A Symbolic Derivation of a Characteristic Length Scale from a Toy Dimensionless Oscillator Model: R0 = 2λ_C (Twice the Reduced Compton Wavelength)

Abstract

This note presents a purely symbolic derivation, starting from an ad hoc dimensionless dynamical equation a = 1/x^2 - x^2 with equilibrium point x_eq = 1. Using only the algebraic relations f = c/(pi*R0), E = h*f, h = 2*pi*hbar, and E = m*c^2, a characteristic length scale R0 is derived without substituting any numerical values until the final reference calculation. The result is R0 = 2*hbar/(m*c) = 2*lambda_bar_C, exactly twice the reduced Compton wavelength. The dimensionless model itself is a heuristic construction and does not correspond to any established force law in physics (it is neither an inverse-square force alone nor a standard harmonic restoring force). The derived frequency omega = 2 and period T = pi are exact only in the small-oscillation (linearized) limit around the equilibrium point; for finite oscillation amplitude the period deviates from pi due to the asymmetry of the effective potential U(x) = x^3/3 + 1/x. This note documents the algebraic derivation transparently, including this scope-of-validity caveat, for reference and further discussion.