On the uncertainty relations for an electron in a constant magnetic field

TL;DR

This paper examines the uncertainty relations between the azimuthal angle and the z-component of angular momentum for an electron in a magnetic field, highlighting conditions where standard relations do not hold.

Contribution

It derives a generalized uncertainty relation for arbitrary observables and clarifies the limitations of the standard relation for angle and angular momentum.

Findings

Standard uncertainty relation does not apply to the pair (φ, Lz) in certain states.

Derived a more general uncertainty relation valid for arbitrary observables.

Illustrated the failure of the standard relation with a parameter-dependent state.

Abstract

We discuss the uncertainty relation for the azimuthal angle $\phi$ and the $z$-component of the angular momentum $L_z$. To this end we derive the uncertainty relation for an arbitrary pair of observables and discuss the conditions for its validity. By means of a simple parameter-dependent state we illustrate the well known fact that the standard uncertainty relation for the coordinate and its conjugate momentum does not apply the pair of observables $\phi-L_z$. This analysis is motivated by a discussion of the motion of an electron in a constant magnetic field appeared recently in this journal (Eur. J. Phys. \textbf{33} (2012) 1147) where the author assumed the validity of the standard uncertainty relation for the pair $\phi-L_z$.