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 M\"akel\"a 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.