Investigation on the Mechanical Model and Quantization of Black Holes
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Investigation on the Mechanical Model and Quantization of Black Holes Zhao-Xi Li 1 & Ji-Jian Jiang 2 Received: 18 June 2020 / Accepted: 17 August 2020/ # Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
Quantum mechanics model of Schwarzschild black hole is obtained by putting the classical Hamilton into Schrödinger equation. The results show that quantum mechanics model of Schwarzschild black hole is a harmonic oscillator containing perturbation in the general case. It is related to the microscopic quantum potential. Under special cases, it can be simplified into a standard model of the harmonic oscillator. The energy spectrum, event horizon area and wave function of Schwarzschild black hole are strictly obtained by the second quantization. The quantization of black hole is also proved. Keywords Black hole . Hamiltonian . Quantum mechanics model . Second quantization . Wave function
1 Introduction Black hole and its Hawking radiation have become hot topics in theoretical physics based on the pioneering researchers of Bekenstein [1, 2] and Hawking [3]. As is well known, the study on black hole is focusing on its thermodynamics [4, 5]. Meanwhile black hole is also an ideal model of understanding quantum gravity. Black hole radiation is essentially a macroscopic quantum effect [6–9]. Because the black hole is dominated by gravitational interaction, its quantum behavior also reflects the quantum properties of gravity in a certain sense [10–15]. Therefore, quantization of black hole provides an ideal model for researching quantum gravity. Building quantum mechanics model and studying its quantization of black holes is very important in further understanding the nature of black hole and quantum properties of gravity [16–22].
* Ji-Jian Jiang [email protected]
1
School of Science, Shandong Jianzhu University, Jinan 250101, China
2
School of Physics and Electronic Engineering, Heze University, Heze 274015, China
International Journal of Theoretical Physics
In 1974, Bekenstein firstly proposed the following expression about horizon eigenvalue of black hole [23] An ¼ γnl2pl ;
ð1Þ
where n is a positive integer, γ is a pure number, and Planck length is denoted by pffiffiffiffiffiffiffiffiffiffiffiffi lpl ¼ ℏGc−3 . There are two quantization schemes on black hole. One is to use observables of black hole through canonical transformation to realize its quantization [19–22]. The other is to put the classical Hamilton [16–18, 24] into the Schrödinger equation. The abovementioned methods have made great contributions to achieve the quantization of black hole. Considering large eigenvalue, J. Louko group used WKB approximation to find that the behavior of the energy eigenvalue can be expressed as E2WKB ∼2n þ const þ 0ð1Þ:
ð2Þ
Meanwhile, Louko and Mäkelä also obtained the area spectrum of Event horizon of Schwarzschild black hole, which can be written as [17] An ∼32πkl 2pl þ const þ 0ð1Þ:
ð3Þ
Here, k is integer. When k is a large value, 0(1) tends to 0. Unfortunately, the researchers did not give a direct, clear and s
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