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College Physics

One method to derive the form of general operators in occupation particle number representation

Ren Zheng-xu

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

The expansion of single- particle transition operator( defined as the product of creation and annihilation operator,namely a+iajin single- particle state is obtained through the form of single- operator in occupation particle number representation,and the commutation relations for transition operator and creation( annihilation) operator are also given,then based on these conclusions,the form of general two- and n- particle operators in occupation particle number representation can be derived succinctly without distinguishing the statistical property of identical particle system.

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arXiv:2305.18113•May 29, 2023•0 citations
The expansion of single- particle transition operator( defined as the product of creation and annihilation operator,namely a+iajin single- particle state is obtained through the form of single- operator in occupation particle number representation,and the commutation relations for transition operator and creation( annihilation) operator are also given,then based on these conclusions,the form of general two- and n- particle operators in occupation particle number representation can be derived succinctly without distinguishing the statistical property of identical particle system.
Abstract Focusing on systems of many identical particles, Chapter 2 introduces appropriate operators to describe their properties in terms of Schwinger’s “measurement symbols.” The latter are then factorized into “creation” and “annihilation” operators, whose fundamental properties and commutation/anticommutation relations are derived in conjunction with the Pauli exclusion principle. This leads to “second quantization” with the Hamiltonian, number, linear and angular momentum operators expressed in terms of the annihilation and creation operators, as well as the occupation number representation. Finally, the concept of coherent states, as eigenstates of the annihilation operator, having minimum uncertainty, is introduced and discussed in detail.
Abstract We propose a new generalization of the standard (anti-)commutation relations for creation and annihilation operators of bosons and fermions. These relations preserve the usual symmetry properties of bosons and fermions. Only the standard (anti-)commutator relation involving one creation and one annihilation operator is deformed by introducing fractional coefficients, containing a positive integer parameterp. The Fock space is determined by the classical definition. The new relations are chosen in such a way that the total occupation number in the system has the maximum valuep. From the actions of creation and annihilation operators in the Fock space, a group theoretical framework is determined, and from here the correspondence with known particle statistics is established.
In this paper, we propose a new way to represent graphs in quantum space. In that approach, we replace the rows of the adjacency matrix of the graph by state vectors in the occupation number representation. Unlike the traditional definition of graph states, we actually let the occupation number of a single-particle state denote the number of edges between each two adjacent vertices. This allows us to avoid taking into account the interaction between each two particles. Based on the creation and annihilation operators, we propose the edge creation and annihilation operators. With these two operators, we can implement the fundamental operation of adding and removing edges and vertices in a graph. Then all additional operations in the graph such as vertex contractions can be defined. Our method can be used to represent both simple and multigraphs. Directed and undirected graphs are also compatible with our approach. The method of representation proposed in this paper enriches the theory of graph representation in quantum space.
A basis of products of one-particle vectors (symmetric or antisymmetric for bosons or fermions) is defined, and related to the occupation-number basis in an abstract Fock space, a generalization of the wave-mechanical configuration representation of the space of second quantization. Creation and destruction operators are simply defined in the product-vector basis and shown to have their usual properties in occupation-number representation. Algebraic properties of product vectors are developed and illustrated. The representation of operators in the product-vector basis is described, together with a brief discussion of the effects on these operators of transformations from one set of one-particle vectors to another. A simple treatment of density and reduced density operators for systems of bosons or fermions is given in the product-vector basis. The relationship between degeneracy in the Fock-space spectrum of an additive operator NA1 and degeneracy in the spectrum of one-particle operator A1 is obtained.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation (a†a^\daggera†) and annihilation (aaa) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator (a†x+ay)n(a^\dagger x+ay)^n(a†x+ay)n with nnn as the exponent and x, yx,\,yx,y are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.