Tight-binding model and Harper's equations are partial cases of the Caldirola-Montaldi procedure

In the present paper the tight-binding model and Harper’s equations are derived directly from the Schrodinger equation by extending the Caldirola-Montaldi model in space. Then we apply approximation methods in order to determine the eigenvalues and eigenfunctions of Harper’s type equations which appear recently of significant physical interest (quantized Hall effect). Also, we calculate the Zener-tunnelling probability in the presence of electric and magnetic fields. Finally, we study the Caldirola-Montaldi model for free electrons in a uniform magnetic field.