§ 20. Quantum Mechanics

Quantum mechanics is based on Planck's blackbody derivation of the energy element (hγ) but Planck uses the kinetic energies of the blackbody surface electrons in the derivation of the energy element (e = hγ) since Planck's constant h = 6.6 × 10-34 m2 kg/s contains the unit of the mass (kg) yet light is composed of massless light particles. The quantum mechanic wave packet that is used to represent a particle structure is formed by the superpositioning of ten probability waves (fig 13) but doubling the lengths of the probability waves that are used to construct a single wave packet results in the formation of a second wave packet. Increasing the lengths of the original probability waves by 100 times produces 100 successive wave packets. The quantum mechanic wave packet does not depict a particle structure. Davisson–Germer (1927) electron scattering experiment is used to justify electron wave interference but the destruction of electrons to form the non-electron fringes of the electron scattering pattern represents the arbitrary destruction of electrons which violates energy conservation. de Broglie's electron matter wave is used to represent the structure of a Bohr atom (fig 14) but de Broglie's continuous electron matter wave conflicts with the particle structure of an electron since an electron that has a particles structure depicted with a matter wave simultaneously exists on both sides of an atom. The atomic electron matter wave contained in a circular path cannot be represented in a Cartesian, cylindrical or spherical coordinate system which results in the atomic electron matter wave transformed into a particle-in-a-box electron matter wave (fig 15) represented with Schrödinger's wave equation,

-(h2/2m)∇"Ψ(x,y,z) + V(x,y,z) + V(x,y,z)Ψ(x,y,z) = EΨ(x,y,z)...................................................58

Schrodinger's wave equation is used to derive a wave function

Ψ = Σ c u exp[(2πEt/h + θ)i]............(Schrodinger, p. 1066)......................................................59

"The wave-function physical means and determines a continuous distribution of electricity in space, the fluctuations of which determine the radiation by laws of ordinary electrodynamics." (Schrödinger, Abstract).

"The fluctuation of the charge will be governed by the Eq. 28, applied to the special case of the hydrogen atom. To find the radiation, that by electrodynamics will originate from these fluctuating charges, we have simply to calculate the rectangular components of the total electric moment by multiplying (28) by x, y, z respectively, then integrating over space, e.g." (Schrodinger, p. 1066).

"1. The theory which is reported in the following pages is based on the very interesting and fundamental researches of L. de Broglie' on what he called "phase-waves" ("ondes de phase") and thought to be associated with the motion'of material points, especially with the motion of an electron or proton. The point of view taken here, which was first published in a series of German papers, is rather that material points consist of, or are nothing but, wave-systems. This extreme conception may be wrong, indeed it does not offer as yet the slightest explanation of why only such wave-systems seem to be realized in nature as correspond to mass-points of definite mass and charge. On the other hand the opposite point of view, which neglects altogether the waves discovered by L. de Broglie and treats only the motion of material points, has led to such grave difficulties in the theory of atomic mechanics —and this after century-long development and refinement— that it seems not only not dangerous but even desirable, for a time at least, to lay an exaggerated stress on its counterpart. In doing this we must of course realize that a thorough correlation of all features of physical phenomena can probably be afforded only by a harmonic union of these two extremes." (Schrödinger, p. 1049-50).

"As an alternative, in 1926 German physicist Max Born sharply refined Schrodinger's interpretation of an electron wave, and it is his interpretation--amplified by Bohr and his colleagues--that is still with us today......He asserted that an electron wave must be interpreted from the standpoint of probability." (Greene, p. 105).

"Just a few months after de Broglie's suggestion, Schrodinger took the decisive step toward this end by determining an equation that governs the shape and the evolution of probability waves, or as they came to be known, wave functions." (Greene, p. 107).

"Schrodinger, de Broglie, and Born explained this phenomenon by associating a probability wave to each electron." (Greene, p. 109).

Schrödinger is structurally representing an electron with a massless electric wave depicted with the wave function but a massless electric wave conflicts with the structure of an electron that has a mass. Schrödinger's electric wave is replaced with an electron probability wave, and represented with a spherical coordinate system but an electron position probability can only represent a positive value or zero and cannot depict a negative value that is required in representing destructive wave interference used to derive the equations of the atomic orbitals, and the representation of a plane wave equation of the probability wave that has a constant maximum amplitude in a spherical coordinate system results in a mathematical implosion that is used with Lagrangian polynomials to derive the equations of the atomic orbitals.

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Quantum mechanics uses a gauge transformation of Maxwell's equations.

"A similar, but more subtle and deep, situation arises in electrodynamics where one can express the (physical) electric and magnetic fields in terms of scalar (ɸ(r,t)) and vector (A(r,t)) potentials via

B(r,t) = ∇ x A(r,t)......................................................................................60

E(r,t) = - ∇ɸ(r,t) - d/dtA(r,t).......................................................................61

....Such a change in potentials is called a gauge transformation, and will be seen to play and important role in the quantum mechanical treatment of charged particle interactions." (Robinett, p.447); (Cohen-Tannoudji, p. 315).

The quantum mechanics gauge transformation is based on Maxwell's equations but representing Maxwell's equations with a gauge potential does not change the fact that Maxwell's equations are derived using Faraday's induction effect that is not luminous nor can the potential of a massless and expanding electromagnetic field be used to represent the particle structure of an atom, ion, nuclei, neutron, proton or electron that has a mass.

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