diff --git a/Chapter-4/Chapter-4.tex b/Chapter-4/Chapter-4.tex index 0cf5bee..038d38b 100644 --- a/Chapter-4/Chapter-4.tex +++ b/Chapter-4/Chapter-4.tex @@ -19,7 +19,7 @@ Let, % \begin{equation} \label{eq:zeldovich} - \braket{\phi_1}{\phi_2} + \braket{\phi_1 | \phi_2} =\lim_{\mu\to 0}\int_{0}^{\infty}e^{-\mu r^2}\phi_1(r)\phi_2(r)\,dr \end{equation} % @@ -28,7 +28,7 @@ their asymptotic forms, $N_1 e^{ip_1r}$ and $N_2 e^{ip_2r}$ respectively. Then t integral can be broken into two. % \begin{spliteq} - \braket{\phi_1}{\phi_2} + \braket{\phi_1 | \phi_2} & \approx \lim_{\mu\to 0}\int_{0}^{R}e^{-\mu r^2}\phi_1(r)\phi_2(r)\,dr \\ & \quad +N_1N_2\lim_{\mu\to 0} \int_{R}^{\infty}e^{-\mu r^2}e^{ip_1r}e^{ip_2r}\,dr \\ @@ -40,7 +40,7 @@ the first term, this simplifies as follows. % \begin{equation} \label{eq:easy} - \braket{\phi_1}{\phi_2} \approx \int_{0}^{R}\phi_1(r)\phi_2(r)\,dr + \braket{\phi_1 | \phi_2} \approx \int_{0}^{R}\phi_1(r)\phi_2(r)\,dr +\frac{\ii N_1 N_2 e^{\ii(p_1+p_2)R}}{p_1+p_2} \end{equation} %