Improve knowledge base

1 files changed, 12 insertions, 10 deletions

diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md
index 1214703..28885af 100644
--- a/source/know/concept/maxwell-bloch-equations/index.md
+++ b/source/know/concept/maxwell-bloch-equations/index.md
@@ -17,8 +17,8 @@ where $$\varepsilon_g$$ and $$\varepsilon_e$$ are the time-independent eigenener
 and the weights $$c_g$$ and $$c_g$$ are functions of $$t$$:
 
 $$\begin{aligned}
-    \ket{\Psi}
-    &= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar}
+    \ket{\Psi(t)}
+    &= c_g(t) \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e(t) \ket{e} e^{-i \varepsilon_e t / \hbar}
 \end{aligned}$$
 
 This system is being perturbed by an electromagnetic wave
@@ -32,8 +32,8 @@ $$\begin{aligned}
 Where the forward-propagating component $$\vb{E}^{+}$$
 is a modulated plane wave $$\vb{E}_0^{+} e^{-i \omega t}$$
 with slowly-varying amplitude $$\vb{E}_0^{+}(t)$$,
-and similarly $$\vb{E}^{-}(t) \equiv \vb{E}_0^{-}(t) e^{i \omega t}$$;
-since $$\vb{E}$$ is real, $$\vb{E}_0^{+} \!=\! (\vb{E}_0^{-})^*$$.
+and similarly $$\vb{E}^{-}(t) \equiv \vb{E}_0^{-}(t) e^{i \omega t}$$.
+Since $$\vb{E}$$ is real, $$\vb{E}_0^{+} \!=\! (\vb{E}_0^{-})^*$$.
 
 For $$\ket{\Psi}$$ as defined above,
 the pure [density operator](/know/concept/density-operator/)
@@ -92,7 +92,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 However, the light wave affects the electron,
-so the actual electromagnetic dipole moment $$\vb{p}$$ is as follows,
+so the true electromagnetic dipole moment $$\vb{p}$$ is as follows,
 using [Laporte's selection rule](/know/concept/selection-rules/)
 to remove diagonal terms by assuming that
 the electron's orbitals are spatially odd or even:
@@ -106,9 +106,9 @@ $$\begin{aligned}
     \\
     &= q \Big( \rho_{ge} \matrixel{e}{\vu{x}}{g} + \rho_{eg} \matrixel{g}{\vu{x}}{e} \Big)
     \\
-    &= \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t)
+    &= \vb{p}_0^{-} \rho_{ge} + \vb{p}_0^{+} \rho_{eg}
     \\
-    &\equiv \vb{p}^{-}(t) + \vb{p}^{+}(t)
+    &\equiv \vb{p}^{-} + \vb{p}^{+}
 \end{aligned}$$
 
 Where we have split $$\vb{p}$$ analogously to $$\vb{E}$$
@@ -117,8 +117,9 @@ Its equation of motion can then be found from the optical Bloch equations:
 
 $$\begin{aligned}
     \dv{\vb{p}^{+}}{t}
-    = \vb{p}_0^{+} \dv{\rho_{eg}}{t}
-    = - \vb{p}_0^{+} \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
+    &= \vb{p}_0^{+} \dv{\rho_{eg}}{t}
+    \\
+    &= - \vb{p}_0^{+} \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
     + \frac{i}{\hbar} \vb{p}_0^{+} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \Big( \rho_{gg} - \rho_{ee} \Big)
 \end{aligned}$$
 
@@ -147,7 +148,8 @@ we find its equation of motion to be:
 $$\begin{aligned}
     \dv{d}{t}
     &= \dv{\rho_{ee}}{t} - \dv{\rho_{gg}}{t}
-    = 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
+    \\
+    &= 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
     + \frac{i 2}{\hbar} \Big( \vb{p}^{-} \cdot \vb{E}^{+} - \vb{p}^{+} \cdot \vb{E}^{-} \Big)
 \end{aligned}$$