More improvements to knowledge base

8 files changed, 307 insertions, 209 deletions

diff --git a/source/blog/2022/email-server-revisited/index.md b/source/blog/2022/email-server-revisited/index.md
index 20eff24..9713b77 100644
--- a/source/blog/2022/email-server-revisited/index.md
+++ b/source/blog/2022/email-server-revisited/index.md
@@ -127,7 +127,7 @@ Here, `rsa-sha256` is the signature algorithm
 (this is the best available, because DKIM is ancient),
 and `relaxed/relaxed` is the so-called *canonicalization* method,
 which is applied before signing and verification,
-to prevents failures if e.g. the email's whitespace gets changed in transit.
+to prevent failures if e.g. the email's whitespace gets changed in transit.
 
 
 
@@ -135,8 +135,8 @@ to prevents failures if e.g. the email's whitespace gets changed in transit.
 
 OpenSMTPD needs to send all outbound mail through `dkimproxy.out`.
 In `/etc/smtpd/smtpd.conf`, we tell it that all emails coming from the MUA
-must be relayed through `localhost:10027`, and then, after DKIM signing,
-picked up again on `localhost:10028`:
+must be relayed through `localhost:10027`,
+and then picked up again on `localhost:10028` after DKIM signing:
 ```sh
 # Outbound
 listen on eth0 port 465 smtps       pki "example.com" auth <passwds> tag "TRUSTED"
diff --git a/source/know/concept/central-limit-theorem/index.md b/source/know/concept/central-limit-theorem/index.md
index e933ee7..42bc05b 100644
--- a/source/know/concept/central-limit-theorem/index.md
+++ b/source/know/concept/central-limit-theorem/index.md
@@ -9,7 +9,7 @@ layout: "concept"
 ---
 
 In statistics, the **central limit theorem** states that
-the sum of many independent variables tends towards a normal distribution,
+the sum of many independent variables tends to a normal distribution,
 even if the individual variables $$x_n$$ follow different distributions.
 
 For example, by taking $$M$$ samples of size $$N$$ from a population,
diff --git a/source/know/concept/fredholm-alternative/index.md b/source/know/concept/fredholm-alternative/index.md
index c954272..fdc90be 100644
--- a/source/know/concept/fredholm-alternative/index.md
+++ b/source/know/concept/fredholm-alternative/index.md
@@ -14,7 +14,7 @@ It is an *alternative* because it gives two mutually exclusive options,
 given here in [Dirac notation](/know/concept/dirac-notation/):
 
 1.  $$\hat{L} \Ket{u} = \Ket{f}$$ has a unique solution $$\Ket{u}$$ for every $$\Ket{f}$$.
-2.  $$\hat{L}^\dagger \Ket{w} = 0$$ has non-zero solutions.
+2.  $$\hat{L}^\dagger \Ket{w} = 0$$ has nonzero solutions.
     Then regarding $$\hat{L} \Ket{u} = \Ket{f}$$:
     1.  If $$\Inprod{w}{f} = 0$$ for all $$\Ket{w}$$, then it has infinitely many solutions $$\Ket{u}$$.
     2.  If $$\Inprod{w}{f} \neq 0$$ for any $$\Ket{w}$$, then it has no solutions $$\Ket{u}$$.
@@ -31,7 +31,7 @@ this theorem can alternatively be stated as follows using the determinant:
 1.  If $$\mathrm{det}(\hat{L}) \neq 0$$, then $$\hat{L} \vec{u} = \vec{f}$$
     has a unique solution $$\vec{u}$$ for every $$\vec{f}$$.
 2.  If $$\mathrm{det}(\hat{L}) = 0$$,
-    then $$\hat{L}^\dagger \vec{w} = \vec{0}$$ has non-zero solutions.
+    then $$\hat{L}^\dagger \vec{w} = \vec{0}$$ has nonzero solutions.
     Then regarding $$\hat{L} \vec{u} = \vec{f}$$:
     1.  If $$\vec{w} \cdot \vec{f} = 0$$ for all $$\vec{w}$$, then it has
         infinitely many solutions $$\vec{u}$$.
@@ -48,7 +48,7 @@ Then for the equation $$\hat{M} \Ket{u} = \Ket{f}$$, we can say that:
 1.  If $$\lambda$$ is *not* an eigenvalue,
     then there is a unique solution $$\Ket{u}$$ for each $$\Ket{f}$$.
