The Cosmic Game ( PDFDrive.com )

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Cosmic Game © Douglass A. White, 2012 v151207 175 Significance The relation between properties of mass and their associated physical constants. Every massive object is believed to exhibit all five properties. However, due to extremely large or extremely small constants, it is generally impossible to verify more than two or three properties for any object. • The Schwarzschild radius (r s ) represents the ability of mass to cause curvature in space and time. • The standard gravitational parameter (μ) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies. • Inertial mass (m) represents the Newtonian response of mass to forces. • Rest energy (E 0 ) represents the ability of mass to be converted into other forms of energy. • The Compton wavelength (λ) represents the quantum response of mass to local geometry. Reduced Compton wavelength When the Compton wavelength is divided by 2̟, one obtains a smaller or “reduced” Compton wavelength: λ / 2̟ = ħ / m c The reduced Compton wavelength is a natural representation for mass on the quantum scale, and as such, it appears in many of the fundamental equations of quantum mechanics. The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle:
Cosmic Game © Douglass A. White, 2012 v151207 176 It appears in the Dirac equation (the following is an explicitly covariant form employing the Einstein summation convention): The reduced Compton wavelength also appears in Schrödinger's equation, although its presence is obscured in traditional representations of the equation. The following is the traditional representation of Schrödinger's equation for an electron in a hydrogen-like atom: Dividing through by ħc, and rewriting in terms of the fine structure constant α, one obtains: Relationship between the reduced and non-reduced Compton wavelength The reduced Compton wavelength is a natural representation for mass on the quantum scale. Equations that pertain to mass in the form of mass, like Klein-Gordon and Schrödinger's, use the reduced Compton wavelength. The non-reduced Compton wavelength is a natural representation for mass that has been converted into energy. Equations that pertain to the conversion of mass into energy, or to the wavelengths of photons interacting with mass, use the non-reduced Compton wavelength. A particle of rest mass m has a rest energy of E = mc 2 . The non-reduced Compton wavelength for this particle is the wavelength of a photon of the same energy. For photons of frequency f, energy is given by which yields the non-reduced Compton wavelength formula if solved for λ.
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