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Starting from the zero modes of the Dirac-Weyl equation for Landau levels inthe symmetric gauge, we propose a novel mechanism to construct the eigenvaluesand its eigenfunctions. We show that the problem may be addressed withoutnumerical calculation and only solving the Dirac-Weyl equation for the zeromodes. Specifically, the eigenstates associated to the negative magnetic fieldconfigurations may be constructed from the zero mode with positive chirality.In addition, we obtain that the eigenstates associated to the positive magneticfield configurations may be constructed from the zero mode with negativechirality. Finally, we show that our mechanism may be used to obtain theeigenvalues and eigenfunctions of the Hamiltonian corresponding to bilayergraphene system.

Spin-1/2 Landau levels in the symmetric gauge from the zero energy modes