 2.  If $$\lambda$$ is an eigenvalue, then $$\hat{M}^\dagger \Ket{w} = 0$$
-    has non-zero solutions. Then:
+    has nonzero solutions. Then:
     1.  If $$\Inprod{w}{f} = 0$$ for all $$\Ket{w}$$, then there are
         infinitely many solutions $$\Ket{u}$$.
     2.  If $$\Inprod{w}{f} \neq 0$$ for any $$\Ket{w}$$, then there are no
diff --git a/source/know/concept/matsubara-greens-function/index.md b/source/know/concept/matsubara-greens-function/index.md
index fd46abf..5e753db 100644
--- a/source/know/concept/matsubara-greens-function/index.md
+++ b/source/know/concept/matsubara-greens-function/index.md
@@ -11,7 +11,7 @@ layout: "concept"
 The **Matsubara Green's function** is an
 [imaginary-time](/know/concept/imaginary-time/) version
 of the real-time [Green's functions](/know/concept/greens-functions/).
-We define as follows in the imaginary-time
+We define it as follows in the imaginary-time
 [Heisenberg picture](/know/concept/heisenberg-picture/):
 
 $$\begin{aligned}
diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md
index 0252b5c..1214703 100644
--- a/source/know/concept/maxwell-bloch-equations/index.md
+++ b/source/know/concept/maxwell-bloch-equations/index.md
@@ -11,96 +11,43 @@ categories:
 layout: "concept"
 ---
 
-For an electron in a two-level system with time-independent states
-$$\ket{g}$$ (ground) and $$\ket{e}$$ (excited),
-consider the following general solution
-to the time-dependent Schrödinger equation:
+For an electron in a two-orbital system $$\{\ket{g}, \ket{e}\}$$,
+the Schrödinger equation has the following general solution,
+where $$\varepsilon_g$$ and $$\varepsilon_e$$ are the time-independent eigenenergies,
+and the weights $$c_g$$ and $$c_g$$ are functions of $$t$$:
 
 $$\begin{aligned}
     \ket{\Psi}
-    &= c_g \ket{g} \exp(-i E_g t / \hbar) + c_e \ket{e} \exp(-i E_e t / \hbar)
+    &= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar}
 \end{aligned}$$
 
-Perturbing this system with
-an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
-introduces a time-dependent sinusoidal term $$\hat{H}_1$$ to the Hamiltonian.
-In the [electric dipole approximation](/know/concept/electric-dipole-approximation/),
-$$\hat{H}_1$$ is given by:
+This system is being perturbed by an electromagnetic wave
+with [electric field](/know/concept/electric-field/) $$\vb{E}$$ given by:
 
 $$\begin{aligned}
-    \hat{H}_1(t)
-    = - \hat{\vb{p}} \cdot \vb{E}(t)
-    \qquad \qquad
-    \vu{p}
-    \equiv q \vu{x}
-    \qquad \qquad
     \vb{E}(t)
-    = \vb{E}_0 \cos(\omega t)
-\end{aligned}$$
-
-Where $$\vb{E}$$ is an [electric field](/know/concept/electric-field/),
-and $$\hat{\vb{p}}$$ is the dipole moment operator.
-From [Rabi oscillation](/know/concept/rabi-oscillation/),
-we know that the time-varying coefficients $$c_g$$ and $$c_e$$
-can then be described by:
-
-$$\begin{aligned}
-    \dv{c_g}{t}
-    &= i \frac{q \matrixel{g}{\vu{x}}{e} \cdot \vb{E}_0}{2 \hbar} \exp\!\big( i \omega t \!-\! i \omega_0 t \big) \: c_e
-    \\
-    \dv{c_e}{t}
-    &= i \frac{q \matrixel{e}{\vu{x}}{g} \cdot \vb{E}_0}{2 \hbar} \exp\!\big(\!-\! i \omega t \!+\! i \omega_0 t \big) \: c_g
-\end{aligned}$$
-
-We want to rearrange these equations a bit.
-Therefore, we split the electric field $$\vb{E}$$ like so,
-where the amplitudes $$\vb{E}_0^{-}$$ and $$\vb{E}_0^{+}$$ may be slowly varying:
-
-$$\begin{aligned}
-    \vb{E}(t)
-    = \vb{E}^{-}(t) + \vb{E}^{+}(t)
-    = \vb{E}_0^{-} \exp(i \omega t) + \vb{E}_0^{+} \exp(-i \omega t)
-\end{aligned}$$
-
-Since $$\vb{E}$$ is real, $$\vb{E}_0^{+} = (\vb{E}_0^{-})^*$$.
