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Uncertainties $(\Delta x)^2$ and $(\Delta p)^2$ are analytically derived inan $N$-coupled harmonic oscillator system when spring and coupling constantsare arbitrarily time-dependent and each oscillator is in an arbitrary excitedstate. When $N = 2$, those uncertainties are shown as just arithmetic averageof uncertainties of two single harmonic oscillators. We call this property as"sum rule of quantum uncertainty". However, this arithmetic average property isnot generally maintained when $N \geq 3$, but it is recovered in $N$-coupledoscillator systems if and only if $(N-1)$ quantum numbers are equal. Thegeneralization of our results to a more general quantum system is brieflydiscussed.

Sum Rule of Quantum Uncertainties: Coupled Harmonic Oscillator System with Time-Dependent Parameters