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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 ...
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]
Specific periodic one-dimensional equations include the Kronig–Penney model and Mathieu's equation. Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is applied to spectral geometry. [20] [21] [22]
Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wave function.
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. [1]
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.
The Hamiltonian of the one-dimensional Ising model on a lattice of L sites with free boundary conditions is = =, …, +, where J and h can be any number, since in this simplified case J is a constant representing the interaction strength between the nearest neighbors and h is the constant external magnetic field applied to lattice sites.