Moment quantization

Both the orbital and the effective spinning motions of the electron have associated angular moments quantized in units of ii = 1.055 x 10-34 Js. It is an elementary exercise in physics to show that the relationship between the magnetic dipole moment /< and the angular momentum L for a moving particle of mass m and charge Q is... 

It is well established that wavelet based methods are more efiicient than Fourier analysis in representing transient, short scale, effects (Chui (1992)). However, for a given quantum system, what are its most important, localized, features One important set of localized structures correspond to all of the turning points, including those in the complex plane (or turning hyper-surfaces, for multidimensional problems). Indeed, from a Moment Quantization perspective, for (multidimensional) rational fraction, bound state potentials, they completely define the system through their analytic contribution. In several recent articles. Handy and Murenzi (1996 - 1999) have established the complementary relationship between CWT and Moment Quantization analysis. 

Continuous Wavelet Transform (CWT) theory, and other related multiscale methods, come about naturally within a Moment Quantization (Sec. 1.2) based formulation for the Schrodinger equation. From this perspective, the more important structures are the scaling transforms, in terms... 

Turning Point Quantization and Scalet- Wavelet Analysis 209 1.2 Moment Quantization... 

As noted in the Introduction, in this presentation, we will limit our formalism and analysis to one dimensional, rational fraction, bound state potentials, for simplicity. Our intention is to motivate what we perceive to be the principal importance of Continuous Wavelet Transform (CWT) theory in quantum mechanics, that of facilitating the multiscale analysis of singular systems, particularly those associated with multiple (complex) turning point interactions. The understanding of these issues rests on a clear appreciation of the significant role Moment Quantization methods bear on the multiscale analysis of quantum operators. 

The determination of the discrete energy states, through the ME representation, is referred to as Moment Quantization. As previously mentioned, various methods have been proposed by Blankenbeckler (1980), Killinbeck et al (1985), and Fernandez (1993). However, those proposed by Handy, Bessis, and co-workers (1985, 1988a,b), have the greatest relevance to wavelet analysis, as well as introduce some fundamentally important theoretical features. 

These considerations are significant for two reasons. The first (discussed more fully later on), is that one can show that the wavefunction - wavelet reconstruction (inversion) relation converges fastest at the inflection points, Xi- The second is that Handy 8md Bessis Moment Quantization method explicitly targets the nodal structure of the wavefunction. We explain this below. 

In addition to the Hankel-Hadamard moment quantization method for the bosonic ground state energy (as well as other states, provided their nodes axe known), two other, powerful, moment quantization methods have been recently developed and used to generate the energy and wavefunction for arbitrary states. Both use the same basis representation, that corresponding to the Multiscale Reference Function (MRF) formalism. 

Although all of the preceding methods work very well, and have proven, thus far, more computationally expedient than the wavelet formalism to be presented, they do not explicitly involve a scale parameter dependence, as does the wavelet representation. We present this, from a Moment Quantization perspective. 

The scalet equation representation incorporates the Moment Quantization formalism, with its explicit anal rtic (regular) dependence on the kinetic energy expansion parameter, c, and all the (complex) turning points. 

The previous analysis does not deal with designing the best dual-wavelet combination for a given problem. Clearly, the more localized in space and scale (around the turning points) is the wavelet transform, the more efficient it will be for implementing the above TPQ analysis. Methods for doing this, based on our imderlying Moment Quantization perspective, are being developed. 

Handy, C. R. and Murenzi, M., (1999) Moment Quantization and (A-adic) Discrete-Continuous Wavelet Transform Theory , J. Phys. A Math. Gen. 32, 8111. 

The calculated energy of interaction of an atomic moment and the Weiss field (0.26 uncoupled conduction electrons per atom) for magnetic saturation is 0.135 ev, or 3070 cal. mole-1. According to the Weiss theory the Curie temperature is equal to this energy of interaction divided by 3k, where k is Boltzmann s constant. The effect of spatial quantization of the atomic moment, with spin quantum number S, is to introduce the factor (S + 1)/S that is, the Curie temperature is equal to nt S + l)/3Sk. For iron, with 5 = 1, the predicted value for the Curie constant is 1350°K, in rough agreement with the experimental value, 1043°K. 

Some years later a more thorough discussion of the motion of pairs of electrons in a metal was given by Cooper,7 as well as by Abrikosov8 and Gor kov,9 who emphasized that the effective charge in superconductivity is 2e, rather than e. The quantization of flux in units hc/2e in superconducting metals has been verified by direct experimental measurement of the magnetic moments induced in thin films.10 Cooper s discussion of the motion of electron pairs in interaction with phonons led to the development of the Bardeen-Cooper-Schrieffer (BCS) theory, which has introduced great clarification in the field of superconductivity.2... 

When a magnetic field is applied to an electron or nuclear spin, the spin quantization axis is defined by the field direction. Spin magnetic moments... 

In contrast to ESR spectroscopy, which can only be used to study species with unpaired electrons, NMR spectroscopy is applicable to the investigation of all polymer samples. Nuclei with non-zero total nuclear spin (e.g., 1H, l3C, 19F, 14N) will have a magnetic moment which will interact with an external magnetic field resulting in quantized energy levels. Transitions between these energy levels form the basis of NMR spectroscopy. 1H and 13C... 

At temperatures above absolute zero, all atoms in molecules are in continuous vibration with respect to each other [26]. Infrared spectroscopy is an absorption spectroscopy. Two primary conditions must be fulfilled for infrared absorption to occur. First, the energy of the radiation must coincide with the energy difference between excited and ground states of the molecule, i.e., it is quantized (Fig. 14.2). Radiant energy will then be absorbed by the molecule, increasing its natural vibration. Second, the vibration must entail a change in the electrical dipole moment (Fig. 14.3). 

In addition, a relativistic treatment of the electron introduces a fourth quantum number, the spin, m, with ms = j. This is because every electron has associated with it a magnetic moment which it quantized in one of two possible orientations parallel with or opposite to an applied magnetic field. 

The Point Charge Concept and the Related Divergence Quantized Charged Equilibrium B.4.1. Conditions on Spin and Magnetic Moment... 

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