-Similarly, we define the transition dipole moment $$\vb{p}_0^{-}$$:
-
-$$\begin{aligned}
-    \vb{p}_0^{-}
-    \equiv q \matrixel{e}{\vu{x}}{g}
-    \qquad \qquad
-    \vb{p}_0^{+}
-    \equiv (\vb{p}_0^{-})^*
-    = q \matrixel{g}{\vu{x}}{e}
-\end{aligned}$$
-
-With these, the equations for $$c_g$$ and $$c_e$$ can be rewritten as shown below.
-Note that $$\vb{E}^{-}$$ and $$\vb{E}^{+}$$ include the driving plane wave, and the
-[rotating wave approximation](/know/concept/rotating-wave-approximation/) is still made:
-
-$$\begin{aligned}
-    \dv{c_g}{t}
-    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e
-    \\
-    \dv{c_e}{t}
-    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g
+    &\equiv \vb{E}^{-}(t) + \vb{E}^{+}(t)
 \end{aligned}$$
 
-
-
-## Optical Bloch equations
+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, $$\vb{E}_0^{+} \!=\! (\vb{E}_0^{-})^*$$.
 
 For $$\ket{\Psi}$$ as defined above,
-the corresponding pure [density operator](/know/concept/density-operator/)
-$$\hat{\rho}$$ is as follows:
+the pure [density operator](/know/concept/density-operator/)
+$$\hat{\rho}$$ is as follows,
+with $$\omega_0 \equiv (\varepsilon_e \!-\! \varepsilon_g) / \hbar$$
+being the transition's resonance frequency:
 
 $$\begin{aligned}
     \hat{\rho}
     = \ket{\Psi} \bra{\Psi}
     =
     \begin{bmatrix}
-        c_e c_e^* & c_e c_g^* \exp(-i \omega_0 t) \\
-        c_g c_e^* \exp(i \omega_0 t) & c_g c_g^*
+        c_e c_e^* & c_e c_g^* e^{-i \omega_0 t} \\
+        c_g c_e^* e^{i \omega_0 t} & c_g c_g^*
     \end{bmatrix}
     \equiv
     \begin{bmatrix}
@@ -109,139 +56,59 @@ $$\begin{aligned}
     \end{bmatrix}
 \end{aligned}$$
 
-Where $$\omega_0 \equiv (E_e \!-\! E_g) / \hbar$$ is the resonance frequency.
-We take the $$t$$-derivative of the matrix elements,
-and insert the equations for $$c_g$$ and $$c_e$$:
+Under the [electric dipole approximation](/know/concept/electric-dipole-approximation/)
+and [rotating wave approximation](/know/concept/rotating-wave-approximation/),
+it can be shown that $$\hat{\rho}$$ is governed by
+the [optical Bloch equations](/know/concept/optical-bloch-equations/):
 
 $$\begin{aligned}
     \dv{\rho_{gg}}{t}
-    &= \dv{c_g}{t} c_g^* + c_g \dv{c_g^*}{t}
-    \\
-    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e c_g^*
-    - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g c_e^*
+    &= \gamma_e \rho_{ee} - \gamma_g \rho_{gg}
+    + \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
     \\
     \dv{\rho_{ee}}{t}
-    &= \dv{c_e}{t} c_e^* + c_e \dv{c_e^*}{t}
-    \\
-    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g c_e^*
-    - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e c_g^*
+    &=  \gamma_g \rho_{gg} - \gamma_e \rho_{ee}
+    + \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
     \\
     \dv{\rho_{ge}}{t}
-    &= \dv{c_g}{t} c_e^* \exp(i \omega_0 t) + c_g \dv{c_e^*}{t} \exp(i \omega_0 t) + i \omega_0 c_g c_e^* \exp(i \omega_0 t)
-    \\
-    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \: c_e c_e^*
-    - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \: c_g c_g^*
-    + i \omega_0 c_g c_e^* \exp(i \omega_0 t)
+    &= - \Big( \gamma_\perp - i \omega_0 \Big) \rho_{ge}
+    + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \Big( \rho_{ee} - \rho_{gg} \Big)
     \\
     \dv{\rho_{eg}}{t}
-    &= \dv{c_e}{t} c_g^* \exp(-i \omega_0 t) + c_e \dv{c_g^*}{t} \exp(-i \omega_0 t) - i \omega_0 c_e c_g^* \exp(- i \omega_0 t)
-    \\
-    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \: c_g c_g^*
-    - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \: c_e c_e^*
-    - i \omega_0 c_e c_g^* \: \exp(- i \omega_0 t)
+    &= - \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
+    + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \Big( \rho_{gg} - \rho_{ee} \Big)
 \end{aligned}$$
 
-Recognizing the density matrix elements allows us
-to reduce these equations to:
+Where we have defined the transition dipole moment $$\vb{p}_0^{-}$$,
+with $$q < 0$$ the electron charge:
 
 $$\begin{aligned}
-    \dv{\rho_{gg}}{t}
-    &= \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
-    \\
-    \dv{\rho_{ee}}{t}
-    &= \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
-    \\
-    \dv{\rho_{ge}}{t}
-    &= i \omega_0 \rho_{ge} + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \big( \rho_{ee} - \rho_{gg} \big)
-    \\
-    \dv{\rho_{eg}}{t}
-    &= - i \omega_0 \rho_{eg} + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \big( \rho_{gg} - \rho_{ee} \big)
-\end{aligned}$$
-
-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](/know/concept/einstein-coefficients/).
