Singularity, quantum waves

Another possibility [16,17] for testing the reality of the quantum waves derives directly from de Broglie causal theory. As we have seen, in this approach, the quantum particle is composed of a wave plus a singularity. These two composing entities have different properties when interacting with matter or with the surrounding subquantum medium. 

Finally a few words would be perhaps not superfluous about the relation of our results to the quantum mechanics of wave fields. It might appear that all calculations based on p.m. in the latter theory are not justifiable from the mathematical standpoint. But it must be remarked that even the operators themselves used in this theory are quite singular and at present do not accept rigorous mathematical taeatment. In such a state of affairs it would be premature to discuss the validity of p,m. in particular. 

As can be seen from the equations (21)-(22) and (23)-(24), there is an essential difference between the representations of plane and multipole waves of photons. In particular, a monochromatic plane wave of photons is specihed by only two different quantum numbers a = x, y, describing the linear polarization in Cartesian coordinates. In turn, the monochromatic multipole photons are described by much more quantum numbers. Even in the simplest case of the electric dipole radiation when X = E and j = 1, we have three different states of multipole photons in (23) with m = 0, 1. Besides that, the plane waves of photons have the same polarization a everywhere, while the states of multipole photons have given m. It is seen from (24) that, in this case, the polarization described by the spin index p can have different values at different distances from the singular point. In Section V we discuss the polarization properties of the multipole radiation in greater detail. 

The anthropic principle is by no means the only baseless theoretical concept of standard cosmology. Perhaps equally prominent and equally spurious is quantum gravity. Although nobody has formulated such a theory it is being referenced as imbued with the capacity to generate universal wave functions that solve the problems around space-time singularities and cosmic probabilities. This could be the only instance where the solutions of a non-existent equation override empirical observations. 

So far we have specified the structure of the S/kPF and SAP pair energies by means of the sets of standard quantum numbers T and T, respectively. However, in studies of the asymptotic behavior of the SAP energies it turns out to be useful to supplement this specification in an identical way as Kutzelnigg and Morgan have done in the case of two-electron states of He-like systems [21], To characterize the behavior of the first-order wave function at the singular point, these authors have ingeniously employed the idea of natural and unnatural parity of two-electron states, which has been used in atomic spectroscopy studies for the first time by Baneqee [28]. The parity of a two-electron state corresponding to the quantum number L is defined as... 

The solutions for k = 1 have a singularity at the origin. This could be a serious problem in quantum chemistry where Gaussian basis functions are commonly used to expand the wave function. The cusp that has to be represented in the Schrodinger solutions is now replaced by a singularity, which will inevitably make greater demands on the flexibility of the basis. 

Wave Mechanics n [plural but singular or plural in con-struction] (1926) The branch of physics which describes the behavior of small particles by assigning wavelike properties to them, also known as Quantum Mechanics. 

Equation (24) can even be modified to allow for the calculation of total cross sections for polar molecules, without resorting to a full laboratory-frame formulation of the scattering problem including dynamical treatment of the rotational degrees of freedom. The idea is to introduce rotational quantum numbers only into the channel momenta in the three-dimensional Bom term in equation (24) (which removes the singularity in the amplitude for forward scattering) but to compute the partial-wave sums entirely within the adiabatic-nuclei picture. This hybrid treatment is justified because the low partial-wave terms which contribute to the sum in equation (24) are insensitive to rotational quantum numbers." ... 

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