Expand knowledge base

2 files changed, 838 insertions, 0 deletions

diff --git a/content/know/concept/harmonic-oscillator/index.pdc b/content/know/concept/harmonic-oscillator/index.pdc
new file mode 100644
index 0000000..5d97a24
--- /dev/null
+++ b/content/know/concept/harmonic-oscillator/index.pdc
@@ -0,0 +1,292 @@
+---
+title: "Harmonic oscillator"
+firstLetter: "H"
+publishDate: 2021-06-02
+categories:
+- Physics
+- Mathematics
+
+date: 2021-03-09T20:35:14+01:00
+draft: false
+markup: pandoc
+---
+
+# Harmonic oscillator
+
+A **harmonic oscillator** obeys
+the simple 1D version of [Hooke's law](/know/concept/hookes-law/):
+to displace the system away from its equilibrium,
+the needed force $F_d(x)$ scales linearly with the displacement $x(t)$:
+
+$$\begin{aligned}
+    F_d(x) = k x
+\end{aligned}$$
+
+Where $k$ is a system-specific proportionality constant,
+called the **spring constant**,
+since a spring is a good example of a harmonic oscillator,
+at least for small displacements.
+Hooke's law is also often stated for
+the restoring force $F_r(x)$ instead:
+
+$$\begin{aligned}
+    F_r(x) = - k x
+\end{aligned}$$
+
+Let a mass $m$ be attached to the end of the spring.
+After displacing it, we let it go $F_d = 0$,
+so Newton's second law for the restoring force $F_r$ demands that:
+
+$$\begin{aligned}
+    F_r = m x''
+\end{aligned}$$
+
+But $F_r = - k x$,
+meaning $m x'' = - k x$,
+leading to the following equation for $x(t)$:
+
+$$\begin{aligned}
+    \boxed{
+        x'' + \omega_0^2 x = 0
+    }
+\end{aligned}$$
+
+Where $\omega_0 \equiv \sqrt{k / m}$ is the **natural frequency** of the system.
+This differential equation has the following general solution:
+
+$$\begin{aligned}
+    \boxed{
+        x(t)
+        = C_1 \sin\!(\omega_0 t) + C_2 \cos\!(\omega_0 t)
+    }
+\end{aligned}$$
+
+Where $C_1$ and $C_2$ are constants determined by the initial conditions.
+For example, for $x(0) = 1$ and $x'(0) = 0$, the solution becomes:
+
+$$\begin{aligned}
+    x(t) = \cos\!(\omega_0 t)
+\end{aligned}$$
+
+When using Lagrangian mechanics or Hamiltonian mechanics,
+we need to know the potential energy $V(x)$
+added to the system by a displacement to $x$.
+This equals the work done by the displacement,
+and is therefore given by:
+
+$$\begin{aligned}
+    V(x) = \int_0^x F_d(x) \:dx = \frac{1}{2} k x^2 = \frac{1}{2} m \omega_0^2 x^2
+\end{aligned}$$
+
+
+## Damped oscillation
+
+If there is a **friction force** $F_f$ affecting the system,
+then the oscillation amplitude will decrease,
+or it might not oscillate at all.
+We define $F_f$ using a **viscous damping coefficient** $c$:
+
+$$\begin{aligned}
+    F_f = - c x'
+\end{aligned}$$
+
+Both $F_r$ and $F_f$ are acting on the system,
+so Newton's second law states that:
+
+$$\begin{aligned}
+    m x'' = - c x' - k x
+\end{aligned}$$
+
+This can be rewritten in the following conventional form
+by defining the **damping coefficient** $\zeta \equiv c / (2 \sqrt{m k})$,
+which determines the expected behaviour of the system:
+
+$$\begin{aligned}
+    \boxed{
+        x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = 0
+    }
+\end{aligned}$$
+
+The general solution is found from the roots $u$ of the auxiliary quadratic equation:
+
+$$\begin{aligned}
+    u^2 + 2 \zeta \omega_0 u + \omega_0^2 = 0
+\end{aligned}$$
+
+The discriminant $D = 4 \zeta^2 \omega_0^2 - 4 \omega_0^2$
+tells us that the behaviour changes substantially
+depending on the damping coefficient $\zeta$,
+with three possibilities: $\zeta < 1$ or $\zeta = 1$ or $\zeta > 1$.
+
+If $\zeta < 1$, there is **underdamping**:
+the system oscillates with exponentially decaying
+amplitude and reduced frequency $\omega_1 \equiv \omega_0 \sqrt{1 - \zeta^2}$.
+The general solution is:
+
+$$\begin{aligned}
+    \boxed{
+        x(t)
+        = \big( C_1 \sin\!(\omega_1 t) + C_2 \cos\!(\omega_1 t) \big) \exp\!(- \zeta \omega_0 t)
+    }
+\end{aligned}$$
+
+If $\zeta = 1$, there is **critical damping**:
+the system returns to its equilibrium point in minimum time.
