We present 1d-qt-ideal-solver, an open-source Python library for simulating one-dimensional quantum tunneling dynamics under idealized coherent conditions. The solver implements the split-operator method with second-order Trotter-Suzuki factorization, utilizing FFT-based spectral differentiation for the kinetic operator and complex absorbing potentials to eliminate boundary reflections. Numba just-in-time compilation achieves performance comparable to compiled languages while maintaining code accessibility. We validate the implementation through two canonical test cases: rectangular barriers modeling field emission through oxide layers and Gaussian barriers approximating scanning tunneling microscopy interactions. Both simulations achieve exceptional numerical fidelity with machine-precision energy conservation over femtosecond-scale propagation. Comparative analysis employing information-theoretic measures and nonparametric hypothesis tests reveals that rectangular barriers exhibit moderately higher transmission coefficients than Gaussian barriers in the over-barrier regime, though Jensen-Shannon divergence analysis indicates modest practical differences between geometries. Phase space analysis confirms complete decoherence when averaged over spatial-temporal domains. The library name reflects its scope: idealized signifies deliberate exclusion of dissipation, environmental coupling, and many-body interactions, limiting applicability to qualitative insights and pedagogical purposes rather than quantitative experimental predictions. Distributed under the MIT License, the library provides a deployable tool for teaching quantum mechanics and preliminary exploration of tunneling dynamics.
We present 1d-qt-ideal-solver, an open-source Python library for simulating one-dimensional quantum tunneling dynamics under idealized coherent conditions. The solver implements the split-operator method with second-order Trotter-Suzuki factorization, utilizing FFT-based spectral differentiation for the kinetic operator and complex absorbing potentials to eliminate boundary reflections. Numba just-in-time compilation achieves performance comparable to compiled languages while maintaining code accessibility. We validate the implementation through two canonical test cases: rectangular barriers modeling field emission through oxide layers and Gaussian barriers approximating scanning tunneling microscopy interactions. Both simulations achieve exceptional numerical fidelity with machine-precision energy conservation over femtosecond-scale propagation. Comparative analysis employing information-theoretic measures and nonparametric hypothesis tests reveals that rectangular barriers exhibit moderately higher transmission coefficients than Gaussian barriers in the over-barrier regime, though Jensen-Shannon divergence analysis indicates modest practical differences between geometries. Phase space analysis confirms complete decoherence when averaged over spatial-temporal domains. The library name reflects its scope: idealized signifies deliberate exclusion of dissipation, environmental coupling, and many-body interactions, limiting applicability to qualitative insights and pedagogical purposes rather than quantitative experimental predictions. Distributed under the MIT License, the library provides a deployable tool for teaching quantum mechanics and preliminary exploration of tunneling dynamics.
This non-peer-reviewed exploratory preprint tests whether a coupled quantum Hamiltonian with one additional spatial coordinate can produce barrier transmission distinct from ordinary one-dimensional quantum mechanics when both calculations share exactly the same observable slice V(x,0). A prescribed self-similar Gaussian barrier is compared using three calculations: fixed-width 1D, shrinking identical-slice 1D, and full coupled (x,w) scattering with harmonic confinement. The shrinking 1D slice is itself non-monotonic, so an upward turn is not uniquely extra-dimensional. The full model nevertheless gives a quantitatively different curve, with a sampled minimum near V0=18 and T=9.87e-3 at V0=64 versus 6.09e-9 for the identical slice. This is a reproducible coupled-channel result within a mathematical toy Hamiltonian, not evidence that nature has an accessible fourth spatial dimension. The barrier family is prescribed rather than derived from a constitutive field law. The record includes the manuscript and a supplement containing code, complete numerical results, metadata, and convergence checks. ChatGPT and Sarvam AI assisted with brainstorming and language development. ChatGPT also assisted with literature triage, control design, numerical implementation and auditing, and drafting. AI systems are not authors. Chaman Prakash Kanth remains responsible for the manuscript, assumptions, metadata, and publication decision.
