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

TL;DR

This paper derives analytical expressions for quantum uncertainties in time-dependent coupled harmonic oscillators, revealing a sum rule for two oscillators and conditions under which it extends to larger systems.

Contribution

It introduces the sum rule of quantum uncertainty for coupled oscillators and identifies conditions for its validity in systems with more than two oscillators.

Findings

Uncertainties are analytically derived for time-dependent coupled oscillators.

The sum rule holds exactly for two oscillators as an average of individual uncertainties.

The sum rule extends to larger systems only when certain quantum numbers are equal.

Abstract

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 $N \geq 3$, 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.