+The general solution is given by:
+
+$$\begin{aligned}
+    \boxed{
+        x(t)
+        = \big( C_1 + C_2 t \big) \exp\!(- \omega_0 t)
+    }
+\end{aligned}$$
+
+If $\zeta > 1$, there is **overdamping**:
+the system returns to equilibrium slowly.
+The general solution is as follows,
+where $\omega_1 \equiv \omega_0 \sqrt{\zeta^2 - 1}$:
+
+$$\begin{aligned}
+    \boxed{
+        x(t)
+        = \big( C_1 \exp\!(\omega_1 t) + C_2 \exp\!(- \omega_1 t) \big) \exp\!(- \zeta \omega_0 t)
+    }
+\end{aligned}$$
+
+
+## Forced oscillation
+
+In the differential equations given above,
+the right-hand side has always been zero,
+meaning that the oscillator is not affected by any external forces.
+What if we put a function there?
+
+$$\begin{aligned}
+    x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = f(t)
+\end{aligned}$$
+
+Obviously, there exist infinitely many $f(t)$ to choose from,
+and each needs a separate analysis.
+However, there is one type of $f(t)$ that deserves special mention,
+namely sinusoids:
+
+$$\begin{aligned}
+    \boxed{
+        x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = \frac{F}{m} \cos\!(\omega t + \chi)
+    }
+\end{aligned}$$
+
+Where $F$ is a constant force, $\chi$ is an arbitrary phase,
+and the frequency $\omega$ is not necessarily $\omega_0$.
+We solve this case for $x(t)$ in detail.
+Consider the complex version of the equation:
+
+$$\begin{aligned}
+    X'' + 2 \zeta \omega_0 X' + \omega_0^2 X = \frac{F}{m} \exp\!\big(i (\omega t + \chi)\big)
+\end{aligned}$$
+
+Then $x(t) = \Re\{X(t)\}$.
+Inserting the ansatz $X(t) = C \exp\!(i \omega t)$,
+for some constant $C$:
+
+$$\begin{aligned}
+    - C \omega^2 + C 2 i \zeta \omega_0 \omega + C \omega_0^2 = \frac{F}{m} \exp\!(i \chi)
+\end{aligned}$$
+
+Where $\exp\!(i \omega t)$ has already been divided out.
+We isolate this equation for $C$:
+
+$$\begin{aligned}
+    C
+    = \frac{F}{m \big((\omega_0^2 - \omega^2) + 2 i \zeta \omega_0 \omega\big)} \exp\!(i \chi)
+    = \frac{F \big((\omega_0^2 - \omega^2) - 2 i \zeta \omega_0 \omega\big)}
+    {m \big((\omega_0^2 - \omega^2)^2 + 4 \zeta^2 \omega_0^2 \omega^2\big)}
+    \exp\!(i \chi)
+\end{aligned}$$
+
+We would like to rewrite this in polar form $C = r \exp\!(i \theta)$,
+which turns out to be as follows:
+
+$$\begin{aligned}
+    C
+    &= \frac{F}{m \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \zeta^2 \omega_0^2 \omega^2}}
+    \exp\!\bigg(i \chi - i \arctan\!\Big(\frac{2 \zeta \omega_0 \omega}{\omega_0^2 - \omega^2}\Big)\bigg)
+\end{aligned}$$
+
+For brevity, let us define the **impedance** $Z$
+and the **phase shift** $\phi$
+in the following way:
+
+$$\begin{aligned}
+    Z
+    \equiv \sqrt{(\omega_0^2 - \omega^2)^2 / \omega^2 + 4 \zeta^2 \omega_0^2}
+    \qquad \quad
+    \phi
+    \equiv \arctan\!\Big(\frac{2 \zeta \omega_0 \omega}{\omega_0^2 - \omega^2}\Big)
+\end{aligned}$$
+
+Returning to the original ansatz $X(t) = C \exp\!(i \omega t)$,
+we take its real part to find $x(t)$:
+
+$$\begin{aligned}
+    \boxed{
+        x(t)
+        = \frac{F}{m \omega Z} \sin\!(\omega t + \chi - \phi)
+    }
+\end{aligned}$$
+
+Two things are noteworthy here.
+Firstly, $f(t)$ and $x(t)$ are out of phase by $\phi$; there is some lag.
+This is caused by damping, because if $\zeta = 0$, it disappears $\phi = 0$.
+
+Secondly, the amplitude of $x(t)$ depends on $\omega$ and $\omega_0$.
+This brings us to **resonance**,
+where the amplitude can become extremely large.
+Actually, resonance has two subtly different definitions,
+depending on which one of $\omega$ and $\omega_0$ is a free parameter,
+and which one is fixed.
