The Fundamental Kinetic Energy Uncertainty in Free Particle Momentum

The motion of a free quantum particle is governed by strict energy and momentum conservation laws. This work proposes that a free particle is subject to inherent uncertainty in its position and momentum, as described by the Heisenberg Uncertainty Principle, leading to a minimum kinetic energy constraint. We derive that a fundamental momentum uncertainty establishes a lower bound on kinetic energy, connecting it to Planck’s constant (hhh). This constraint refines the quantum-toclassical transition, impacts the interpretation of quantum electrodynamics (QED), and provides insights into the behavior of particles at low kinetic energies.Furthermore, we introduce Arrabal’s Critical Velocity, a universal velocity threshold:vcrit=2hmv_{\text{crit}} = \sqrt{\frac{2h}{m}}vcrit=m2hwhich determines the transition from quantum to classical behavior for any particle. This equation allows us to calculate the critical velocity for all particles, helping to establish a fundamental limit where quantum coherence is lost. We suggest experimental approaches that could validate the predictedkinetic energy threshold and refine our understanding of quantum constraints in free-particle motion.