The planar Landau system which describes the quantum mechanical motion of a charged particle in a plane with a uniform magnetic field perpendicular to the plane, is explored within pedagogical settings aimed at the beginning graduate level. The system is known to possess the Euclidean symmetry in two dimensions with central extension . In this paper, we revisit the well-known energy eigenvalues of the system, known as the Landau levels, by exploiting the related symmetry algebra. Specifically, we utilize the Casimir operator and the commutation relations of the generators of the group. More importantly, an algebraic formalism on this topic based on Schwinger’s oscillator model of angular momentum is also presented. The dimensions of irreducible representations of the group and their implications on the degeneracy of Landau levels is discussed.

在针对初等毕业水平的教学环境中，探索了平面Landau系统，该系统描述了带电粒子在垂直于该平面的均匀磁场的平面中的量子力学运动。已知该系统具有中心扩展的二维欧几里得对称性。在本文中，我们通过利用相关的对称代数，重新访问了系统的著名能量本征值，即Landau能级。具体来说，我们利用卡西米尔运算符和该组生成器的换向关系。更重要的是，还提出了基于Schwinger角动量振荡器模型的关于该主题的代数形式主义。不可约表示的维数 讨论了该小组及其对Landau等级退化的影响。