+
+If the natural $\omega_0$ is fixed and the driving $\omega$ is variable,
+we find for which $\omega$ resonance occurs by minimizing the amplitude denominator $\omega Z$.
+We thus find:
+
+$$\begin{aligned}
+    0
+    = \dv{(\omega Z)}{\omega}
+    = \frac{- 4 \omega_0^2 \omega + 4 \omega^3 + 8 \zeta^2 \omega_0^2 \omega}{2 \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \zeta^2 \omega_0^2 \omega^2}}
+    \quad \implies \quad
+    \boxed{
+        \omega = \omega_0 \sqrt{1 - 2 \zeta^2}
+    }
+\end{aligned}$$
+
+Meaning the resonant $\omega$ is lower than $\omega_0$,
+and resonance can only occur if $\zeta < 1 / \sqrt{2}$.
+
+However, if the driving $\omega$ is fixed and the natural is $\omega_0$ is variable,
+the problem is bit more subtle:
+the damping coefficient $\zeta = c / (2 m \omega_0)$
+depends on $\omega_0$.
+This leads us to:
+
+$$\begin{aligned}
+    0
+    = \dv{(\omega Z)}{\omega_0}
+    = \frac{4 \omega_0^3 - 4 \omega^2 \omega_0}{2 \sqrt{(\omega_0^2 - \omega^2)^2 + c^2 \omega^2 / m^2}}
+    \quad \implies \quad
+    \boxed{
+        \omega_0 = \omega
+    }
+\end{aligned}$$
+
+Surprisingly, the damping does not affect $\omega_0$, if $\omega$ is given.
+However, in both cases, the damping *does* matter for the eventual amplitude:
+$c \to 0$ leads to $x \to \infty$,
+and resonance disappears or becomes negligible for $c \to \infty$.
+
+
+
+## References
+1.  M.L. Boas,
+    *Mathematical methods in the physical sciences*, 2nd edition,
+    Wiley.
diff --git a/content/know/concept/selection-rules/index.pdc b/content/know/concept/selection-rules/index.pdc
new file mode 100644
index 0000000..5da97e7
--- /dev/null
+++ b/content/know/concept/selection-rules/index.pdc
@@ -0,0 +1,546 @@
+---
+title: "Selection rules"
+firstLetter: "S"
+publishDate: 2021-06-02
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-05-29T14:42:08+02:00
+draft: false
+markup: pandoc
+---
+
+# Selection rules
+
+In quantum mechanics, it is often necessary to evaluate
+matrix elements of the following form,
+where $\ell$ and $m$ respectively represent
+the total angular momentum and its $z$-component:
+
+$$\begin{aligned}
+    \matrixel{f}{\hat{O}}{i}
+    = \matrixel{n_f \ell_f m_f}{\hat{O}}{n_i \ell_i m_i}
+\end{aligned}$$
+
+Where $\hat{O}$ is an operator, $\ket{i}$ is an initial state, and
+$\ket{f}$ is a final state (usually at least; $\ket{i}$ and $\ket{f}$
+can be any states). **Selection rules** are requirements on the relations
+between $\ell_i$, $\ell_f$, $m_i$ and $m_f$, which, if not met,
+guarantee that the above matrix element is zero.
+Note that $n_f$ and $n_i$ typically do not matter in this context,
+so they will be omitted from now on.
+
+
+## Parity rules
+
+Let $\hat{O}$ denote any operator which is odd under spatial inversion
+(parity):
+
+$$\begin{aligned}
+    \hat{\Pi}^\dagger \hat{O} \hat{\Pi} = - \hat{O}
+\end{aligned}$$
+
+Where $\hat{\Pi}$ is the parity operator.
+We wrap this property of $\hat{O}$
+in the states $\ket{\ell_f m_f}$ and $\ket{\ell_i m_i}$:
+
+$$\begin{aligned}
+    \matrixel{\ell_f m_f}{\hat{O}}{\ell_i m_i}
+    &= - \matrixel{\ell_f m_f}{\hat{\Pi}^\dagger \hat{O} \hat{\Pi}}{\ell_i m_i}
+    \\
+    &= - \matrixel{\ell_f m_f}{(-1)^{\ell_f} \hat{O} (-1)^{\ell_i}}{\ell_i m_i}
+    \\
+    &= (-1)^{\ell_f + \ell_i + 1} \matrixel{\ell_f m_f}{\hat{O}}{\ell_i m_i}
+\end{aligned}$$
+
+Which clearly can only be true if the exponent is even,
+so $\Delta \ell \equiv \ell_f - \ell_i$ must be odd.
+This leads to the following selection rule,
+often referred to as **Laporte's rule**:
+
+$$\begin{aligned}
+    \boxed{
+        \Delta \ell \:\:\text{is odd}
+    }
+\end{aligned}$$
+
+If this is not the case,
+then the only possible way that the above equation can be satisfied
+is if the matrix element vanishes $\matrixel{\ell_f m_f}{\hat{O}}{\ell_i m_i} = 0$.
