所谓特征值与平均值,就是发散级数在取不同截断时得到的值与级数平均值的关系。

级数和积分,在收敛时是趋于同一结果,在发散时表现的才会大不相同。但是,在尺度很大的时候,这种不断的小差别,即便是收敛级数,在步长取最小值后,差别也会越来越大。这可以通过物理实验进行检验。

附:

普朗克长度:

\[\ell_p=\sqrt{\frac{\hbar G}{c^3}\]

这项单位首先由顺便测试一下本站latex的公式序号是否工作正常:

Let’s calculate the simplest black hole which is the Schwarzschild black hole. The metric of the black hole reads \begin{equation} ds^2=-\bigg(1-\frac{2M}{r}\bigg)dt^2+\bigg(1-\frac{2M}{r}\bigg)^{-1}dr^2+r^2d\theta+r^2\sin^2\theta d\Phi^2 \end{equation} where M is the mass of the black hole. We can see that this metric describes a statistical black hole. We put it into Klein-Gordon equation \begin{equation} \frac{1}{\sqrt{-g}}\frac{tial}{tial x^\mu}\bigg(\sqrt{-g}g^{\mu\nu}\frac{tial \phi}{tial x^\nu}\bigg)-\frac{\mu_0 ^2 c^2}{\hbar^2}\phi=0 \end{equation} where \(g=-r^4\sin^2\theta\) is the determinant of the metric. then we can get \begin{equation}\label{eq:CG-LONG} -\bigg(1-\frac{2M}{r}\bigg)^{-1}\frac{tial^2\phi}{tial t^2}+\frac{1}{r^2}\frac{tial}{tial r}\bigg[(r^2-2Mr)\frac{tial \phi}{tial r}\bigg]+\frac{1}{r^2\sin\theta}\frac{tial}{tial\theta}\bigg(\sin\theta\frac{tial \phi}{tial \theta}\bigg)+\frac{1}{r^2\sin^2\theta}\frac{tial^2\phi}{tial\Phi^2}=\frac{\mu_0^2}{\hbar^2}\phi \end{equation} We set the wave function as \begin{equation} \phi=e^{-i\omega t}Y_{lm}(\theta,\Phi)\psi(r) \end{equation} put it into \eqref{eq:CG-LONG},then we get \begin{equation}\label{eq:jingxiangfangcheng} \frac{d}{dr}\bigg[(r^2-2Mr)\frac{d\psi}{dr}\bigg]+\bigg[\frac{r^3\omega^2}{r-2M}-\frac{\mu_0^2}{\hbar^2}r^2-l(l+1)\bigg]\psi=0 \end{equation} Using Wentzel-Kramers-Brillouin (WKB) approximation,let \begin{equation} \psi(r)=e^{\frac{i}{h}s(r)} \end{equation} put it into \eqref{eq:jingxiangfangcheng}, then we have \begin{equation} k^2=\bigg(1-\frac{2M}{r}\bigg)^{-1}\bigg[\omega^2\bigg(1-\frac{2M}{r}\bigg)^{-1}-\frac{\mu_0^2}{\hbar^2}-\frac{l(l+1)}{r^2}\bigg] \end{equation} which is the wave vector. Here we use the semi-classical quantization condition \begin{equation}\label{eq:semi-classical-vactor} n\pi=\int_{r_{H}+h}^Lk(r,l,\omega)dr,n \in \mathbb{N_+} \end{equation}