Quantum Foundations in Energy-Efficiency Theory: From Probability Density to Measurement

We present a unified energy-ontological interpretation of non-relativistic quantum mechanics within the framework of Energy-Efficiency Theory (EET). Starting from three physically grounded axioms, we define the constrained potential U(r,t) as the spatial density of localized bound-state energy, and derive the continuity equation for the normalized constrained potential ρ(r,t)=U(r,t)/E_c. We prove that ρ(r,t) satisfies the quantum probability continuity equation, establishing the exact correspondence ρ(r,t)=|ψ|^2. We introduce the dimensionless energy ratio η = Ė_resp / Ė_main, which quantifies the balance between energy allocated to spatial exploration (response) and constraint maintenance. From first principles, we derive a modified Schrödinger equation with a real effective potential that preserves probability conservation for all values of η. Using an exact analytical solution for Gaussian wave packets, we derive the scaling relations Δx ∝ η^{-1/2} and Δp ∝ η^{1/2}, which imply the Heisenberg uncertainty principle ΔxΔp ≥ ħ/2 independent of η. Wave-particle duality is interpreted as the continuous sliding of η between exploration-dominated (wave-like, η>1) and maintenance-dominated (particle-like, η