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

Abstract

Starting from the Wheeler-DeWitt equation for the Schwarzschild black hole interior, which is derived from a Hamiltonian formulated in terms of canonical phase space coordinates, we show that by applying a simple reparametrization, this equation can be expressed as the eigenvalue equation of a quantum linear harmonic oscillator. Within the standard quantization framework, we find that the resulting wave function diverges in the region of the classical singularity, and the expectation value of the Kretschmann scalar is undefined for all states within the black hole. However, when we apply the minimal uncertainty approach to the quantization process, we obtain a wave function that is both well-defined and square-integrable. Additionally, the expectation value of the Kretschmann scalar for these states remains finite throughout the black hole's interior, suggesting that the classical singularity is resolved in this approach, replaced it by a minimum radius.