An infinite square-well potential as a limiting case of a finite square-well potential in a minimal-length scenario

Abstract

One of the most widely problem studied in quantum mechanics is of an infinite square-well potential. In a minimal-length scenario its study requires additional care because the boundary conditions at the walls of the well are not well fixed. In order to avoid this we solve the finite square-well potential whose the boundary conditions are well fixed, even in a minimal-length scenario, and then we take the limit of the potential going to infinity to find the eigenfunctions and the energy equation for the infinite square-well potential. Although the first correction for the energy eigenvalues is the same one has been found in the literature, our result shows that the eigenfunctions have the first derivative continuous at the square-well walls what is in disagreement those previous work. That is because in the literature the authors have neglected the hyperbolic solutions and have assumed the discontinuity of the first derivative of the eigenfunctions at the walls of the infinite square-well which is not correct. As we show, the continuity of the first derivative of the eigenfunctions at the square-well walls guarantees the continuity of the probability current density and the unitarity of the time evolution

arXiv commentAccepted for publication in the IJMPA

References in corpus (25)

• A Large Mass Hierarchy from a Small Extra Dimension
• The Hierarchy Problem and New Dimensions at a Millimeter
• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• Quantum gravity and minimum length
• Bounds on Universal Extra Dimensions
• Testable scenario for Relativity with minimum-length
• Universality of Quantum Gravity Corrections
• Discreteness of Space from the Generalized Uncertainty Principle
• On Gravity and the Uncertainty Principle
• Nonpointlike Particles in Harmonic Oscillators
• Minimal Length Uncertainty Relation and Hydrogen Atom
• Self-adjoint extensions of operators and the teaching of quantum mechanics
• Signatures in the Planck Regime
• Phenomenological Implications of the Generalized Uncertainty Principle
• A Higher Order GUP with Minimal Length Uncertainty and Maximal Momentum
• Minimum Length from Quantum Mechanics and Classical General Relativity
• Some Aspects of Minimal Length Quantum Mechanics
• New Approach to Nonperturbative Quantum Mechanics with Minimal Length Uncertainty
• A class of GUP solutions in deformed quantum mechanics
• On the Minimal Length Uncertainty Relation and the Foundations of String Theory
• Canonical Quantization and Impenetrable Barriers
• Hermitian versus non-Hermitian representations for minimal length uncertainty relations
• Quantum Wells and the Generalized Uncertainty Principle
• Dirac $\delta$-function potential in quasiposition representation of a minimal-length scenario
• Near-Dirichlet quantum dynamics for a $p^3$-corrected particle on an interval

Cited by in corpus (1)

• Remarks on the quasi-position representation in models of generalized uncertainty principle