More improvements to knowledge base

1 files changed, 13 insertions, 12 deletions

diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md
index ba8a677..0252b5c 100644
--- a/source/know/concept/maxwell-bloch-equations/index.md
+++ b/source/know/concept/maxwell-bloch-equations/index.md
@@ -12,13 +12,13 @@ layout: "concept"
 ---
 
 For an electron in a two-level system with time-independent states
-$$\Ket{g}$$ (ground) and $$\Ket{e}$$ (excited),
+$$\ket{g}$$ (ground) and $$\ket{e}$$ (excited),
 consider the following general solution
-to the full Schrödinger equation:
+to the time-dependent Schrödinger equation:
 
 $$\begin{aligned}
-    \Ket{\Psi}
-    &= c_g \: \Ket{g} \exp(-i E_g t / \hbar) + c_e \: \Ket{e} \exp(-i E_e t / \hbar)
+    \ket{\Psi}
+    &= c_g \ket{g} \exp(-i E_g t / \hbar) + c_e \ket{e} \exp(-i E_e t / \hbar)
 \end{aligned}$$
 
 Perturbing this system with
@@ -87,15 +87,16 @@ $$\begin{aligned}
 \end{aligned}$$
 
 
+
 ## Optical Bloch equations
 
-For $$\Ket{\Psi}$$ as defined above,
+For $$\ket{\Psi}$$ as defined above,
 the corresponding pure [density operator](/know/concept/density-operator/)
 $$\hat{\rho}$$ is as follows:
 
 $$\begin{aligned}
     \hat{\rho}
-    = \Ket{\Psi} \Bra{\Psi}
+    = \ket{\Psi} \bra{\Psi}
     =
     \begin{bmatrix}
         c_e c_e^* & c_e c_g^* \exp(-i \omega_0 t) \\
@@ -159,11 +160,10 @@ $$\begin{aligned}
 
 These equations are correct if nothing else is affecting $$\hat{\rho}$$.
 But in practice, these quantities decay due to various processes,
-e.g. spontaneous emission (see [Einstein coefficients](/know/concept/einstein-coefficients/)).
+e.g. [spontaneous emission](/know/concept/einstein-coefficients/).
 
-Let $$\rho_{ee}$$ decays with rate $$\gamma_e$$.
-Since the total probability $$\rho_{ee} + \rho_{gg} = 1$$,
-we thus have:
+Suppose $$\rho_{ee}$$ decays with rate $$\gamma_e$$.
+Because the total probability $$\rho_{ee} + \rho_{gg} = 1$$, we have:
 
 $$\begin{aligned}
     \Big( \dv{\rho_{ee}}{t} \Big)_{e}
@@ -220,10 +220,11 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Many authors simplify these equations a bit by choosing
+Some authors simplify these equations a bit by choosing
 $$\gamma_g = 0$$ and $$\gamma_\perp = \gamma_e / 2$$.
 
 
+
 ## Including Maxwell's equations
 
 This two-level system has a dipole moment $$\vb{p}$$ as follows,
@@ -286,7 +287,7 @@ $$\begin{aligned}
 We can rewrite the first two terms in the following intuitive form,
 which describes a decay with
 rate $$\gamma_\parallel \equiv \gamma_g + \gamma_e$$
-towards an equilbrium $$d_0$$:
+towards an equilibrium $$d_0$$:
 
 $$\begin{aligned}
     2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}