Simulating Quantum Harmonic Oscillators Under Exponentially Modeled Forces: Deriving the Critical Stability Limit via the Lambert W Function and the Implications for Bound Quantum Systems under Spatial Asymmetry
This project investigates how quantum bound states behave when a standard physical model,the one-dimensional Quantum Harmonic Oscillator (QHO),is exposed to an extreme asymmetric force modeled by an exponential potential. This perturbation breaks the spatial symmetry of the system and serves as an idealized model for studying quantum systems under intense localized external stress fields, such as strong gravitational acceleration or structural stresses in semiconductor nanostructures. The project successfully marries theoretical mathematical derivations with an automated Python simulation environment. Mechanistically, as the asymmetric strength parameter increases, the uniform energy levels characteristic of a standard harmonic oscillator begin to distort and converge. Eventually, the system crosses a precise mathematical threshold where the confining potential well disappears entirely. At this exact branch boundary, the excited states completely evaporate, and the ground-state wavefunction collapses tightly against the domain boundary.
This project investigates how quantum bound states behave when a standard physical model,the one-dimensional Quantum Harmonic Oscillator (QHO),is exposed to an extreme asymmetric force modeled by an exponential potential. This perturbation breaks the spatial symmetry of the system and serves as an idealized model for studying quantum systems under intense localized external stress fields, such as strong gravitational acceleration or structural stresses in semiconductor nanostructures. The project successfully marries theoretical mathematical derivations with an automated Python simulation environment. Mechanistically, as the asymmetric strength parameter increases, the uniform energy levels characteristic of a standard harmonic oscillator begin to distort and converge. Eventually, the system crosses a precise mathematical threshold where the confining potential well disappears entirely. At this exact branch boundary, the excited states completely evaporate, and the ground-state wavefunction collapses tightly against the domain boundary.
The quantum harmonic oscillator is one of the few exactly solvable quantum systems whose spectral and semigroup properties are completely explicit. In realistic settings, however, external fields, anharmonic interactions, lattice defects, and engineered trapping profiles introduce perturbed potentials that require rigorous operator-theoretic tools to analyze stability, self-adjointness, spectral deformation, and the validity of perturbation expansions. This paper develops a functional-analytic and operator-theoretic framework for one-dimensional harmonic oscillators with additive perturbations W(x), focusing on (i) self-adjointness via Kato-Rellich and quadratic form methods, (ii) discrete-spectrum stability and eigenvalue bounds through the minmax principle and compactness arguments, (iii) analytic perturbation theory for isolated eigenvalues and the computation of first-order energy shifts for representative perturbations, and (iv) semigroup/resolvent estimates that quantify robustness of dynamics under perturbations. In addition, we propose an uncertainty-aware parameterization of perturbed potentials using intuitionistic fuzzy sets and fuzzy graph/hypergraph abstractions, linking operator stability certificates to entropy-style and stability-style diagnostics inspired by prior fuzzy-systems work. Representative figures and tables illustrate potential profiles, spectral schematics, energy shifts, and a structured workflow connecting operator estimates to computation. The resulting manuscript provides a Word-ready, mathematics-forward template for rigorous spectral analysis of perturbed quantum oscillators while also offering practical, interpretable computational guidance.
We present quantum algorithms for the simulation of quantum systems in one spatial dimension, which result in quantum speedups that range from superpolynomial to polynomial. We first describe a method to simulate the evolution of the quantum harmonic oscillator (QHO) based on a refined analysis of the Trotter-Suzuki formula that exploits the Lie algebra structure. For total evolution time t and precision ε>0, the complexity of our method is $ O(\exp(γ\sqrt{\log(N/ε)}))$, where γ>0 is a constant and N is the quantum number associated with an "energy cutoff" of the initial state. Remarkably, this complexity is subpolynomial in N/ε. We also provide a method to prepare discrete versions of the eigenstates of the QHO of complexity polynomial in log(N)/ε, where N is the dimension or number of points in the discretization. This method may be of independent interest as it provides a way to prepare, e.g., quantum states with Gaussian-like amplitudes. Next, we consider a system with a quartic potential. Our numerical simulations suggest a method for simulating the evolution of sublinear complexity O~(N1/3+o(1)), for constant t and ε. We also analyze complex one-dimensional systems and prove a complexity bound O~(N), under fairly general assumptions. Our quantum algorithms may find applications in other problems. As an example, we discuss the fractional Fourier transform, a generalization of the Fourier transform that is useful for signal analysis and can be formulated in terms of the evolution of the QHO.
This paper presents a comprehensive simulation of non-linear harmonic oscillators, incorporating quantum-mechanical effects to investigate collapse events and predict quantum entanglement. The core mechanism centers around utilizing entanglement to model probabilities of collapse, offering a significant advancement over existing simulation techniques. The study explores the potential of this approach to provide a deeper understanding of quantum phenomena and their implications within the context of complex systems. This work addresses a significant gap in simulation capabilities, focusing on a rigorous, quantum-driven model capable of capturing the nuances of collapse analysis.
Information‐based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length ( x c ). In the symmetric case, a wide range of x c has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary‐time propagation scheme. For asymmetric confinement, a variation‐induced exact diagonalization procedure is also introduced, which produces very high quality results. One finds that, in symmetric confinement, after a certain characteristic x c , all these properties converge to respective values of a free harmonic oscillator. In the asymmetric situation, excited‐state energies always pass through a maximum. For this potential, the classical turning point decreases, whereas well depth increases with the strength of asymmetry. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter. image
We investigate the competition between attractive spin-spin interactions and spin-separating external forces in the ground state of a one-dimensional Fermi-Hubbard model. We consider a lattice with open boundary conditions, subject to a linear external potential whose gradient is opposite for the two spin components, so that each spin species sees a potential minimum at a different end of the lattice. Using density-matrix renormalization group (DMRG) simulations, we map the ground-state density distributions and the number of doubly occupied sites as a function of the potential gradient β and interaction strength. We identify three distinct regimes separated by critical threshold gradients: (i) a small-β regime where fermion pairing remains robust against the external potential; (ii) an intermediate-β phase-separated regime characterized by a staircase-like decrease in the doublon number, corresponding to the successive, one-by-one breaking of bound pairs; and (iii) a large-β regime where the two spin components are completely spatially separated. We complement the numerical results with a phenomenological model and a local-density approximation analysis, from which we derive closed-form analytical estimates for these critical threshold values. We also verify that the staircase structure persists under additional harmonic confinement. Our results are directly testable in cold-atom experiments, and demonstrate that a spin-dependent linear potential enables precise, integer-level control of the number of bound fermion pairs.
This paper investigates the application of non-linear harmonic oscillators within the framework of quantum field theory, seeking to provide a deeper understanding of fundamental forces and their dynamics. The core focus is to develop a novel method, termed the "quantum-geometric" algorithm, that leverages geometric properties to represent and model quantum fields. Traditional approaches often struggle with the inherent complexities of quantum mechanics, and this research proposes a shift towards a more intuitive and geometric representation, potentially offering new insights into the nature of quantum phenomena. We present a preliminary algorithm and discuss its implications for both theoretical modeling and the potential for future advancements in quantum field theory.