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Progress of Theoretical Physics

A Diagonalization Method of a Hamiltonian in Second-Quantization. I

Raoelina Andriambololona

Published Apr 1, 1969• 1 citation
Publisher

Abstract

A method to obtain all the operators diagonalizing a given Hamiltonian in second-quan-tization theory is presented. The importance not only of the dynamical commutation relations but also of the kinematical and mutual commutation relations is pointed out. Different classes of Hamiltonians (with examples), programs, degree of a program, order of calculation are introduced. Finally, the practical utilization of the method is given.

DOI
10.1143/ptp.41.1064

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A method to obtain all the operators diagonalizing a given Hamiltonian in second-quan-tization theory is presented. The importance not only of the dynamical commutation relations but also of the kinematical and mutual commutation relations is pointed out. Different classes of Hamiltonians (with examples), programs, degree of a program, order of calculation are introduced. Finally, the practical utilization of the method is given.
The unitary matrix that brings a Hermitian H into block-diagonal form can be uniquely determined under very simple and transparent conditions. Int his work the block-diagonalization problem is investigated in the framework of the second quantization formalism. Starting with an operator Ḩ which in any n-particle Fock space has a well-defined matrix representation an attemt was made to answer the question whether the transformation matrices T which can be separately given in the various n-particle spaces can be considered matrix representations of the same operator. T̂. Interestingly, the very important result was reached that the block-diagonalization operator T̂ exists and is unique. As a particular example, attention was concentrated on the case of an operator Ĥ given by a one-particle operator. In this case the block-diagonalization operator can be constructed and given in explicit from. This approach is applied to the theory of Green’s functions where the block-diagonalization of the Hamiltonian has interesting consequences that are illustrated in some details.
The formalism of the second quantization is developed entirely within the framework of the Dirac-Jordan representation theory. The equivalence of this formalism and the methods of ordinary coordinate-spin space is shown, and the equations of "connection" are fully developed. It is shown that the "creation" and "annihilation" operators, and indeed all operators which do not commute with the quantized Hamiltonian and hence whose representatives cannot be explicitly given in a configuration space of a fixed number of dimensions are easily treated in terms of the general representation theory and arise quite naturally. Both types of statistics are considered, and the Fock-Dirac density matrix and self-consistent field are treated as a simple illustration.
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.
A practical method is developed to deal with the second quantization of the many-body system containing the composite particles.In our treatment,the modes associated with composite particles are regarded approximately as independent ones compared with those of unbound particles.The field operators of the composite particles thus arise naturally in the second quantization Hamiltonian.To be emphasized,the second quantization Hamiltonian has the regular structures which correspond clearly to different physical processes.
The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focuses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti) ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general.