Compatibility between the Second Law of Thermodynamics and the Quantum Uncertainty Principle in the Harmonic Oscillator Model

Boltzmann’s statistical interpretation of the second law of thermodynamics indicates that systems evolve from less probable to more probable states. In an isolated system, entropy reaches its maximum at equilibrium, corresponding to the greatest number of accessible microstates. The quantum uncertainty principle, a fundamental feature of quantum mechanics, reflects the intrinsic probabilistic nature of physical systems. This study proposes that the equilibrium state defined by maximal entropy and microstate accessibility-also represents a condition of maximal intrinsic uncertainty. We examine the connection between Boltzmann’s entropy framework and the quantum mechanical uncertainty relation between position and momentum, focusing on localized harmonic oscillator systems. Properties of the uncertainty product are analyzed in detail. Both the uncertainty principle and the second law of thermodynamics are rooted in probabilistic foundations. Our findings suggest that the uncertainty principle provides a conceptual perspective on thermodynamic behavior.