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arXiv:2204.09044 · Zeitschrift für angewandte Mathematik und Physik

Bi-coherent states as generalized eigenstates of the position and the momentum operators

Fabio Bagarello, Francesco Gargano

Published May 16, 2022• 3 citations

Abstract

Abstract In this paper, we show that the position and the derivative operators, $${{\hat{q}}}$$ q^ and $${{\hat{D}}}$$ D^ , can be treated as ladder operators connecting various vectors of two biorthonormal families, $${{{\mathcal {F}}}}_\varphi $$ Fφ and $${{{\mathcal {F}}}}_\psi $$ Fψ . In particular, the vectors in $${{{\mathcal {F}}}}_\varphi $$ Fφ are essentially monomials inx, $$x^k$$ xk , while those in $${{{\mathcal {F}}}}_\psi $$ Fψ are weak derivatives of the Dirac delta distribution, $$\delta ^{(m)}(x)$$ δ(m)(x) , times some normalization factor. We also show how bi-coherent states can be constructed for these $${{\hat{q}}}$$ q^ and $${{\hat{D}}}$$ D^ , both as convergent series of elements of $${{{\mathcal {F}}}}_\varphi $$ Fφ and $${{{\mathcal {F}}}}_\psi $$ Fψ , or using two different displacement-like operators acting on the two vacua of the framework. Our approach generalizes well- known results for ordinary coherent states.

DOI: 10.1007/s00033-022-01759-z

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Abstract In this paper, we show that the position and the derivative operators,

{{\hat{q}}}

q^ and

{{\hat{D}}}

D^ , can be treated as ladder operators connecting various vectors of two biorthonormal families,

{{\hat{D}}}

Fφ and

{{{\mathcal {F}}}}_\psi

Fψ . In particular, the vectors in

{{{\mathcal {F}}}}_\varphi

Fφ are essentially monomials inx,

x^k

xk , while those in

{{{\mathcal {F}}}}_\psi

Fψ are weak derivatives of the Dirac delta distribution,

\delta ^{(m)}(x)

δ(m)(x) , times some normalization factor. We also show how bi-coherent states can be constructed for these

{{\hat{q}}}

q^ and

{{\hat{D}}}

D^ , both as convergent series of elements of

{{\hat{D}}}

Fφ and

{{{\mathcal {F}}}}_\psi

Fψ , or using two different displacement-like operators acting on the two vacua of the framework. Our approach generalizes well- known results for ordinary coherent states.

Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.

Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.

Abstract We introduce and study some special classes of ladder operators in finite-dimensional Hilbert spaces. In particular we consider a truncated version of quons, their pseudo-version, and a third family of operators acting on a closed chain. In this latter situation, we discuss the existence of what could be considered discrete coherent states, as suitable eigenvectors of the annihilation operator of the chain. We see that, under reasonable assumptions, a resolution of the identity can be recovered, involving these states, together with a biorthogonal family of vectors, which turn out to be eigenstates of the raising operator of the chain.

In this work we make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) As eigenstates of a deformed annihilation operator and ii) by application of a deformed displacement operator to the vacuum state. We also construct the coherent states for the same systems using the ladder operators obtained by traditional methods with the knowledge of the eigenfunctions and eigenvalues of the corresponding Schrödinger equation. We show that both methods yield coherent states with identical algebraic structure.

We consider the Hamiltonian

H

of a particle in one dimension with a position dependent mass for which we apply the recent strategy of the so-called {\em abstract ladder operators}, in the attempt to find its eigenvalues and eigenvectors. We don't assume that

H

is self-adjoint, while we focus on the case of a factorizable operator. We show then that pseudo-bosonic operators play a relevant role in this analysis, and we construct bi-coherent states attached to these operators. Explicit examples are discussed.

In the paper we developed a procedure for constructing generalized coherent states with shifted argument, as a result of the action of the generalized displacement operator. This was based on the action of a pair of nonlinear ladder operators, which generate nonlinear coherent states. To examine the properties of coherent states with shifted argument, the rules of the normal operator ordering technique (DOOT) were used. The results obtained were verified for a series of particular cases, obtaining expressions consistent with those in the literature. The expressions obtained will be, among others, useful in quantum optics, e.g. for calculating the Wigner operator in the representation of coherent states.