Detailed solution of the problem of Landau states in a symmetric gauge

Abstract

Understanding the quantum problem of a free charged particle undergoing two-dimensional motion in a perpendicular uniform, constant magnetic field is necessary to the comprehension of some very important phenomena in physics. In particular, a grasp of the nature of the Landau states in a symmetric gauge is crucial to explain the underlying principles of quantum Hall effects. In this work we provide a step-by-step solution of this quantum problem in a pedagogical fashion that is easy to follow by an audience of undergraduate students and prospective physics teachers. This approach should enable undergraduate students to comprehend all the technical mathematical details involved in the process. Such details are routinely missing from mainstream quantum mechanics textbooks. In particular, this study allows a broad audience of students and teachers to gradually absorb knowledge not only on basic principles of quantum mechanics, but also on various special mathematical functions that are encountered in the process.

Quantum Charged Particle in a Flat Box under Static Electromagnetic Field with Landau’s Gauge and Special Case with Symmetric Gauge

“…. However, as one can see from (47) or (49), this solution is not of separable variable type in these coordinates, and this is consistent with the fact that the eigenvalue problem (29) written in polar coordinates is not of separable variable type in these coordinates. Therefore, the solution (47) cannot be equivalent to Landau-Laughlin solutions.…”

“…One needs to mention that Laughlin [28] gave a solution to this problem which is equivalent Landau' solution, and this equivalence was demonstrated by Orion [29]. Their solutions are of separable variable type in the polar coordinates ϕ and ρ in the space x-y (…”

Quantum Zeno Isolation of Quantum Dots

“…Thus, the reservoir eigen-functions are no longer plane waves as in Eq. ( 2), but discrete Landau levels which can be represented by the modes of a two-dimensional harmonic oscillator in polar coordinates r and ϕ [38] ψ n,l (r, ϕ) = f n,l (r)e ilϕ .…”

A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions

“…In addition, continuum problems, such as the free particle in one, two, and three dimensions, the one-dimensional Cartesian linear potential, the continuum of the Coulomb problem in two and three dimensions, and the continuum of the Cartesian one-dimensional Morse potential, can also be solved using confluent hypergeometric functions. Moreover, we note that there is a very nice discussion of Landau levels that also employs confluent hypergeometric functions [18]. The confluent hypergeometric equation also arises in optics [15,[19][20][21][22], classical electrodynamics [6,19,23], classical waves [7,24,25], diffusion [26], fluid flow [27], heat transfer [28], general relativity [29][30][31][32], semiclassical quantum mechanics [33], quantum chemistry [34,35], graphic design [36], and many other areas.…”