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Journal of Taiyuan University of Science and Technology

Canonical Transformation and Diagonalization of the Hamiltonian of n-mode Two-body Coupling Interaction

Lai Yun-zhong

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

By utilizing the canonical transformation,a concise diagonalization approach for Hamiltonian of the n-mode Bose and Fermi and Two-body Coupling Interaction is given.The issue of Hamiltonian diagonalization is transformed into a solution of matrix eigenvalue problem.By solving the secular equation,the value of the spectrum is obtained.Finally,we take a example to verify the reliability of this method and discuss the differences of the diagonalization berween Bose and Fermi system Hamiltonian.

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By utilizing the canonical transformation,a concise diagonalization approach for Hamiltonian of the n-mode Bose and Fermi and Two-body Coupling Interaction is given.The issue of Hamiltonian diagonalization is transformed into a solution of matrix eigenvalue problem.By solving the secular equation,the value of the spectrum is obtained.Finally,we take a example to verify the reliability of this method and discuss the differences of the diagonalization berween Bose and Fermi system Hamiltonian.
The problem of the exact diagonalization of the Hamiltonian of an assembly of N bilinearly interacting bosons is discussed in what concerns the eigenvalues as well as for the expression of the new boson operators in terms of the old ones. The method is equivalent to the standard equation-of-motion approach, nevertheless sensibly more concise. Three sets of operational rules are indicated, and their use is exhibited in some examples. In some particular cases of practical importance (for example, when all the coefficients of the Hamiltonian are real), the research of the eigenvalues has been compacted as much as possible.
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.
A system of linearly coupled quantum harmonic oscillators can be diagonalized when the system is dynamically stable using a Bogoliubov canonical transformation. However, this is just a particular case of more general canonical transformations that can be performed even when the system is dynamically unstable. Specific canonical transformations can transform a quadratic Hamiltonian into a normal form, which greatly helps to elucidate the underlying physics of the system. Here, we provide a self-contained review of the normal form of a quadratic Hamiltonian as well as step-by-step instructions to construct the corresponding canonical transformation for the most general case. Among other examples, we show how the standard two-mode Hamiltonian with a quadratic position coupling presents, in the stability diagram, all the possible normal forms corresponding to different types of dynamical instabilities.
In a system of coupled harmonic oscillators, the interaction can be represented by a real, symmetric and positive definite interaction matrix.The quantization of a Hamiltonian describing such a system has been done in the canonical case.In this paper, we take a more general approach and look at the system as a Wigner quantum system.Hereby, one does not assume the canonical commutation relations, but instead one just requires the compatibility between the Hamilton and Heisenberg equations.Solutions of this problem are related to the Lie superalgebras gl(1|n) and osp(1|2n).We determine the spectrum of the considered Hamiltonian in specific representations of these Lie superalgebras and discuss the results in detail.We also make the connection with the well-known canonical case.