Mathieu Potential Research Articles

Mass-velocity correction to the solution of the Schrodinger equation for the Mathieu potential

Perturbation theory is used to calculate the mass-velocity correction to the energy solution of the Schrodinger equation for the Mathieu potential. It shows that this relativistic effect causes a shrinkage of the energy spectrum.

Ionic, Covalent, and Metallic Surface States of a One-Dimensional Mathieu Potential with Arbitrary Termination

A model of surface states on a one-dimensional crystal with sinusoidal potential has been simply solved for arbitrary values of the sinusoidal amplitude $q$, lattice termination ${z}_{0}$, surface step height $\ensuremath{\xi}$, and band-gap index $m$. The Schr\"odinger equation inside the crystal is equivalent to the Mathieu equation, whose eigenfunctions and eigenvalues are well known. Surface states are constructed as hybrids of bulk states [incorporating the proper nearly free-electron (NFE) and linear-combinations-of-atomic-orbitals (LCAO) limits], and they move towards the centers of the band gaps the stronger the surface "perturbation." It is shown that ionic surface states occur for $m=2$, ${z}_{0}=\frac{1}{4}\ensuremath{\pi}$, and $m=2$, ${z}_{0}=\frac{3}{4}\ensuremath{\pi}$; covalent surface states occur for $m=1$, ${z}_{0}=\frac{1}{2}\ensuremath{\pi}$; and metallic (or virtual) surface states occur in the allowed band between $m=0$ and $m=1$. Shockley states (looked for at ${z}_{0}=0$, all $m$) do not appear. This work has been favorably related to the author's semiclassical ionic model, to previous NFE and LCAO approximation methods, to the theories of Shockley and Statz, and to real surfaces in accordance with low-energy-electron-diffraction studies. In addition, it has uncovered a new linkage between the NFE and LCAO terminologies.

Energy Bands in Periodic Lattices—Green's Function Method

The mathematical basis of calculations of energy bands in periodic lattices using the Green's function method is presented and the method's usefulness discussed. The original formulation of the method by Kohn and Rostoker is modified to achieve more efficient and accurate evaluation of structure constants using symmetry considerations and the full Ewald summation procedure. Formulas are derived giving the wave function both inside and outside the sphere inscribed in the unit cell. The method is demonstrated with the 3-dimensional Mathieu potential. Convergence is found to be very rapid both in this test case and in practical calculations on metals, and accurate energies and wave functions can be obtained without elaborate calculation even at points of low symmetry within the Brillouin zone.