PP03 Simple Harmonic Motion in Classical and Quantum Phase Space


Simple Harmonic Motion in Classical and Quantum Phase Space

Martin Gibson, August 1, 2013

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In classical mechanics, natural, including man-made, repetitive motion sustained over extended time frames can be studied using the model of Simple Harmonic Motion (SHM) in which an oscillator and its support framework is deemed to be a closed system in that energy, momentum and related properties of the motion are conserved. In fact no system is completely closed and the energy of such motion is in some measure damped or otherwise lost to the background of the system. Still SHM can be closely approximated by driving the oscillation with controlled energy input. The oscillation of displacement and momentum in SHM can be accounted for succinctly with the use of planar phase space (PS2) modeling in which the correspondence of the dynamics of linear oscillation and Uniform Circular Motion (UCM) is utilized. Accounting for energy, force, action, and power oscillation is better handled by graphic modeling of sine and cosine wave functions.

In the first section, these models are briefly recapitulated. In the second section they are applied to an analysis of the energy of the oscillation, through the Hamiltonian and Lagrangian approach, with the graphic development of the action, via both Lagrange and Maupertuis, and the power of the oscillation, and their semi-periodic maximum moments. The two models are synthesized in a three-dimensional phase space that we are calling PS3. In the process of this synthesis it is shown that there is necessarily a component of the dynamics that is present even in the absence of oscillation, an inertial invariant that is both a scalar and vector potential. Finally, the point oscillator of the initial development is replaced by torsion oscillation as a disk, which can be represented by PS2. The synthesis also suggests a quantum application of the modeling. Section three shows the development of rotation of the action and power moments with attendant torques which sustain the oscillation and result in the property of angular momentum, with an invariant Lagrangian as well as Hamiltonian. A review of the nature of body forces and stress or surface forces models PS3 as a system of rotating stress force and corresponding strain, and the model is fully developed as an emergent quantum phenomena driven by an expanding spacetime fabric (STF) coupled with necessary geometric constraints. The neutron is shown to be the resonant state of PS3. Spin and charge as elaborations of the angular momentum is developed, along with beta-decay for both ordinary and anti matter.

The Verification section derives a gravitational quantum, Newton’s gravitational constant and law, ties beta-decay to cosmic expansion and thereby predicts the Hubble rate, which is shown to be an exponential rate. In the process the reason for the neutron-electron mass ratio is developed along with the nature of the missing mass of beta decay. Finally, the value of elementary charge is derived, with some interesting observations about the structure of the fine structure constant.

The Conclusion section waxes philosophical, concludes the PS3 model deserves a proper vetting, and the Asides offer supporting information.