-
-Suppose $$\rho_{ee}$$ decays with rate $$\gamma_e$$.
-Because the total probability $$\rho_{ee} + \rho_{gg} = 1$$, we have:
-
-$$\begin{aligned}
-    \Big( \dv{\rho_{ee}}{t} \Big)_{e}
-    = - \gamma_e \rho_{ee}
-    \quad \implies \quad
-    \Big( \dv{\rho_{gg}}{t} \Big)_{e}
-    = \gamma_e \rho_{ee}
-\end{aligned}$$
-
-Meanwhile, for whatever reason,
-let $$\rho_{gg}$$ decay into $$\rho_{ee}$$ with rate $$\gamma_g$$:
-
-$$\begin{aligned}
-    \Big( \dv{\rho_{gg}}{t} \Big)_{g}
-    = - \gamma_g \rho_{gg}
-    \quad \implies \quad
-    \Big( \dv{\rho_{gg}}{t} \Big)_{g}
-    = \gamma_g \rho_{gg}
-\end{aligned}$$
-
-And finally, let the diagonal (perpendicular) matrix elements
-both decay with rate $$\gamma_\perp$$:
-
-$$\begin{aligned}
-    \Big( \dv{\rho_{eg}}{t} \Big)_{\perp}
-    = - \gamma_\perp \rho_{eg}
+    \vb{p}_0^{-}
+    \equiv q \matrixel{e}{\vu{x}}{g}
     \qquad \qquad
-    \Big( \dv{\rho_{ge}}{t} \Big)_{\perp}
-    = - \gamma_\perp \rho_{ge}
-\end{aligned}$$
-
-Putting everything together,
-we arrive at the **optical Bloch equations** governing $$\hat{\rho}$$:
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \dv{\rho_{gg}}{t}
-            &= \gamma_e \rho_{ee} - \gamma_g \rho_{gg}
-            + \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
-            \\
-            \dv{\rho_{ee}}{t}
-            &=  \gamma_g \rho_{gg} - \gamma_e \rho_{ee}
-            + \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
-            \\
-            \dv{\rho_{ge}}{t}
-            &= - \Big( \gamma_\perp - i \omega_0 \Big) \rho_{ge}
-            + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \Big( \rho_{ee} - \rho_{gg} \Big)
-            \\
-            \dv{\rho_{eg}}{t}
-            &= - \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
-            + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \Big( \rho_{gg} - \rho_{ee} \Big)
-        \end{aligned}
-    }
+    \vb{p}_0^{+}
+    \equiv (\vb{p}_0^{-})^*
+    = q \matrixel{g}{\vu{x}}{e}
 \end{aligned}$$
 
-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,
-where we use [Laporte's selection rule](/know/concept/selection-rules/)
-to remove diagonal terms, by assuming that
-the electron's orbitals are odd or even:
+However, the light wave affects the electron,
+so the actual 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:
 
 $$\begin{aligned}
     \vb{p}
-    &= \matrixel{\Psi}{\hat{\vb{p}}}{\Psi}
+    &= q \matrixel{\Psi}{\vu{x}}{\Psi}
     \\
     &= q \Big( c_g c_g^* \matrixel{g}{\vu{x}}{g} + c_e c_e^* \matrixel{e}{\vu{x}}{e}
-    + c_g c_e^* \matrixel{e}{\vu{x}}{g} \exp(i \omega_0 t) + c_e c_g^* \matrixel{g}{\vu{x}}{e} \exp(-i \omega_0 t) \Big)
+    + c_g c_e^* \matrixel{e}{\vu{x}}{g} e^{i \omega_0 t} + c_e c_g^* \matrixel{g}{\vu{x}}{e} e^{-i \omega_0 t} \Big)
     \\
     &= q \Big( \rho_{ge} \matrixel{e}{\vu{x}}{g} + \rho_{eg} \matrixel{g}{\vu{x}}{e} \Big)
-    = \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t)
-    \equiv \vb{p}^{-}(t) + \vb{p}^{+}(t)
+    \\
+    &= \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t)
+    \\
+    &\equiv \vb{p}^{-}(t) + \vb{p}^{+}(t)
 \end{aligned}$$
 
 Where we have split $$\vb{p}$$ analogously to $$\vb{E}$$
@@ -256,7 +123,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Some authors do not bother multiplying $$\rho_{ge}$$ by $$\vb{p}_0^{+}$$.