+We can derive an analogous rule for
+any operator $\hat{E}$ which is even under parity:
+
+$$\begin{aligned}
+    \hat{\Pi}^\dagger \hat{E} \hat{\Pi} = \hat{E}
+    \quad \implies \quad
+    \boxed{
+        \Delta \ell \:\:\text{is even}
+    }
+\end{aligned}$$
+
+
+## Dipole rules
+
+Arguably the most common operator found in such matrix elements
+is a position vector operator, like $\vu{r}$ or $\hat{x}$,
+and the associated selection rules are known as **dipole rules**.
+
+For the $z$-component of angular momentum $m$ we have the following:
+
+$$\begin{aligned}
+    \boxed{
+        \Delta m = 0 \:\:\mathrm{or}\: \pm 1
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-dipole-m"/>
+<label for="proof-dipole-m">Proof</label>
+<div class="hidden">
+<label for="proof-dipole-m">Proof.</label>
+We know that the angular momentum $z$-component operator $\hat{L}_z$ satisfies:
+
+$$\begin{aligned}
+    \comm*{\hat{L}_z}{\hat{x}} = i \hbar \hat{y}
+    \qquad
+    \comm*{\hat{L}_z}{\hat{y}} = - i \hbar \hat{x}
+    \qquad
+    \comm*{\hat{L}_z}{\hat{z}} = 0
+\end{aligned}$$
+
+We take the first relation,
+and wrap it in $\bra{\ell_f m_f}$ and $\ket{\ell_i m_i}$, giving:
+
+$$\begin{aligned}
+    i \hbar \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
+    &= \matrixel{\ell_f m_f}{\hat{L}_z \hat{x}}{\ell_i m_i} - \matrixel{\ell_f m_f}{\hat{x} \hat{L}_z}{\ell_i m_i}
+    \\
+    &= \hbar m_f \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i} - \hbar m_i \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
+    \\
+    &= \hbar (m_f - m_i) \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
+\end{aligned}$$
+
+Next, we do the same thing with the second relation, for $[\hat{L}_z, \hat{y}]$, giving:
+
+$$\begin{aligned}
+    - i \hbar \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
+    &= \matrixel{\ell_f m_f}{\hat{L}_z \hat{y}}{\ell_i m_i} - \matrixel{\ell_f m_f}{\hat{y} \hat{L}_z}{\ell_i m_i}
+    \\
+    &= \hbar m_f \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i} - \hbar m_i \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
+    \\
+    &= \hbar (m_f - m_i) \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
+\end{aligned}$$
+
+Respectively isolating the two above results for $\hat{x}$ and $\hat{y}$,
+we arrive at these equations:
+
+$$\begin{aligned}
+    \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
+    &= i (m_f - m_i) \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
+    \\
+    \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
+    &= - i (m_f - m_i) \matrixel{\ell_f m_f}{\hat{x}}{\ell_i m_i}
+\end{aligned}$$
+
+By inserting the first into the second,
+we find (part of) the selection rule:
+
+$$\begin{aligned}
+    \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
+    &= (m_f - m_i)^2 \matrixel{\ell_f m_f}{\hat{y}}{\ell_i m_i}
+\end{aligned}$$
+
+This can only be true if $\Delta m = \pm 1$,
+unless the inner products of $\hat{x}$ and $\hat{y}$ are zero,
+in which case we cannot say anything about $\Delta m$ yet.
+Assuming the latter, we take the inner product of
+the commutator $\comm*{\hat{L}_z}{\hat{z}} = 0$, and find:
+
+$$\begin{aligned}
+    0
+    &= \matrixel{\ell_f m_f}{\hat{L}_z \hat{z}}{\ell_i m_i} - \matrixel{\ell_f m_f}{\hat{z} \hat{L}_z}{\ell_i m_i}
+    \\
+    &= \hbar m_f \matrixel{\ell_f m_f}{\hat{z}}{\ell_i m_i} - \hbar m_i \matrixel{\ell_f m_f}{\hat{z}}{\ell_i m_i}
+    \\
+    &= \hbar (m_f - m_i) \matrixel{\ell_f m_f}{\hat{z}}{\ell_i m_i}
+\end{aligned}$$
+
+If $\matrixel{f}{\hat{z}}{i} \neq 0$, we require $\Delta m = 0$.
+The previous requirement was $\Delta m = \pm 1$,
+implying that $\matrixel{f}{\hat{x}}{i} = \matrixel{f}{\hat{y}}{i} = 0$
+whenever $\matrixel{f}{\hat{z}}{i} \neq 0$.
+Only if $\matrixel{f}{\hat{z}}{i} = 0$
+does the previous rule $\Delta m = \pm 1$ hold,
+in which case the inner products of $\hat{x}$ and $\hat{y}$ are nonzero.
