Maximally Localized States in Quantum Mechanics with a Modified Commutation Relation to All Orders

Abstract

We construct the states of maximal localization taking into account a modification of the commutation relation between position and momentum operators to all orders of the minimum length parameter. To first order, the algebra we use reproduces the one proposed by Kempft, Mangano and Mann. It is emphasized that a minimal length acts as a natural regulator for the theory, thus eliminating the otherwise ever appearing infinities. So, we use our results to calculate the first correction to the Casimir Effect due to the minimal length. We also discuss some of the physical consequences of the existence of a minimal length, culminating in a proposal to reformulate the very concept of "position measurement".

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Cited by in corpus (16)

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• Exact solutions of the (2+1) -dimensional Dirac oscillator under a magnetic field in the presence of a minimal length in the noncommutative phase-space
• Dirac $\delta$-function potential in quasiposition representation of a minimal-length scenario
• Quantum, noncommutative and MOND corrections to the entropic law of gravitation
• Some consequences of GUP induced ultraviolet wavevector cutoff in one-dimensional Quantum Mechanics
• Relativistic Approach to the Hydrogen Atom in a Minimal Length Scenario
• Nonstandard Deformed Oscillators from $q$- and $p,q$-Deformations of Heisenberg Algebra
• New version of pseudo-hermiticity in the two-sided deformation of Heisenberg algebra
• New Deformed Heisenberg Algebra from the $\mu$-Deformed Model of Dark Matter
• Particle in a cavity in one-dimensional bandlimited quantum mechanics
• Remarks on the quasi-position representation in models of generalized uncertainty principle