-In any case, we arrive at:
+In our case, we arrive at a prototype of the first of three Maxwell-Bloch equations:
 
 $$\begin{aligned}
     \boxed{
@@ -266,8 +133,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where we have defined the **population inversion** $$d \in [-1, 1]$$ as follows,
-which quantifies the electron's excitedness:
+Where we have defined the **population inversion** $$d \in [-1, 1]$$ like so,
+to quantify the electron's "excitedness" i.e. its localization to $$\ket{e}$$:
 
 $$\begin{aligned}
     d
@@ -325,8 +192,8 @@ $$\begin{aligned}
 {% include proof/end.html id="proof-inversion-decay" %}
 
 
-With this, the equation for the population inversion $$d$$
-takes the following final form:
+With this, the equation for the population inversion $$d$$ takes the form below,
+namely the second Maxwell-Bloch equation's prototype:
 
 $$\begin{aligned}
     \boxed{
@@ -337,9 +204,11 @@ $$\begin{aligned}
 
 Finally, we would like a relation between the polarization
 and the electric field $$\vb{E}$$,
-for which we turn to [Maxwell's equations](/know/concept/maxwells-equations/).
-We start from Faraday's law,
-and split $$\vb{B} = \mu_0 (\vb{H} + \vb{M})$$:
+for which we turn to [Maxwell's equations](/know/concept/maxwells-equations/);
+we will effectively derive a modified form of
+the [electromagnetic wave equation](/know/concept/electromagnetic-wave-equation/).
+Starting from Faraday's law
+and splitting $$\vb{B} = \mu_0 (\vb{H} + \vb{M})$$:
 
 $$\begin{aligned}
     \nabla \cross \vb{E}
@@ -391,7 +260,8 @@ $$\begin{aligned}
 Where $$\varepsilon_r \equiv 1 + \chi_e$$ is the medium's relative permittivity.
 The speed of light $$c^2 = 1 / (\mu_0 \varepsilon_0)$$,
 and the refractive index $$n^2 = \mu_r \varepsilon_r$$,
-where $$\mu_r = 1$$ due to our assumption that $$\vb{M} = 0$$, so:
+where $$\mu_r = 1$$ due to our assumption that $$\vb{M} = 0$$,
+so the third Maxwell-Bloch equation's prototype is:
 
 $$\begin{aligned}
     \boxed{
@@ -436,11 +306,8 @@ $$\begin{aligned}
 
 ## References
 1.  F. Kärtner,
-    [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-977-ultrafast-optics-spring-2005/lecture-notes/),
-    2005, MIT.
+    [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/pages/lecture-notes/),
+    2005, Massachusetts Institute of Technology.
 2.  H. Haken,
     *Light: volume 2: laser light dynamics*,
     1985, North-Holland.
