Revisiting the compatibility problem between the gauge principle and the observability of the canonical orbital angular momentum in the Landau problem

TL;DR

This paper investigates the longstanding issue of whether the canonical orbital angular momentum quantum number in the Landau problem is observable, examining both particle and wave aspects to reconcile gauge invariance with physical observability.

Contribution

It provides a more comprehensive analysis of the gauge compatibility problem by considering both particle-like and wave-like properties of Landau electrons.

Findings

Reveals the non-trivial rotational dynamics of Landau electrons depending on quantum number m.

Clarifies the gauge invariance of the canonical OAM in the Landau problem.

Offers insights into the observability of quantum numbers in gauge theories.

Abstract

As is widely-known, the eigen-functions of the Landau problem in the symmetric gauge are specified by two quantum numbers. The first is the familiar Landau quantum number $n$, whereas the second is the magnetic quantum number $m$, which is the eigen-value of the canonical orbital angular momentum (OAM) operator of the electron. The eigen-energies of the system depend only on the first quantum number $n$, and the second quantum number $m$ does not correspond to any direct observables. This seems natural since the canonical OAM is generally believed to be a {\it gauge-variant} quantity, and observation of a gauge-variant quantity would contradict a fundamental principle of physics called the {\it gauge principle}. In recent researches, however, Bliohk et al. analyzed the motion of helical electron beam along the direction of a uniform magnetic field, which was mostly neglected in past analyses of the Landau states. Their analyses revealed highly non-trivial $m$-dependent rotational dynamics of the Landau electron, but the problem is that their papers give an impression that the quantum number $m$ in the Landau eigen-states corresponds to a genuine observable. This compatibility problem between the gauge principle and the observability of the quantum number $m$ in the Landau eigen-states was attacked in our previous letter paper. In the present paper, we try to give more convincing answer to this delicate problem of physics, especially by paying attention not only to the {\it particle-like} aspect but also to the {\it wave-like} aspect of the Landau electron.