WKB Approximation for a Deformed Schrodinger-like Equation and its Applications to Quasinormal Modes of Black Holes and Quantum Cosmology

TL;DR

This paper develops a WKB approximation method for a deformed Schrödinger-like equation relevant in quantum gravity models, applying it to black hole quasinormal modes, quantum cosmology, and potential quantum gravity effects.

Contribution

It introduces a generalized WKB approach for deformed quantum equations, including connection formulas and quantization rules, with applications to black holes and cosmology.

Findings

Calculated quantum gravity effects on black hole quasinormal modes

Estimated black hole area quantum via Bohr's principle

Analyzed inflation probability in quantum cosmology

Abstract

In this paper, we use the WKB approximation method to approximately solve a deformed Schrodinger-like differential equation: $\left[ -\hbar^{2} \partial_{\xi}^{2}g^{2}\left( -i\hbar\alpha\partial_{\xi}\right) -p^{2}\left( \xi\right) \right] \psi\left( \xi\right) =0$, which are frequently dealt with in various effective models of quantum gravity, where the parameter $\alpha$ characterizes effects of quantum gravity. For an arbitrary function $g\left( x\right) $ satisfying several properties proposed in the paper, we find the WKB solutions, the WKB connection formulas through a turning point, the deformed Bohr--Sommerfeld quantization rule, and the deformed tunneling rate formula through a potential barrier. Several examples of applying the WKB approximation to the deformed quantum mechanics are investigated. In particular, we calculate the bound states of the P\"{o}schl-Teller potential and estimate the effects of quantum gravity on the quasinormal modes of a Schwarzschild black hole. Moreover, the area quantum of the black hole is considered via Bohr's correspondence principle. Finally, the WKB solutions of the deformed Wheeler--DeWitt equation for a closed Friedmann universe with a scalar field are obtained, and the effects of quantum gravity on the probability of sufficient inflation is discussed in the context of the tunneling proposal.