Quantum Black Hole as a Harmonic Oscillator from the Perspective of the Minimum Uncertainty Approach

Abstract

Starting from the eigenvalue equation for the mass of a black hole derived by Mäkelä and Repo, we show that, by reparametrizing the radial coordinate and the wave function, it can be rewritten as the eigenvalue equation of a quantum harmonic oscillator. We then study the interior of a Schwarzschild black hole using two quantization approaches. In the standard quantization, the area and mass spectra are discrete, characterized by a quantum number $n$, but the wave function is not square-integrable, limiting its physical interpretation. In contrast, a minimal-uncertainty quantization approach yields an area spectrum that grows as $n^2$, and consequently the mass $M$ also increases. In this framework, the wave function is finite and square-integrable, with convergence requiring that the deformation parameter $\beta$ be regulated by a discrete quantum number $m$. The wave function exhibits quantum tunneling connecting the black hole interior with both its exterior and a white hole region, effects that disappear in the limit $\beta \to 0$. These results demonstrate how minimal-length effects both regularize the wave function and modify the semiclassical structure of the black hole.

References in corpus (15)

• Loop Quantum Dynamics of the Schwarzschild Interior
• Polymer Quantum Mechanics and its Continuum Limit
• Phenomenological loop quantum geometry of the Schwarzschild black hole
• Black hole interior from loop quantum gravity
• Interior of a Schwarzschild black hole revisited
• Black hole entropy quantization
• Black hole singularity resolution via the modified Raychaudhuri equation in loop quantum gravity
• Wave functions for the Schwarschild black hole interior
• Path integral polymer propagator of relativistic and non-relativistic particles
• Deformed algebra and the effective dynamics of the interior of black holes
• The minimal length: a cut-off in disguise?
• Black hole interior quantization: a minimal uncertainty approach
• Modified entropies as the origin of generalized uncertainty principles
• On the horizon area of effective loop quantum black holes
• Semiclassical resolution of the black hole singularity inspired in the minimal uncertainty approach

Cited by in corpus (1)

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