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Journal of Guizhou University of Technology

Exact solution for quantum oscillators of one dimensional coupled identical diatomic chain

Kejia Liu

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

In this paper,a quantum transformation for diagonalizing the Hamiltonian of a ring of N coupled diatomic identical oscillators is found,from which the dispersion relation of the system is obtained.For a large N,the dispersion relation obtained is different from what is given by the Newtonian equation of motion.The reason for the difference in dispersion relation is discussed,and the deficiency of the classical method is pointed out.

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In this paper,a quantum transformation for diagonalizing the Hamiltonian of a ring of N coupled diatomic identical oscillators is found,from which the dispersion relation of the system is obtained.For a large N,the dispersion relation obtained is different from what is given by the Newtonian equation of motion.The reason for the difference in dispersion relation is discussed,and the deficiency of the classical method is pointed out.
We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.
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.
Abstract The resolution of the Schrödinger equation for the translation-invariant N-body harmonic oscillator Hamiltonian in D dimensions with one-body and two-body interactions is performed by diagonalizing a matrix J of order N − 1 . It has been previously established that the diagonalization can be analytically performed in specific situations, such as for N ≤ 5 or for N identical particles. We show that the matrix J is diagonal, and thus the problem can be analytically solved, for any number of arbitrary masses provided some specific relations exist between the coupling constants and the masses. We present analytical expressions for the energies under those constraints.
In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.