从最小不确定度方法视角看量子黑洞作为谐振子
Quantum Black Hole as a Harmonic Oscillator from the Perspective of the Minimum Uncertainty Approach
AI总结：
本研究将黑洞质量本征值方程重写为谐振子形式，采用最小不确定度量子化方法，得到平方可积波函数及随n²增长的面积谱，并揭示量子隧穿连接内外区域，展示了最小长度效应的正则化作用。
摘要：
从Mäkelä和Repo推导的黑洞质量本征值方程出发，我们证明，通过重新参数化径向坐标和波函数，该方程可以改写为量子谐振子的本征值方程。随后，我们使用两种量子化方法研究Schwarzschild黑洞的内部。在标准量子化中，面积和质量谱是离散的，由量子数$n$表征，但波函数不是平方可积的，限制了其物理解释。相比之下，最小不确定度量子化方法产生随$n^2$增长的面积谱，因此质量$M$也随之增加。在此框架下，波函数是有限且平方可积的，其收敛性要求形变参数$β$由离散量子数$m$调节。波函数展现出量子隧穿效应，连接黑洞内部与其外部及白洞区域，这些效应在$β\ o 0$极限下消失。这些结果展示了最小长度效应如何既正则化波函数，又修改黑洞的半经典结构。
英文摘要：
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 $β$ 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 $β\to 0$. These results demonstrate how minimal-length effects both regularize the wave function and modify the semiclassical structure of the black hole.
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