+</div>
+</div>
+
+Meanwhile, for the total angular momentum $\ell$ we have the following:
+
+$$\begin{aligned}
+    \boxed{
+        \Delta \ell = \pm 1
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-dipole-l"/>
+<label for="proof-dipole-l">Proof</label>
+<div class="hidden">
+<label for="proof-dipole-l">Proof.</label>
+We start from the following relation
+(which is already quite a chore to prove):
+
+$$\begin{aligned}
+    \comm{\hat{L}^2}{\comm*{\hat{L}^2}{\vu{r}}}
+    = 2 \hbar^2 (\vu{r} \hat{L}^2 + \hat{L}^2 \vu{r})
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-dipole-l-comm"/>
+<label for="proof-dipole-l-comm">Proof</label>
+<div class="hidden">
+<label for="proof-dipole-l-comm">Proof.</label>
+To begin with, we want to find the commutator of $\hat{L}^2$ and $\hat{x}$:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\hat{x}}
+    &= \comm*{\hat{L}_x^2}{\hat{x}} + \comm*{\hat{L}_y^2}{\hat{x}} + \comm*{\hat{L}_z^2}{\hat{x}}
+    = \comm*{\hat{L}_y^2}{\hat{x}} + \comm*{\hat{L}_z^2}{\hat{x}}
+    \\
+    &= \hat{L}_y \comm*{\hat{L}_y}{\hat{x}} + \comm*{\hat{L}_y}{\hat{x}} \hat{L}_y
+    + \hat{L}_z \comm*{\hat{L}_z}{\hat{x}} + \comm*{\hat{L}_z}{\hat{x}} \hat{L}_z
+\end{aligned}$$
+
+Evaluating these commutators gives us:
+
+$$\begin{aligned}
+    \comm*{\hat{L}_y}{\hat{x}}
+    &= \comm*{\hat{z} \hat{p}_x}{\hat{x}} - \comm*{\hat{x} \hat{p}_z}{\hat{x}}
+    = \hat{z} \comm*{\hat{p}_x}{\hat{x}} + \comm*{\hat{z}}{\hat{x}} \hat{p}_x
+    - \hat{x} \comm*{\hat{p}_z}{\hat{x}} - \comm*{\hat{x}}{\hat{x}} \hat{p}_z
+    = - i \hbar \hat{z}
+    \\
+    \comm*{\hat{L}_z}{\hat{x}}
+    &= \comm*{\hat{x} \hat{p}_y}{\hat{x}} - \comm*{\hat{y} \hat{p}_x}{\hat{x}}
+    = \hat{x} \comm*{\hat{p}_y}{\hat{x}} + \comm*{\hat{x}}{\hat{x}} \hat{p}_y
+    - \hat{y} \comm*{\hat{p}_x}{\hat{x}} - \comm*{\hat{y}}{\hat{x}} \hat{p}_x
+    = i \hbar \hat{y}
+\end{aligned}$$
+
+Which we then insert back into the original equation, yielding:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\hat{x}}
+    &= i \hbar (- \hat{L}_y \hat{z} - \hat{z} \hat{L}_y + \hat{L}_z \hat{y} + \hat{y} \hat{L}_z)
+\end{aligned}$$
+
+This can be simplified by introducing some more commutators:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\hat{x}}
+    &= i \hbar \big( \!-\! ( \comm*{\hat{L}_y}{\hat{z}} + \hat{z} \hat{L}_y ) - \hat{z} \hat{L}_y
+    + ( \comm*{\hat{L}_z}{\hat{y}} + \hat{y} \hat{L}_z ) + \hat{y} \hat{L}_z \big)
+\end{aligned}$$
+
+Evaluating these commutators gives us:
+
+$$\begin{aligned}
+    \comm*{\hat{L}_y}{\hat{z}}
+    &= \comm*{\hat{z} \hat{p}_x}{\hat{z}} - \comm*{\hat{x} \hat{p}_z}{\hat{z}}
+    = \hat{z} \comm*{\hat{p}_x}{\hat{z}} + \comm*{\hat{z}}{\hat{z}} \hat{p}_x
+    - \hat{x} \comm*{\hat{p}_z}{\hat{z}} - \comm*{\hat{x}}{\hat{z}} \hat{p}_z
+    = i \hbar \hat{x}
+    \\
+    \comm*{\hat{L}_z}{\hat{y}}
+    &= \comm*{\hat{x} \hat{p}_y}{\hat{y}} - \comm*{\hat{y} \hat{p}_x}{\hat{y}}
+    = \hat{x} \comm*{\hat{p}_y}{\hat{y}} + \comm*{\hat{x}}{\hat{y}} \hat{p}_y
+    - \hat{y} \comm*{\hat{p}_x}{\hat{y}} - \comm*{\hat{y}}{\hat{y}} \hat{p}_x
+    = - i \hbar \hat{x}
+\end{aligned}$$
+
+Substituting these then leads us to the first milestone of this proof:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\hat{x}}
+    &= i \hbar \big( \!-\! i \hbar \hat{x} - \hat{z} \hat{L}_y - \hat{z} \hat{L}_y
+    - i \hbar \hat{x} + \hat{y} \hat{L}_z + \hat{y} \hat{L}_z \big)
+    \\
+    &= 2 i \hbar (\hat{y} \hat{L}_z - \hat{z} \hat{L}_y - i \hbar \hat{x})
+\end{aligned}$$
+
+Repeating this process for $\comm*{\hat{L}^2}{\hat{y}}$ and $\comm*{\hat{L}^2}{\hat{z}}$,
+we find analogous expressions:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\hat{y}}