-3.  H.J. Metcalf, P. van der Straten,
-    *Laser cooling and trapping*,
-    1999, Springer.
diff --git a/source/know/concept/optical-bloch-equations/index.md b/source/know/concept/optical-bloch-equations/index.md
new file mode 100644
index 0000000..fe74b7e
--- /dev/null
+++ b/source/know/concept/optical-bloch-equations/index.md
@@ -0,0 +1,231 @@
+---
+title: "Optical Bloch equations"
+sort_title: "Optical Bloch equations"
+date: 2023-01-19
+categories:
+- Physics
+- Quantum mechanics
+- Two-level system
+layout: "concept"
+---
+
+For an electron in a two-level system with time-independent states
+$$\ket{g}$$ (ground) and $$\ket{e}$$ (excited),
+consider the following general solution
+to the time-dependent Schrödinger equation:
+
+$$\begin{aligned}
+    \ket{\Psi}
+    &= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar}
+\end{aligned}$$
+
+Perturbing this system with
+an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
+introduces a time-dependent sinusoidal term $$\hat{H}_1$$ to the Hamiltonian.
+In the [electric dipole approximation](/know/concept/electric-dipole-approximation/),
+$$\hat{H}_1$$ is given by:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    = - \hat{\vb{p}} \cdot \vb{E}(t)
+    \qquad \qquad
+    \vu{p}
+    \equiv q \vu{x}
+    \qquad \qquad
+    \vb{E}(t)
+    = \vb{E}_0 \cos(\omega t)
+\end{aligned}$$
+
+Where $$\vb{E}$$ is an [electric field](/know/concept/electric-field/),
+and $$\hat{\vb{p}}$$ is the dipole moment operator.
+From [Rabi oscillation](/know/concept/rabi-oscillation/),
+we know that the time-varying coefficients $$c_g$$ and $$c_e$$
+can then be described by:
+
+$$\begin{aligned}
+    \dv{c_g}{t}
+    &= i \frac{q \matrixel{g}{\vu{x}}{e} \cdot \vb{E}_0}{2 \hbar} \: e^{i (\omega - \omega_0) t} \: c_e
+    \\
+    \dv{c_e}{t}
+    &= i \frac{q \matrixel{e}{\vu{x}}{g} \cdot \vb{E}_0}{2 \hbar} \: e^{- i (\omega - \omega_0) t} \: c_g
+\end{aligned}$$
+
+Where $$\omega_0 \equiv (\varepsilon_e \!-\! \varepsilon_g) / \hbar$$ is the resonance frequency.
+We want to rearrange these equations a bit,
+so we split the field $$\vb{E}$$ as follows,
+where the amplitudes $$\vb{E}_0^{-}$$ and $$\vb{E}_0^{+}$$
+may be slowly-varying with respect to the carrier wave $$e^{\pm i \omega t}$$:
+
+$$\begin{aligned}
+    \vb{E}(t)
+    &\equiv \vb{E}^{-}(t) + \vb{E}^{+}(t)
+    \\
+    &\equiv \vb{E}_0^{-} e^{i \omega t} + \vb{E}_0^{+} e^{-i \omega t}
+\end{aligned}$$
+
+Since $$\vb{E}$$ is real, $$\vb{E}_0^{+} = (\vb{E}_0^{-})^*$$.
+Similarly, we define the transition dipole moment $$\vb{p}_0^{-}$$:
+
+$$\begin{aligned}
+    \vb{p}_0^{-}
+    \equiv q \matrixel{e}{\vu{x}}{g}
+    \qquad \qquad
+    \vb{p}_0^{+}
+    \equiv (\vb{p}_0^{-})^*
+    = q \matrixel{g}{\vu{x}}{e}
+\end{aligned}$$
+
+With these, the equations for $$c_g$$ and $$c_e$$ can be rewritten as shown below.
