to the plane of the metal, which implies the physically questionable notion of a field either emanating from or converging to an axis at the centre of the loop. That this configuration did not coincide precisely with the experiment was immaterial — the concept of quantization was properly established and what mattered was that the Landau gauge allowed Laughlin to proceed analytically with the argument. Similarly, Halperin [@halperin1982quantized] followed up with a thin film with annular geometry, and adopted the gauge choice $\vec{\bf A} = [B_0r/2 + \Phi/(2 \pi r)] \hat{\bf \theta}$ to produce a uniform field $B_0$ in the $z$-direction along with a central flux of magnitude $\Phi$.

Other, “physical” geometries, suitable to so-called “quantum dots,” were adopted thereafter — see for example, Refs. \[\] and \[\]. In this paper we will review and expand upon some of these calculations, representing charged particles in a confined geometry subjected to a constant magnetic field. A very good reference for the quantum dot work is Chakraborty’s book, [*Quantum Dots*]{},[@chakraborty99] which also contains a number of very useful reprinted articles. As mentioned above, we will focus on the orbital motion of the charged particle (hereafter these will be electrons) and neglect the 