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Chinese Physics Letters

Exact Solution for a Chain of Coupled Oscillators with Two Types of Atom

Liu Ke-Jia, Huai-Xin Lu, Zhang Yong-De

Published Apr 1, 2000• 0 citations
Publisher

Abstract

A quantum transformation that can diagonalize the Hamiltonian of a ring of N coupled diatomic oscillators is found. By means of the transformation, the dispersion relation of the atomic system is obtained. For large N it turns out that the dispersion relation resembles to the phonon spectrum of the lattice vibration derived by Newton's equation of motion.

DOI
10.1088/0256-307x/17/4/003

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A quantum transformation that can diagonalize the Hamiltonian of a ring of N coupled diatomic oscillators is found. By means of the transformation, the dispersion relation of the atomic system is obtained. For large N it turns out that the dispersion relation resembles to the phonon spectrum of the lattice vibration derived by Newton's equation of motion.
A unitary operator which can directly diagonalize the Hamiltonian of a ring of coupled oscillators is found. With use of the technique of integration within an ordered product of operators we see that includes a squeezing transformation. For large it turns out that, except a zero-mode, the ring Hamiltonian has the same spectrum as the one-dimensional monatomic lattice model plus the Born–Von–Karmen boundary condition. The density matrix of this model is calculated by the transformation, which further leads us to derive the heat capacity of the coupled oscillators in the large limit.
The problem of the quantum description of the damping of a 1D chain of coupled oscillators with periodic boundary conditions in the Markovian regime is solved. The equations of motion of the normal modes are derived, and the evolution in time of their mean values and variances is explicitly studied. Furthermore, we investigate the effect of the diffusion and the friction on the dynamics of the chain. It is shown that when the off-diagonal elements of the diffusion and friction tensors are taken into account, the most notable contribution of these processes is observed in the mode in which all the oscillators undergo an in-phase vibration.
It is one of the important problems to solve the Schrodinger equation for the coupled harmonic oscillators.Here the Heisenberg equations of motion for the coupled harmonic oscillators are written in the form of Schrodinger type equation,which can be decoupled by certain transformation.Further,the Hamiltonian can be diagonalized.For the anisotropic harmonic oscillator in a background magnetic field,the diagonalization form of the Hamiltonian,the energy eigenvalues and the eigenstates are derived.Such a process of derivation has clear physical picture and is easy to understand.
Mechanical motions of atoms near their equilibrium geometry are approximated by a universal mapping on a collection of independent harmonic oscillators. The respective frequencies are characteristic to the material, for example, lattice phonons or molecular vibrations. The quantum description of atomic motions involves the solution of the Schrödinger equation for the harmonic oscillator in the space of proper wave functions. The energy levels obtained are shown to be equally spaced where the level spacing is proportional to the oscillator frequency. The stationary solutions are identified in terms of Hermite polynomials, demonstrating remarkable differences from the classical harmonic oscillator. The importance of the zero-point energy is emphasized in the context of the stability of chemical bonds, where the Harmonic approximation is shown to be reasonably valid for typical interatomic bonds in standard thermal conditions. This explains the relevance of the harmonic approximation for analyzing the absorption spectrum of infrared radiation by molecules.