+Note that $$\vb{E}^{-}$$ and $$\vb{E}^{+}$$ include the driving plane wave, and the
+[rotating wave approximation](/know/concept/rotating-wave-approximation/) is still made:
+
+$$\begin{aligned}
+    \dv{c_g}{t}
+    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} e^{- i \omega_0 t} \: c_e
+    \\
+    \dv{c_e}{t}
+    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} e^{i \omega_0 t} \: c_g
+\end{aligned}$$
+
+
+For $$\ket{\Psi}$$ as defined above,
+the corresponding pure [density operator](/know/concept/density-operator/)
+$$\hat{\rho}$$ is as follows:
+
+$$\begin{aligned}
+    \hat{\rho}
+    = \ket{\Psi} \bra{\Psi}
+    =
+    \begin{bmatrix}
+        c_e c_e^* & c_e c_g^* e^{-i \omega_0 t} \\
+        c_g c_e^* e^{i \omega_0 t} & c_g c_g^*
+    \end{bmatrix}
+    \equiv
+    \begin{bmatrix}
+        \rho_{ee} & \rho_{eg} \\
+        \rho_{ge} & \rho_{gg}
+    \end{bmatrix}
+\end{aligned}$$
+
+We take the $$t$$-derivative of the matrix elements,
+and insert the equations for $$c_g$$ and $$c_e$$:
+
+$$\begin{aligned}
+    \dv{\rho_{gg}}{t}
+    &= \dv{c_g}{t} c_g^* + c_g \dv{c_g^*}{t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_e c_g^* e^{- i \omega_0 t}
+    - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_g c_e^* e^{i \omega_0 t}
+    \\
+    \dv{\rho_{ee}}{t}
+    &= \dv{c_e}{t} c_e^* + c_e \dv{c_e^*}{t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_g c_e^* e^{i \omega_0 t}
+    - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_e c_g^* e^{- i \omega_0 t}
+    \\
+    \dv{\rho_{ge}}{t}
+    &= \dv{c_g}{t} c_e^* e^{i \omega_0 t} + c_g \dv{c_e^*}{t} e^{i \omega_0 t} + i \omega_0 c_g c_e^* e^{i \omega_0 t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_e c_e^*
+    - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_g c_g^*
+    + i \omega_0 c_g c_e^* e^{i \omega_0 t}
+    \\
+    \dv{\rho_{eg}}{t}
+    &= \dv{c_e}{t} c_g^* e^{-i \omega_0 t} + c_e \dv{c_g^*}{t} e^{-i \omega_0 t} - i \omega_0 c_e c_g^* e^{- i \omega_0 t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_g c_g^*
+    - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_e c_e^*
+    - i \omega_0 c_e c_g^* e^{- i \omega_0 t}
+\end{aligned}$$
+
+Recognizing the density matrix elements allows us
+to reduce these equations to:
+
+$$\begin{aligned}
+    \dv{\rho_{gg}}{t}
+    &= \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
+    \\
+    \dv{\rho_{ee}}{t}
+    &= \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
+    \\
+    \dv{\rho_{ge}}{t}
+    &= i \omega_0 \rho_{ge} + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \big( \rho_{ee} - \rho_{gg} \big)
+    \\
+    \dv{\rho_{eg}}{t}
+    &= - i \omega_0 \rho_{eg} + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \big( \rho_{gg} - \rho_{ee} \big)
+\end{aligned}$$
+
+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](/know/concept/einstein-coefficients/).
+
+Suppose $$\rho_{ee}$$ decays with rate $$\gamma_e$$.
+Because the total probability $$\rho_{ee} + \rho_{gg} = 1$$, we have:
+
+$$\begin{aligned}
+    \Big( \dv{\rho_{ee}}{t} \Big)_{e}
+    = - \gamma_e \rho_{ee}
+    \quad \implies \quad
+    \Big( \dv{\rho_{gg}}{t} \Big)_{e}
+    = \gamma_e \rho_{ee}
+\end{aligned}$$
+
+Meanwhile, for whatever reason,
+let $$\rho_{gg}$$ decay into $$\rho_{ee}$$ with rate $$\gamma_g$$:
+
+$$\begin{aligned}
+    \Big( \dv{\rho_{gg}}{t} \Big)_{g}
+    = - \gamma_g \rho_{gg}
+    \quad \implies \quad
+    \Big( \dv{\rho_{gg}}{t} \Big)_{g}
+    = \gamma_g \rho_{gg}
+\end{aligned}$$
+
+And finally, let the diagonal (perpendicular) matrix elements
+both decay with rate $$\gamma_\perp$$:
+
+$$\begin{aligned}
+    \Big( \dv{\rho_{eg}}{t} \Big)_{\perp}
+    = - \gamma_\perp \rho_{eg}
+    \qquad \qquad
+    \Big( \dv{\rho_{ge}}{t} \Big)_{\perp}
+    = - \gamma_\perp \rho_{ge}
+\end{aligned}$$
+
+Putting everything together,
+we arrive at the **optical Bloch equations** governing $$\hat{\rho}$$:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \dv{\rho_{gg}}{t}
+            &= \gamma_e \rho_{ee} - \gamma_g \rho_{gg}
+            + \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
+            \\
+            \dv{\rho_{ee}}{t}
+            &=  \gamma_g \rho_{gg} - \gamma_e \rho_{ee}
+            + \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
+            \\
+            \dv{\rho_{ge}}{t}
+            &= - \Big( \gamma_\perp - i \omega_0 \Big) \rho_{ge}
+            + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \Big( \rho_{ee} - \rho_{gg} \Big)
+            \\
+            \dv{\rho_{eg}}{t}
+            &= - \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
+            + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \Big( \rho_{gg} - \rho_{ee} \Big)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+Some authors simplify these equations a bit by choosing
+$$\gamma_g = 0$$ and $$\gamma_\perp = \gamma_e / 2$$.
