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In some cases, the Schrödinger equation can be solved analytically on a one-dimensional lattice of finite length [6] [7] using the theory of periodic differential equations. [8] The length of the lattice is assumed to be L = N a {\displaystyle L=Na} , where a {\displaystyle a} is the potential period and the number of periods N {\displaystyle ...
The density of states in a three-dimensional lattice will be the same as in the case of the absence of a lattice. For the three-dimensional case the density of states () is; = . In three-dimensional space the Brillouin zone boundaries are planes. The dispersion relations show conics of the free-electron energy dispersion parabolas for all ...
The Volterra lattice is the set of ordinary differential equations for functions a n: ′ = (+) where n is an integer. Usually one adds boundary conditions: for example, the functions a n could be periodic: a n = a n+N for some N, or could vanish for n ≤ 0 and n ≥ N.
Gold deposited on a stepped Si(553) surface has shown evidence of two simultaneous Peierls transitions. The lattice period is distorted by factors of 2 and 3, and energy gaps open for nearly 1/2-filled and 1/3–1/4 filled bands. The distortions have been studied and imaged using LEED and STM, while the energy bands were studied with ARP. [9]
In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models.It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic isotropic (XXX) Heisenberg model.
The Toda lattice, introduced by Morikazu Toda , is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system. It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics , where the atoms of a crystal automatically form a lattice.
However, to date, no three-dimensional (3D) problem has had a solution that is both complete and exact. [4] Over the last ten years, Aranovich and Donohue have developed lattice density functional theory (LDFT) based on a generalization of the Ono-Kondo equations to three-dimensions, and used the theory to model a variety of physical phenomena.