Explicit quantum circuits for the acoustic and elastic wave equations, obtained by Hamiltonian simulation of their first-order forms, conserve energy and therefore describe closed domains: an outgoing wave reflects off the edge of the computational region or, with periodic boundaries, wraps around and re-enters from the opposite side. We open the domain with the simplest physical absorber, a graded sponge layer in which every wave field is damped at a rate that grows towards the outer edge, and implement the resulting dissipative dynamics on a quantum simulator by Schrödingerisation, which turns a non-unitary linear evolution into a family of unitary ones by means of a single auxiliary variable. We derive the whole chain from the partial differential equation to the circuit, in one, two and three dimensions. An independent finite-difference solver confirms that damping all wave fields absorbs far better than the common velocity-only damping mask. For the sponge the quantum analysis is sharp: recovery is exact on the whole non- negative half of the auxiliary axis at every time; that axis can be truncated periodically at a length set by the largest damping rate and the simulated time; and the success probability is a fixed constant, about 0.4, times the fraction of the energy not yet absorbed, with only one or two auxiliary qubits measured. On the circuit side, the damping step, which generic synthesis compiles into a number of controlled-NOT gates exponential in the number of qubits, factorises exactly into polynomially many multi-controlled phase gates. Gate-level simulations on up to 25 qubits agree with the classical solutions of the damped wave equations.
Quantum tunneling is a phenomenon where particles have a probability of penetrating a potential barrier despite having total energy llower than the barrier height. This study analyzes the tunneling effect by solving the one-dimensional (1D) Schrodinger equation using the finite difference method to obtain the wave function evolution for various potential barrier configurations. The solution is implemented in a Graphical User Interface (GUI) MATLAB to facilitate analysis and visualization, allowing users to interactively adjust potential parameters, energy, and other conditions. Simualation results demonstratehow transmission probability depends on energy, height and width of potential barrier. This GUI provides an intiuitive tool for exploring quantun tunneling, making it valuable for both education and research in quantum physics.
This paper presents a robust computational study of the Time-Dependent Schrödinger Equation (TDSE) dynamics for a spatial Gaussian wave packet confined within a one-dimensional infinite potential well. Employing a discrete Finite Difference Method (FDM) framework, we construct a tridiagonal Hamiltonian matrix to compute the unitary real-time evolution of the system. We systematically analyze two critical quantum phenomena: (1) scattering and quantum tunneling against an engineered central rectangular potential barrier, and (2) temporal dispersion and subsequent fractional quantum revival of the localized state. The numerical solver successfully resolved a distinct fractional revival and constructive self-interference peak at t = 1.416 time units. These computational benchmarks provide a rigorous pedagogical and diagnostic baseline for grid-based matrix methods in low-dimensional quantum mechanics.
This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.
Quantum tunneling is conventionally described by the Schrödinger wave equation with a prescribed static potential barrier, providing accurate transmission probabilities but offering limited insight into the microscopic dynamics of particle–barrier interactions. In this work, we develop a dynamical second-quantization framework in which the barrier is modeled as an ensemble of quantized internal modes rather than as an externally imposed classical potential. The tunneling particle interacts directly with these microscopic barrier excitations through a coupled particle–barrier Hamiltonian, from which the Heisenberg equations of motion are derived. Collective coherent excitations of the barrier modes give rise to an emergent effective barrier that naturally recovers the conventional rectangular barrier and the WKB transmission limit under appropriate conditions. Unlike standard treatments, the present formulation explicitly incorporates microscopic barrier dynamics and provides a unified description of particle–barrier coupling within a second-quantized formalism. The framework further suggests that repeated tunneling events may experience different microscopic interaction histories, motivating a statistical interpretation of tunneling times. Because both the particle and barrier are treated within the same operator formalism, the theory provides a natural foundation for extension to relativistic quantum transport, photonic barriers, cavity quantum electrodynamics, and other structured quantum media.