+    &= 2 i \hbar (\hat{z} \hat{L}_x - \hat{x} \hat{L}_z - i \hbar \hat{y})
+    \\
+    \comm*{\hat{L}^2}{\hat{z}}
+    &= 2 i \hbar (\hat{x} \hat{L}_y - \hat{y} \hat{L}_x - i \hbar \hat{z})
+\end{aligned}$$
+
+Next, we take the commutator with $\hat{L}^2$ of the commutator we just found:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
+    &= 2 i \hbar \big(\comm*{\hat{L}^2}{\hat{y} \hat{L}_z} - \comm*{\hat{L}^2}{\hat{z} \hat{L}_y} - i \hbar \comm*{\hat{L}^2}{\hat{x}}\big)
+    \\
+    &= 2 i \hbar \big( \hat{y} \comm*{\hat{L}^2}{\hat{L}_z} + \comm*{\hat{L}^2}{\hat{y}} \hat{L}_z
+    - \hat{z} \comm*{\hat{L}^2}{\hat{L}_y} - \comm*{\hat{L}^2}{\hat{z}} \hat{L}_y
+    - i \hbar \comm*{\hat{L}^2}{\hat{x}} \big)
+\end{aligned}$$
+
+Where we used that $\comm*{\hat{L}^2}{\hat{L}_y} = \comm*{\hat{L}^2}{\hat{L}_z} = 0$.
+The other commutators look familiar:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
+    &= 2 i \hbar \big( \comm*{\hat{L}^2}{\hat{y}} \hat{L}_z
+    - \comm*{\hat{L}^2}{\hat{z}} \hat{L}_y
+    - i \hbar \comm*{\hat{L}^2}{\hat{x}} \big)
+\end{aligned}$$
+
+By inserting the expressions we found earlier for these commutators, we get:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
+    %&= - 2 \hbar^2 \big( 2 (\hat{z} \hat{L}_x - \hat{x} \hat{L}_z - i \hbar \hat{y}) \hat{L}_z
+    %- 2 (\hat{x} \hat{L}_y - \hat{y} \hat{L}_x - i \hbar \hat{z}) \hat{L}_y
+    %- (\hat{L}^2 \hat{x} - \hat{x} \hat{L}^2) \big)
+    %\\
+    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_x \hat{L}_z - \hat{x} \hat{L}_z^2  - i \hbar \hat{y} \hat{L}_z
+    + \hat{y} \hat{L}_x \hat{L}_y - \hat{x} \hat{L}_y^2 + i \hbar \hat{z} \hat{L}_y \big) \\
+    &\qquad\qquad + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
+\end{aligned}$$
+
+Substituting the well-known commutators
+$i \hbar \hat{L}_y = \comm*{\hat{L}_z}{\hat{L}_x}$ and
+$i \hbar \hat{L}_z = \comm*{\hat{L}_x}{\hat{L}_y}$:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
+    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_x \hat{L}_z + \hat{y} \hat{L}_x \hat{L}_y
+    - \hat{x} \hat{L}_y^2 - \hat{x} \hat{L}_z^2
+    + \hat{z} \comm*{\hat{L}_z}{\hat{L}_x} - \hat{y} \comm*{\hat{L}_x}{\hat{L}_y} \big) \\
+    &\qquad\qquad + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
+    \\
+    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x
+    - \hat{x} \hat{L}_y^2 - \hat{x} \hat{L}_z^2 \big)
+    + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
+\end{aligned}$$
+
+By definition, $\hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2 = \hat{L}^2$,
+which we use to arrive at:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
+    %&= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2
+    %- \hat{x} \hat{L}_x^2 - \hat{x} \hat{L}_y^2 - \hat{x} \hat{L}_z^2 \big)
+    %+ 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
+    %\\
+    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2 - \hat{x} \hat{L}^2 \big)
+    + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
+    \\
+    &= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2 \big)
+    + 2 \hbar^2 \big( \hat{L}^2 \hat{x} + \hat{x} \hat{L}^2 \big)
+\end{aligned}$$
+
+The second term is what we want to prove,
+so the first term must vanish:
+
+$$\begin{aligned}
+    \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2
+    = (\vu{r} \cdot \vu{L}) \hat{L}_x
+    = (\vu{r} \cdot (\vu{r} \cross \vu{p})) \hat{L}_x
+    = (\vu{p} \cdot (\vu{r} \cross \vu{r})) \hat{L}_x
+    = 0
+\end{aligned}$$
+
+Where $\vu{L} = \vu{r} \cross \vu{p}$ by definition,
+and the cross product of a vector with itself is zero.
