Uncertainty relations for the triple components of the hydrogen atom

Abstract

As one of the core principles of quantum mechanics, the uncertainty relation profoundly reveals the inherent non-commutativity of physical quantities in the microscopic world. It provides a theoretical foundation for the precision limits of quantum measurements and has significant applications in fields such as quantum information and precision spectroscopy. Traditional uncertainty relations primarily focus on the mutual constraints between two non-commuting observables. However, for complex quantum systems, uncertainty relations involving multiple physical quantities can more comprehensively characterize the cooperative effects of quantum fluctuations and offer new perspectives for optimizing the allocation of core quantum resources. The hydrogen atom is one of the few strictly solvable systems with hidden symmetries, and the study of its symmetries and conserved quantities has always been a central topic in theoretical physics. In this work, we take the hydrogen atom as the research object and derive the additive and multiplicative uncertainty relations for the three components of the orbital angular momentum, the three components of the quantum Runge–Lenz vector, and the three components of their arbitrary linear combination $\overrightarrow{W}$. The results show that, compared to the usual uncertainty relations involving only two components, the three-component uncertainty relations are related to a constant $2/\sqrt{3}$.