Landau levels for charged particles with anisotropic mass.

The problem of the two-dimensional motion of a charged particle with constant mass in the presence of a uniform constant perpendicular magnetic field features in several undergraduate and graduate quantum physics textbooks. This problem is very important to studies of two-dimensional materials that manifest quantum Hall behavior, as evidenced by several major discoveries over the last few years. Many real experimental samples are more complicated due to the anisotropic mass of the electrons. In this work, we provide the exact solution to this problem by means of a clever scaling of coordinates. Calculations are done for a symmetric gauge of the magnetic field. This study allows a broad audience of students and teachers to understand the mathematical techniques that lead to the solution of this quantum problem. Editor's note: The behavior of an electron confined to move in two dimensions in the presence of a perpendicular magnetic field underlies the quantum Hall effect. This is also an analytically solvable problem, so it is valuable to quantum mechanics pedagogy. This paper reviews the solution and then shows a neat trick for solving the problem when the electron's effective mass is anisotropic, a situation that is fairly common in solid-state systems. [ABSTRACT FROM AUTHOR]