)] the rotational spacings are comparable to the vibrational ones. Tn the harmonic limit V(r) is the potential of an isotropic harmonic oscillator.

S is the scattering vector, Mj is the atomic displacement parameter in this simplified notation assumed to be isotropic, 6 is the scattering angle, and 1 the wavelength of the incident radiation. The atomic displacement depends on the temperature, and hence so does the Debye-Waller factor. If an atom is modeled by a classical oscillator, then the atomic displacement would change linearly with temperature ... 

Consider the waves scattered by isotropic dipole oscillators in the thin slab of matter shown in Fig. 9.3 only part a is of concern at the moment. These waves add vectorially at point P to produce the resultant forward-scattered wave s the important point, which is by no means obvious yet, is that this resultant scattered wave is phase shifted 90° relative to the incident wave (in addition to the phase shift between the oscillators and the incident wave). The background necessary to show this has been presented in Chapter 3 Fig. 3.8 is similar to Fig. 9.3 except that in the former the scatterers were arbitrary particles. The transmitted field t at P is the vector sum of the incident field and the fields scattered by all the oscillators. If we assume that the direction of vibration of the incident wave is not rotated as it propagates through the slab, the transmitted field is given by (3.39) ... 

What are the electric-dipole selection rules for a three-dimensional harmonic oscillator exposed to isotropic radiation ... 

Solve the Schrodinger equation for the two-dimensional isotropic harmonic oscillator using plane polar coordinates. First show that... 

We have considered only one particular degenerate vibrational level. The general case is dealt with by solving the isotropic two-dimensional harmonic oscillator in plane polar coordinates (Problem 6.19). The result is... 

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