+
+
+
+## References
+1.  F. Kärtner,
+    [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/pages/lecture-notes/),
+    2005, Massachusetts Institute of Technology.
+2.  H.J. Metcalf, P. van der Straten,
+    *Laser cooling and trapping*,
+    1999, Springer.
diff --git a/source/know/concept/parsevals-theorem/index.md b/source/know/concept/parsevals-theorem/index.md
index 41e8fed..a7ce0bf 100644
--- a/source/know/concept/parsevals-theorem/index.md
+++ b/source/know/concept/parsevals-theorem/index.md
@@ -17,7 +17,7 @@ where $$A$$, $$B$$, and $$s$$ are constants from the FT's definition:
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
-            \Inprod{f(x)}{g(x)} &= \frac{2 \pi B^2}{|s|} \inprod{\tilde{f}(k)}{\tilde{g}(k)}
+            \inprod{f(x)}{g(x)} &= \frac{2 \pi B^2}{|s|} \inprod{\tilde{f}(k)}{\tilde{g}(k)}
             \\
             \inprod{\tilde{f}(k)}{\tilde{g}(k)} &= \frac{2 \pi A^2}{|s|} \Inprod{f(x)}{g(x)}
         \end{aligned}
@@ -29,7 +29,7 @@ $$\begin{aligned}
 We insert the inverse FT into the definition of the inner product:
 
 $$\begin{aligned}
-    \Inprod{f}{g}
+    \inprod{f}{g}
     &= \int_{-\infty}^\infty \big( \hat{\mathcal{F}}^{-1}\{\tilde{f}(k)\}\big)^* \: \hat{\mathcal{F}}^{-1}\{\tilde{g}(k)\} \dd{x}
     \\
     &= B^2 \int
@@ -65,7 +65,7 @@ $$\begin{aligned}
     &= 2 \pi A^2 \iint f^*(x') \: g(x) \: \delta\big(s (x \!-\! x')\big) \dd{x'} \dd{x}
     \\
     &= \frac{2 \pi A^2}{|s|} \int_{-\infty}^\infty f^*(x) \: g(x) \dd{x}
-    = \frac{2 \pi A^2}{|s|} \Inprod{f}{g}
+    = \frac{2 \pi A^2}{|s|} \inprod{f}{g}
 \end{aligned}$$
 {% include proof/end.html id="proof-fourier" %}
 
diff --git a/source/know/concept/sturm-liouville-theory/index.md b/source/know/concept/sturm-liouville-theory/index.md
index 75daae3..0ac7476 100644
--- a/source/know/concept/sturm-liouville-theory/index.md
+++ b/source/know/concept/sturm-liouville-theory/index.md
@@ -22,7 +22,7 @@ of eigenfunctions.
 
 Consider the most general form of a second-order linear
 differential operator $$\hat{L}$$, where $$p_0(x)$$, $$p_1(x)$$, and $$p_2(x)$$
-are real functions of $$x \in [a,b]$$ which are non-zero for all $$x \in ]a, b[$$:
+are real functions of $$x \in [a,b]$$ which are nonzero for all $$x \in ]a, b[$$:
 
 $$\begin{aligned}
     \hat{L} \{u(x)\} = p_0(x) u''(x) + p_1(x) u'(x) + p_2(x) u(x)
@@ -142,7 +142,7 @@ So even if $$p_0' \neq p_1$$, any second-order linear operator with $$p_0(x) \ne
 can easily be put in self-adjoint form.
 
 This general form is known as the **Sturm-Liouville operator** $$\hat{L}_{SL}$$,
-where $$p(x)$$ and $$q(x)$$ are non-zero real functions of the variable $$x \in [a,b]$$:
+where $$p(x)$$ and $$q(x)$$ are nonzero real functions of the variable $$x \in [a,b]$$:
 
 $$\begin{aligned}
     \boxed{