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Eigenstates of linear combinations of two boson creation and annihilation operators: An algebraic approach

P. Shanta, S. Chaturvedi, Venkatraman Srinivasan

Published Jan 1, 1996• 4 citations
OpenAlex

Abstract

Eigenstates of the linear combinations $a^2+\\beta a^{\\dagger2}$ and $ab+\\beta a^\\dagger b^\\dagger$ of two boson creation and annihilation operators are presented. The algebraic procedure given here is based on the work of Shanta et al. [Phys. Rev. Lett. {\\bf 72}, 1447, 1994] for constructing eigenstates of generalized annihilation operators.

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EIGENSTATES OF LINEAR COMBINATIONS OF TWO-BOSON CREATION AND ANNIHILATION OPERATORS: AN ALGEBRAIC APPROACH

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Eigenstates of the linear combinations a2+βa†2 and ab+βa†b†of two-boson creation and annihilation operators are presented. The algebraic procedure given here is based on the work of Shanta et al. [Phys. Rev. Lett.72, 1447, (1994)] for constructing eigenstates of generalized annihilation operators. Expressions for the overlaps of these states with the number states, the coherent states and the squeezed states are given in a closed form.

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Abstract The eigenvalue problem for arbitrary linear combinations kα + μα† of a boson annihilation operator α and a boson creation operator α† is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ  μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ⩾ 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α† to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α† are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ  μ/k as a parameter.

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Generalized creation and annihilation operators: Physical interpretation and reordering properties

Akiva Novoselsky, Mathilde Vallières, Robert Gilmore +1 more

Matrix elements for fermion and boson operators describing the creation or annihilation of particles with multiple angular-momentum quantum numbers can be factored into products of matrix elements of generalized operators, one for each quantum number. The generalized operators have nonzero matrix elements between states of arbitrary permutational symmetry. This factorization greatly increases the efficiency of shell-model calculations. We provide a physical interpretation for these generalized operators and introduce a simple operator-reordering procedure.

Physical Review A•Jan 1, 1991•4 citations
Eigenstates of the linear combinations a2+betaadagger2a^2+\\beta a^{\\dagger2}a2+betaadagger2 and ab+\\beta a^\\dagger b^\\dagger of two boson creation and annihilation operators are presented. The algebraic procedure given here is based on the work of Shanta et al. [Phys. Rev. Lett. {\bf 72}, 1447, 1994] for constructing eigenstates of generalized annihilation operators.
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.
Quantum matrix algebras s u q (2) are realized in terms of single-state boson operators and their conjugates, ( a + , a ), ( b + , b ), etc., specified as quantum analogs of symplectons. The boson operators obey the linear transformation of S U q (2) such as a ′ + = a + x + a v and a ′ = a + u + a y . Here, x , u , v , y are non-commutative elements which commute with a + and a . Commutation relations for creation and annihilation operators are determined so as to be invariant under the linear transformation. Exchange algebras with the universal R j ′ j ′′ matrix are realized in terms of P ( j ′ , m ′ ; b ), P ( j ′′ , m ′′ ; a ), etc., where P ( j ′ , m ′ ; b ) is the j z ′ = m ′ component of rank- j ′ tensor written as an order-2 j ′ polynomial in b + and b .
Abstract The eigenvalue problem for arbitrary linear combinations kα + μα† of a boson annihilation operator α and a boson creation operator α† is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ  μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ⩾ 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α† to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α† are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ  μ/k as a parameter.
Matrix elements for fermion and boson operators describing the creation or annihilation of particles with multiple angular-momentum quantum numbers can be factored into products of matrix elements of generalized operators, one for each quantum number. The generalized operators have nonzero matrix elements between states of arbitrary permutational symmetry. This factorization greatly increases the efficiency of shell-model calculations. We provide a physical interpretation for these generalized operators and introduce a simple operator-reordering procedure.