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Acta Physica Sinica

Left-hand and right-hand inverse operators of quantum operators and its mathematical property

Shimin Xu, Yunhai Zhang, Xing-Lei Xu, Hongqi Li

Published Jan 1, 2010• 1 citation
Publisher

Abstract

The right-hand inverse operator of the annihilation operators and the left-hand inverse operator of the creation operator are given by using the contour integration. Furthermore, the corresponding relation between the right-hand inverse or left-hand inverse operator derived and the column elementary transformation or row transformation of a matrix is investigated.

DOI
10.7498/aps.59.7575

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The right-hand inverse operator of the annihilation operators and the left-hand inverse operator of the creation operator are given by using the contour integration. Furthermore, the corresponding relation between the right-hand inverse or left-hand inverse operator derived and the column elementary transformation or row transformation of a matrix is investigated.
The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second-quantized, Heisenberg representation, particle-creation, and annihilation operators. The creation operator, ψ †( r , t), when acting to the right on a state of the system, adds a particle to the state at the space-time point r , t; the annihilation operator ψ ( r , t), the adjoint of the creation operator, acting to the right, removes a particle from the state at the point r , t.
It is pointed out that the assertion that the eigenfunction of the creation operator a + is identically zero is not rigorous. On the contrary, the eigenfunction of a + may be discussed by using the contour integral representation of δ function, the property of which and the relation with the eigenstate of the annihilation operator are discussed. Its applications, especially the application in thoroughly studying the inverse operator's properties, are presented. It is shown that the phase operator can also be represented by the inverse operator.
Nonlinear fermions of degree n ( n -fermions) are introduced as particles with creation and annihilation operators obeying the simple nonlinear anticommutation relation AA † + A † n A n = 1. The ( n + 1)th-order nilpotency of these operators follows from the existence of unique A -vacuum. Supposing appropriate ( n + 1)th-order nilpotent para-Grassmann variables and integration rules the sets of n -fermion number states, 'right' and 'left' ladder operator coherent states (CS) and displacement-operator-like CS are constructed. The ( n + 1) × ( n + 1) matrix realization of the related para-Grassmann algebra is provided. General ( n + 1)th-order nilpotent ladder operators of finite-dimensional systems are expressed as polynomials in terms of n -fermion operators. Overcomplete sets of (normalized) 'right' and 'left' eigenstates of such general ladder operators are constructed and their properties are briefly discussed. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.