On a new methodology for handling Landau levels and quantum Hall effect

In this article, the problem of Landau levels and the related quantum Hall effect are discussed by using representation with raising and lowering operators for physical observables and dimensionless occupation number representation for Fock states. In raising and lowering operators, the Hamiltonian for Landau levels under symmetric gauge is released from the apparent x-, y- asymmetry, and it is appropriate to choose H, n_b, J_z as the complete set of commutative observables. The question can be solved at ease in a subspace defined by a definite eigenvalue of the variable n_b = a₁⁺a₁ + a₂⁺a₂, the discrete energy levels need not be attributed to a harmonic oscillator, and their degeneracy turns out to be an essential character of the problem. The essential point here is the identification of a conservative quantity n_b, which resolves the Hilbert space into subspaces of a definite dimension. With a proper Bogoliubov transformation it can be proven that the problem of Landau level is gauge invariant. Furthermore, when represented in raising and lowering operators, the Hamiltonian for quantum Hall effect can be turned into exactly the same form as the Hamiltonian for Landau levels, thus can be solved with the complete set of state-vectors for the latter. This will significantly simplify the treatment of problems related to quantum Hall effect. Our theory provides a universal methodology, which can be helpful for seeking the analytical solutions of various quantum mechanical problems.