Calculation of Band Edge Eigenfunctions and Eigenvalues of Periodic Potentials through the Quantum Hamilton - Jacobi Formalism

X iv :q ua nt -p h/ 03 12 04 1v 1 4 D ec 2 00 3 Calculation of Band Edge Eigenfunctions and Eigenvalues of Periodic Potentials through the Quantum Hamilton Jacobi Formalism ∗ S. Sree Ranjani , A.K. Kapoor ‡ and P.K. Panigrahi § a School of Physics, University of Hyderabad, Hyderabad 500 046, India b Physical Research Laboratory Navrangpura, Ahmedabad, 380009, India Abstract We obtain the band edge eigenfunctions and the eigenvalues of solvable periodic potentials using the quantum Hamilton Jacobi formalism. The potentials studied here are the Lamé and the associated Lamé which belong to the class of elliptic potentials. The formalism requires an assumption about the singularity structure of the quantum momentum function p, which satisfies the Riccati type quantum Hamilton Jacobi equation, p2 − ih̄ d dx p = 2m(E − V (x)) in the complex x plane. Essential use is made of suitable conformal transformations, which leads to the eigenvalues and the eigenfunctions corresponding to the band edges in a simple and straightforward manner. Our study reveals interesting features about the singularity structure of p, responsible in yielding the band edge eigenfunctions and eigenvalues.