Uncertainties \((\Delta x)^2\) and \((\Delta p)^2\) are analytically derived in an N-coupled harmonic oscillator system when spring and coupling constants are arbitrarily time-dependent and each oscillator is in an arbitrary excited state. When \(N = 2\), those uncertainties are shown as just arithmetic average of uncertainties of two single harmonic oscillators. We call this property as “sum rule of quantum uncertainty”. However, this arithmetic average property is not generally maintained when \(N \ge 3\), but it is recovered in N-coupled oscillator systems if and only if \((N-1)\) quantum numbers are equal. The generalization of our results to a more general quantum system is briefly discussed.

中文翻译：

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量子不确定性的总和规则：具有时变参数的耦合谐振子系统

当弹簧和耦合常数随时间而定，并且每个振荡器都处于时域关系时，在N耦合谐波振荡器系统中分析性地得出不确定性\（（\（Delta x）^ 2 \）和\（（Delta p）^ 2 \）。任意的激发态 当\（N = 2 \）时，这些不确定度仅表示为两个单谐波振荡器的不确定度的算术平均值。我们称此性质为“量子不确定性的总和”。但是，当\（N \ ge 3 \）时，通常不保持该算术平均属性，但是当且仅当\（（N-1）\）时，才能在N耦合振荡器系统中恢复该算术平均属性。量子数相等。简要讨论了将结果推广到更通用的量子系统的情况。