+
+This process can be repeated for
+$\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{y}}}$ and
+$\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{z}}}$,
+leading us to:
+
+$$\begin{aligned}
+    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
+    &= 2 \hbar^2 (\hat{x} \hat{L}^2 + \hat{L}^2 \hat{x})
+    \\
+    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{y}}}
+    &= 2 \hbar^2 (\hat{y} \hat{L}^2 + \hat{L}^2 \hat{y})
+    \\
+    \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{z}}}
+    &= 2 \hbar^2 (\hat{z} \hat{L}^2 + \hat{L}^2 \hat{z})
+\end{aligned}$$
+
+At last, this brings us to the desired equation for $\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\vu{r}}}$,
+with $\vu{r} = (\hat{x}, \hat{y}, \hat{z})$.
+</div>
+</div>
+
+We then multiply this relation by $\bra{f} = \bra{\ell_f m_f}$ on the left
+and $\ket{i} = \ket{\ell_i m_i}$ on the right,
+so the right-hand side becomes:
+
+$$\begin{aligned}
+    2 \hbar^2 \matrixel{f}{\vu{r} \hat{L}^2 \!\!+\! \hat{L}^2 \!\vu{r}}{i}
+    %\matrixel{\ell_f m_f}{\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\vu{r}}}}{\ell_i m_i}
+    &= 2 \hbar^2 \big( \matrixel{f}{\vu{r} \hat{L}^2}{i} + \matrixel{f}{\hat{L}^2 \vu{r}}{i} \big)
+    \\
+    &= 2 \hbar^2 \big( \hbar^2 \ell_i (\ell_i \!+\! 1) \matrixel{f}{\vu{r}}{i}
+    + \hbar^2 \ell_f (\ell_f \!+\! 1) \matrixel{f}{\vu{r}}{i} \big)
+    \\
+    &= 2 \hbar^4 \big(\ell_f (\ell_f \!+\! 1) + \ell_i (\ell_i \!+\! 1)\big) \matrixel{f}{\vu{r}}{i}
+\end{aligned}$$
+
+And, likewise, the left-hand side becomes:
+
+$$\begin{aligned}
+    \matrixel{f}{\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\vu{r}}}}{i}
+    &= \matrixel{f}{\hat{L}^2 \comm*{\hat{L}^2}{\vu{r}}}{i}
+    - \matrixel{f}{\comm*{\hat{L}^2}{\vu{r}} \hat{L}^2}{i}
+    \\
+    &= \hbar^2 \ell_f (\ell_f \!+\! 1) \matrixel{f}{\comm*{\hat{L}^2}{\vu{r}}}{i}
+    - \hbar^2 \ell_i (\ell_i \!+\! 1) \matrixel{f}{\comm*{\hat{L}^2}{\vu{r}}}{i}
+    \\
+    &= \hbar^2 \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big) \matrixel{f}{\comm*{\hat{L}^2}{\vu{r}}}{i}
+    \\
+    &= \hbar^2 \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big)
+    \big( \matrixel{f}{\hat{L}^2 \vu{r}}{i} - \matrixel{f}{\vu{r} \hat{L}^2}{i} \big)
+    \\
+    &= \hbar^4 \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big)^2 \matrixel{f}{\vu{r}}{i}
+\end{aligned}$$
+
+Obviously, both sides are equal to each other,
+leading to the following equation:
+
+$$\begin{aligned}
+    2 \ell_f (\ell_f \!+\! 1) + 2 \ell_i (\ell_i \!+\! 1)
+    &= \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big)^2
+\end{aligned}$$
+
+To proceed, we rewrite the right-hand side like so:
+
+$$\begin{aligned}
+    \big( \ell_f (\ell_f \!+\! 1) - \ell_i (\ell_i \!+\! 1) \big)^2
+    &= \big( \ell_f^2 - \ell_i^2 + \ell_f - \ell_i \big)^2
+    \\
+    &= \big( (\ell_f + \ell_i) (\ell_f - \ell_i) + (\ell_f - \ell_i) \big)^2
+    \\
+    &= (\ell_f + \ell_i)^2 (\ell_f - \ell_i)^2 + 2 (\ell_f + \ell_i) (\ell_f - \ell_i)^2 + (\ell_f - \ell_i)^2
+    \\
+    &= \big( (\ell_f + \ell_i)^2 + 2 (\ell_f + \ell_i) + 1 \big) (\ell_f - \ell_i)^2
+    \\
+    &= (\ell_f + \ell_i + 1)^2 (\ell_f - \ell_i)^2
+\end{aligned}$$
+
+And then we do the same to the left-hand side, yielding:
