Uncertainty relations for the triple components of canonical momentum

Abstract Uncertainty relations are one of the fundamental components of quantum theory, typically used to describe the limitations on the range of measurement results for two noncommuting observables. Recently, there has been growing interest in uncertainty relations concerning multiple observables. In this work, we study the uncertainty relations of the three components of the canonical momentum of an electron moving in a magnetic field. We derive the uncertainty relations for both the addition and multiplication of these components. Our results indicate that, compared to uncertainty relations involving only two components, the uncertainty relations for the three components are associated with a constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msqrt> </mml:math> .