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arXiv:2603.18047

Some Studies On Exact Solutions Of Models In Noncommutative Spaces

Manjari Dutta

quant-phhep-thSubmitted Mar 17, 2026• Announced Mar 18, 2026• 0 citations
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Abstract

The central theme of my thesis is to explore various simple prototype models that are exactly solvable in the framework of time dependent noncommutative spaces. By adopting the methodology provided by the Lewis Riesenfeld theory, we developed a procedure for obtaining a class of exact solutions for such model systems. We analyzed these solutions by deriving the energy expectation values analytically and representing those energy dynamics graphically. We also examined the explicit existence of a non-zero Berry geometric phase in the noncommutative framework and analyzed the role of noncommutativity in generating a non-trivial Berry phase when the model Hamiltonian and the noncommutative parameters are periodic in time. Overall, my thesis contributes to a deeper understanding of quantum theory in time dependent noncommutative backgrounds and indicates a strong possibility for developing a consistent quantum theory within such frameworks.

Comments
Ph. D Thesis (2024), Advisor : Prof. Sunandan Gangopadhyay, Collaborator : Dr. Shreemoyee Ganguly

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The central theme of my thesis is to explore various simple prototype models that are exactly solvable in the framework of time dependent noncommutative spaces. By adopting the methodology provided by the Lewis Riesenfeld theory, we developed a procedure for obtaining a class of exact solutions for such model systems. We analyzed these solutions by deriving the energy expectation values analytically and representing those energy dynamics graphically. We also examined the explicit existence of a non-zero Berry geometric phase in the noncommutative framework and analyzed the role of noncommutativity in generating a non-trivial Berry phase when the model Hamiltonian and the noncommutative parameters are periodic in time. Overall, my thesis contributes to a deeper understanding of quantum theory in time dependent noncommutative backgrounds and indicates a strong possibility for developing a consistent quantum theory within such frameworks.
Methods for obtaining exact and approximate solutions of the evolution of quantum-mechanical problems are discussed. The cyclic evolution of quantum systems described by time-periodic Hamiltonians is analyzed. A class of time-periodic Hamiltonians is constructed in the close analytical form. The corresponding cyclic solutions are calculated. Time-dependent Hamiltonians are generated whose expectation values calculated with cyclic solutions are time independent. It is shown that the expectation values of the spin projection calculated with the same cyclic solutions, as well as the probability density of finding a particle at a given space-time point, are also time independent. Therefore, the approach can be used to simulate quantum dynamic potential wells with the particle localization effect. Nonadiabatic geometric phases are expressed in terms of the cyclic solutions. Exactly solvable time-dependent problems are used to construct a universal set of gates for quantum computers. A method for obtaining entanglement operators is discussed.
Abstract In this paper we look for the existence of Berry phase in time dependent harmonic oscillators in noncommutative space. Two systems are considered in our study in the noncommutative framework. The first one is a system in which a scale invariant term is present in the Hamiltonian from the beginning, and the second one is a system in which a scale invariant term emerges due to a change of variables from the noncommutative to the commutative ones. We first compute the eigenstates of both the systems using the Lewis invariant approach. We then employ the Lewis invariant technique to obtain the geometric phase under adiabatic approximation. We also calculate the Berry phase explicitly by choosing appropriate forms of the time dependent parameters appearing in the Hamiltonian. Our analysis surprisingly reveals that a scale invariant time reversal symmetry breaking term may not always lead to a non trivial Berry phase.
We investigate a quantum mechanical system on a noncommutative space for which the structure constant is explicitly time dependent. Any autonomous Hamiltonian on such a space acquires a time-dependent form in terms of the conventional canonical variables. We employ the Lewis-Riesenfeld method of invariants to construct explicit analytical solutions for the corresponding time-dependent Schr\"odinger equation. The eigenfunctions are expressed in terms of the solutions of variants of the nonlinear Ermakov-Pinney equation and discussed in detail for various types of background fields. We utilize the solutions to verify a generalized version of Heisenberg's uncertainty relations for which the lower bound becomes a time-dependent function of the background fields. We study the variance for various states, including standard Glauber coherent states with their squeezed versions and Gaussian Klauder coherent states resembling a quasiclassical behavior. No type of coherent state appears to be optimal in general with regard to achieving minimal uncertainties, as this feature turns out to be background field dependent.
There exist a number of typical and interesting systems or models which possess three-generator Lie-algebraic structure in atomic physics, quantum optics, nuclear physics and laser physics. The well-known fact that all simple 3-generator algebras are either isomorphic to the algebra sl(2,C)sl(2,C)sl(2,C) or to one of its real forms enables us to treat these time-dependent quantum systems in a unified way. By making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation, the present paper obtains exact solutions of the time-dependent Schrödinger equations governing various three-generator quantum systems. For some quantum systems whose time-dependent Hamiltonians have no quasialgebraic structures, we show that the exact solutions can also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator (i.e., the time-independent invariant that commutes with the time-dependent Hamiltonian). The topological property of geometric phase factors in time-dependent systems is briefly discussed.
In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.
A method of constructing periodic time-dependent Hamiltonians admitting exact solutions is used to study the geometric phase. The approach is based on the transformation of soluble time-independent equations into time-dependent ones by employing a set of special time-dependent transformation operators. A class of periodic time-dependent Hamiltonians with cyclic solutions is constructed in a closed analytic form and the nonadiabatic geometric phase is determined in terms of the obtained solutions.