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Journal of Physics A Mathematical and Theoretical

A position dependent mass Hamiltonian and abstract ladder operators

Fabio Bagarello, Emanuele Balistreri, Antonino Faddetta

Published Apr 28, 2026• 3 citations
Publisher

Abstract

Abstract 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 abstract ladder operators , in the attempt to find its eigenvalues and eigenvectors. We do not 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.

DOI
10.1088/1751-8121/ae65e6

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Abstract 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 abstract ladder operators , in the attempt to find its eigenvalues and eigenvectors. We do not 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.
We consider the Hamiltonian HHH 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 HHH 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.
Position-dependent e?ective mass systems are of great signi?cance due to the fact that these models have numerous applications in various areas of physics. The qualitative understanding of a complicated realistic system can be acquired by analyzing the exact solutions of a related simpli?ed model. However, the quantization of position-dependent e?ective mass systems and ?nding their solutions, involve some conceptual and mathematical di?culties of a fundamental nature. The factorization method provides us with a powerful tool for obtaining solutions and the underlying algebraic structure of the exactly solvable systems. The underlying algebra of a system has vast applications in di?erent areas of mathematics and physics, such as it plays an important role in the theory of coherent states. Coherent states are extremely useful in various areas such as quantum mechanics, quantum optics, quantum information and group theory. Inthisthesis, position-dependente?ectivemasssystemsarestudiedinthecontextoftheir quantization, ?nding the solutions, construction of the algebraic structure and associated coherent states. In the ?rst part of the thesis we mainly focus on quantizing and obtaining the exact solutions of the systems with spatially varying mass. The next part deals with the construction of the ladder operators and the inherent algebra of the pertaining systems. The associated coherent states and their properties are presented in the ?nal part of the thesis. Beside the traditional way of obtaining exact solutions by solving the Schrödinger equation there exists another elegant method to solve the systems algebraically by factorizing the corresponding Hamiltonian. This method is based on supersymmetric quantum mechanics and the integrability condition, commonly known as shape invariance. After quantizing the position-dependent e?ective mass system, this factorization technique is used to determine the energy spectrum and the corresponding wave functions. For the sake of completeness the \nmethod of solving a time-independent Schrödinger equation with spatially varying mass is also discussed. Considering a non-linear harmonic oscillator as an illustrative example, it is shown that both the above procedures produce the same results. The property of shape invariance enables us to obtain the ladder operators of the con?ning system. A general recipe for the construction of the ladder operators and inherent algebra for the position-dependent e?ective mass systems is presented. In order to exemplify the general formalism, a non-linear harmonic oscillator together with several other examples of the shape invariant systems with position-dependent e?ective mass is considered. Explicit expressions for the ladder operators and the associated algebra are presented. Usingtheladderoperatorsandtheunderlyingalgebra,thecoherentstatesforthepositiondependent e?ective mass systems are constructed and their properties are analyzed. In particular, we emphasize on various kinds of coherent states for a non-linear harmonic oscillator with spatially varying mass. By realizing SU(1,1) as the dynamic group of the system, the construction of Barut-Girardello coherent states is presented. In addition, an algebraic independent kind of coherent states, namely Gazeau-Klauder coherent states, are also constructed. The statistical properties of Barut-Girardello and Gazeau-Klauder coherent states are investigated by means of the Mandel parameter and the second order correlation function. Moreover, the temporal evolution of the Gazeau-Klauder coherent states is analysed by means of autocorrelation function. It is shown that these states mimic the phenomena of quantum revivals and fractional revivals during their time evolution
In the context of the factorization method, we investigate the pseudo-Hermitian coherent states and their Hermitian counterpart coherent states under the generalized quantum condition in the framework of a position-dependent mass. By considering a specific modification in the superpotential, suitable annihilation and creation operators are constructed in order to reproduce the Hermitian counterpart Hamiltonian in the factorized form. We show that by means of these ladder operators, we can construct a wide range of exactly solvable potentials as well as their accompanying coherent states. Alternatively, we explore the relationship between the pseudo-Hermitian Hamiltonian and its Hermitian counterparts, obtained from a similarity transformation, to construct the associated pseudo-Hermitian coherent states. These latter preserve the structure of Perelomov's states and minimize the generalized position–momentum uncertainty principle. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
We consider a rather general version of ladder operator ZZZ used by some authors in few recent papers, [H0,Z]=λZ[H_0,Z]=λZ[H0​,Z]=λZ for some λ∈Rλ\in\mathbb{R}λ∈R, H0=H0†H_0=H_0^\daggerH0​=H0†​, and we show that several interesting results can be deduced from this formula. Then we extend it in two ways: first we replace the original equality with formula [H0,Z]=λZ[Z†,Z][H_0,Z]=λZ[Z^\dagger, Z][H0​,Z]=λZ[Z†,Z], and secondly we consider [H,Z]=λZ[H,Z]=λZ[H,Z]=λZ for some λ∈Cλ\in\mathbb{C}λ∈C, H≠H†H\neq H^\daggerH=H†. In both cases many applications are discussed. In particular we consider factorizable Hamiltonians and Hamiltonians written in terms of operators satisfying the generalized Heisenberg algebra or the $\D$ pseudo-bosonic commutation relations.
Abstract We analyze a general family of position-dependent mass (PDM) quantum Hamiltonians which are not self-adjoint and include, as particular cases, some Hamiltonians obtained in phenomenological approaches to condensed matter physics. We build a general family of self-adjoint Hamiltonians which are quantum mechanically equivalent to the non-self-adjoint proposed ones. Inspired by the probability density of the problem, we construct an ansatz for the solutions of the family of self-adjoint Hamiltonians. We use this ansatz to map the solutions of the time independent Schrödinger equations generated by the non-self-adjoint Hamiltonians into the Hilbert space of the solutions of the respective dual self-adjoint Hamiltonians. This mapping depends on both the PDM and on a function of position satisfying a condition that assures the existence of a consistent continuity equation. We identify the non-self-adjoint Hamiltonians here studied with a very general family of Hamiltonians proposed in a seminal article of Harrison (1961 Phys. Rev. 123 85 ) to describe varying band structures in different types of metals. Therefore, we have self-adjoint Hamiltonians that correspond to the non-self-adjoint ones found in Harrison’s article.