Physics – Quantum Physics
Scientific paper
2003-12-04
Mod. Phys. Lett. A, 19, No. 27, 2047 (2004).
Physics
Quantum Physics
21 pages, 5 tables
Scientific paper
10.1142/S0217732304014197
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 Lam{\'e} and the associated Lam{\'e} 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, $ p^{2} -i \hbar \frac{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.
Kapoor A. K.
Panigrahi Prasanta K.
Ranjani Sree S.
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