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Fix bra-ket notation

This commit is contained in:
ysyapa 2023-08-28 14:16:23 +00:00
parent 1afefd6a78
commit dfa4a60d10
1 changed files with 3 additions and 3 deletions
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6
Chapter-4/Chapter-4.tex
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@ -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}
%