+
+$$\begin{aligned}
+    2 (\ell_f^2 + \ell_i^2 + \ell_f + \ell_i)
+    &= 2 \ell_f^2 + 2 \ell_i^2 + 2 \ell_f \ell_i - 2 \ell_f \ell_i + 2 \ell_f + 2 \ell_i + 1 - 1
+    \\
+    &= (\ell_f + \ell_i + 1)^2 + \ell_f^2 + \ell_i^2 - 2 \ell_f \ell_i - 1
+    \\
+    &= (\ell_f + \ell_i + 1)^2 + (\ell_f - \ell_i)^2 - 1
+\end{aligned}$$
+
+The equation above has thus been simplified to the following form:
+
+$$\begin{aligned}
+    (\ell_f + \ell_i + 1)^2 + (\ell_f - \ell_i)^2 - 1
+    &= (\ell_f + \ell_i + 1)^2 (\ell_f - \ell_i)^2
+\end{aligned}$$
+
+Rearranging yields a product equal to zero,
+so one or both of the factors must vanish:
+
+$$\begin{aligned}
+    0
+    &= (\ell_f + \ell_i + 1)^2 (\ell_f - \ell_i)^2 - (\ell_f + \ell_i + 1)^2 - (\ell_f - \ell_i)^2 + 1
+    \\
+    &= \big( (\ell_f + \ell_i + 1)^2 - 1 \big) \big( (\ell_f - \ell_i)^2 - 1 \big)
+\end{aligned}$$
+
+The first factor is zero if $\ell_f = \ell_i = 0$,
+in which case the matrix element $\matrixel{f}{\vu{r}}{i} = 0$ anyway.
+The other, non-trivial option is therefore:
+
+$$\begin{aligned}
+    (\ell_f - \ell_i)^2
+    = 1
+\end{aligned}$$
+</div>
+</div>
+
+
+## Superselection rule
+
+Selection rules need not always be about atomic electron transitions.
+According to the **principle of indistinguishability**,
+permutating identical particles never leads to an observable difference.
+In other words, the particles are fundamentally indistinguishable,
+so for any observable $\hat{O}$ and multi-particle state $\ket{\Psi}$, we can say:
+
+$$\begin{aligned}
+    \matrixel{\Psi}{\hat{O}}{\Psi}
+    = \matrixel*{\hat{P} \Psi}{\hat{O}}{\hat{P} \Psi}
+\end{aligned}$$
+
+Where $\hat{P}$ is an arbitrary permutation operator.
+Indistinguishability implies that $\comm*{\hat{P}}{\hat{O}} = 0$
+for all $\hat{O}$ and $\hat{P}$,
+which lets us prove the above equation, using that $\hat{P}$ is unitary:
+
+$$\begin{aligned}
+    \matrixel*{\hat{P} \Psi}{\hat{O}}{\hat{P} \Psi}
+    = \matrixel{\Psi}{\hat{P}^{-1} \hat{O} \hat{P}}{\Psi}
+    = \matrixel{\Psi}{\hat{P}^{-1} \hat{P} \hat{O}}{\Psi}
+    = \matrixel{\Psi}{\hat{O}}{\Psi}
+\end{aligned}$$
+
+Consider a symmetric state $\ket{s}$ and an antisymmetric state $\ket{a}$
+(see [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/)),
+which obey the following for a permutation $\hat{P}$:
+
+$$\begin{aligned}
+    \hat{P} \ket{s}
+    = \ket{s}
+    \qquad
+    \hat{P} \ket{a}
+    = - \ket{a}
+\end{aligned}$$
+
+Any obervable $\hat{O}$ then satisfies the equation below,
+again thanks to the fact that $\hat{P} = \hat{P}^{-1}$:
+
+$$\begin{aligned}
+    \matrixel{s}{\hat{O}}{a}
+    = \matrixel*{\hat{P} s}{\hat{O}}{a}
+    = \matrixel{s}{\hat{P}^{-1} \hat{O}}{a}
+    = \matrixel{s}{\hat{O} \hat{P}}{a}
+    = \matrixel*{s}{\hat{O}}{\hat{P} a}
+    = - \matrixel{s}{\hat{O}}{a}
+\end{aligned}$$
+
+This leads us to the **superselection rule**,
+which states that there can never be any interference
+between states of different permutation symmetry:
+
+$$\begin{aligned}
+    \boxed{
+        \matrixel{s}{\hat{O}}{a}
+        = 0
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  D.J. Griffiths, D.F. Schroeter,
+    *Introduction to quantum mechanics*, 3rd edition,
+    Cambridge.