Expand knowledge base

7 files changed, 527 insertions, 1 deletions

diff --git a/content/know/category/electromagnetism.md b/content/know/category/electromagnetism.md
new file mode 100644
index 0000000..2ab0c8a
--- /dev/null
+++ b/content/know/category/electromagnetism.md
@@ -0,0 +1,9 @@
+---
+title: "Electromagnetism"
+firstLetter: "E"
+date: 2021-07-13T09:47:25+02:00
+draft: false
+layout: "category"
+---
+
+This page will fill itself.
diff --git a/content/know/concept/bose-einstein-distribution/index.pdc b/content/know/concept/bose-einstein-distribution/index.pdc
new file mode 100644
index 0000000..2462c68
--- /dev/null
+++ b/content/know/concept/bose-einstein-distribution/index.pdc
@@ -0,0 +1,83 @@
+---
+title: "Bose-Einstein distribution"
+firstLetter: "B"
+publishDate: 2021-07-11
+categories:
+- Physics
+- Statistics
+- Quantum mechanics
+
+date: 2021-07-11T18:22:44+02:00
+draft: false
+markup: pandoc
+---
+
+# Bose-Einstein statistics
+
+**Bose-Einstein statistics** describe how bosons,
+which do not obey the [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/),
+will distribute themselves across the available states
+in a system at equilibrium.
+
+Consider a single-particle state $s$,
+which can contain any number of bosons.
+Since the occupation number $N_s$ is variable,
+we turn to the [grand canonical ensemble](/know/concept/grand-canonical-ensemble/),
+whose grand partition function $\mathcal{Z_s}$ is as follows,
+where $\varepsilon_s$ is the energy per particle,
+and $\mu$ is the chemical potential:
+
+$$\begin{aligned}
+    \mathcal{Z}_s
+    = \sum_{N_s = 0}^\infty \Big( \exp\!(- \beta (\varepsilon_s - \mu)) \Big)^{N_s}
+    = \frac{1}{1 - \exp\!(- \beta (\varepsilon_s - \mu))}
+\end{aligned}$$
+
+The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/)
+is the Landau potential $\Omega$, given by:
+
+$$\begin{aligned}
+    \Omega_s
+    = - k T \ln{\mathcal{Z_s}}
+    = k T \ln\!\Big( 1 - \exp\!(- \beta (\varepsilon_s - \mu)) \Big)
+\end{aligned}$$
+
+The average number of particles $\expval{N_s}$
+is found by taking a derivative of $\Omega$:
+
+$$\begin{aligned}
+    \expval{N_s}
+    = - \pdv{\Omega_s}{\mu}
+    = k T \pdv{\ln{\mathcal{Z_s}}}{\mu}
+    = \frac{\exp\!(- \beta (\varepsilon_s - \mu))}{1 - \exp\!(- \beta (\varepsilon_s - \mu))}
+\end{aligned}$$
+
+By multitplying both the numerator and the denominator by $\exp\!(\beta(\epsilon_s \!-\! \mu))$,
+we arrive at the standard form of the **Bose-Einstein distribution** $f_B$:
+
+$$\begin{aligned}
+    \boxed{
+        \expval{N_s}
+        = f_B(\varepsilon_s)
+        = \frac{1}{\exp\!(\beta (\varepsilon_s - \mu)) - 1}
+    }
+\end{aligned}$$
+
+This tells the expected occupation number $\expval{N_s}$ of state $s$,
+given a temperature $T$ and chemical potential $\mu$.
+The corresponding variance $\sigma_s^2$ of $N_s$ is found to be:
+
+$$\begin{aligned}
+    \boxed{
+        \sigma_s^2
+        = k T \pdv{\expval{N_s}}{\mu}
+        = \expval{N_s} \big(1 + \expval{N_s}\big)
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  H. Gould, J. Tobochnik,
+    *Statistical and thermal physics*, 2nd edition,
+    Princeton.
diff --git a/content/know/concept/electric-field/index.pdc b/content/know/concept/electric-field/index.pdc
new file mode 100644
index 0000000..ce2c4fc
--- /dev/null
+++ b/content/know/concept/electric-field/index.pdc
@@ -0,0 +1,129 @@
+---
+title: "Electric field"
+firstLetter: "E"
+publishDate: 2021-07-12
+categories:
+- Physics
+- Electromagnetism
+
+date: 2021-07-12T09:46:25+02:00
+draft: false
+markup: pandoc
+---
+
+## Electric field
+
+The **electric field** $\vb{E}$ is a vector field
+that describes electric effects,
+and is defined as the field that
+correctly predicts the Lorentz force
+on a particle with electric charge $q$:
+
+$$\begin{aligned}
+    \vb{F}
+    = q \vb{E}
+\end{aligned}$$
+
+This definition implies that the direction of $\vb{E}$
+is from positive to negative charges,
+since opposite charges attracts and like charges repel.
+
+If two opposite point charges with magnitude $q$
+are observed from far away,
+they can be treated as a single object called a **dipole**,
+which has an **electric dipole moment** $\vb{p}$ defined as follows,
+where $\vb{d}$ is the vector going from
+the negative to the positive charge (opposite direction of $\vb{E}$):
+
+$$\begin{aligned}
+    \vb{p} = q \vb{d}
+\end{aligned}$$
+
+Alternatively, for consistency with [magnetic fields](/know/concept/magnetic-field/),
+$\vb{p}$ can be defined from the aligning torque $\vb{\tau}$
+experienced by the dipole when placed in an $\vb{E}$-field.
+In other words, $\vb{p}$ satisfies:
+
+$$\begin{aligned}
+    \vb{\tau} = \vb{p} \times \vb{E}
+\end{aligned}$$
+
+Where $\vb{p}$ has units of $\mathrm{C m}$.
+The **polarization density** $\vb{P}$ is defined from $\vb{p}$,
+and roughly speaking represents the moments per unit volume:
+
+$$\begin{aligned}
+    \vb{P} \equiv \dv{\vb{p}}{V}
+    \:\:\iff\:\:
+    \vb{p} = \int_V \vb{P} \dd{V}
+\end{aligned}$$
+
+If $\vb{P}$ has the same magnitude and direction throughout the body,
+then this becomes $\vb{p} = \vb{P} V$, where $V$ is the volume.
+Therefore, $\vb{P}$ has units of $\mathrm{C / m^2}$.
+
+A nonzero $\vb{P}$ complicates things,
+since it contributes to the field and hence modifies $\vb{E}$.
+We thus define
+the "free" **displacement field** $\vb{D}$
+from the "bound" field $\vb{P}$
+and the "net" field $\vb{E}$:
+
+$$\begin{aligned}
+    \vb{D} \equiv \varepsilon_0 \vb{E} + \vb{P}
+    \:\:\iff\:\:
+    \vb{E} = \frac{1}{\varepsilon_0} (\vb{D} - \vb{P})
+\end{aligned}$$
+
+Where the **electric permittivity of free space** $\varepsilon_0$ is a known constant.
+It is important to point out some inconsistencies here:
+$\vb{D}$ and $\vb{P}$ contain a factor of $\varepsilon_0$,
+and therefore measure **flux density**,
+while $\vb{E}$ does not contain $\varepsilon_0$,
+and thus measures **field intensity**.
+Note that this convention is the opposite
+of the magnetic analogues $\vb{B}$, $\vb{H}$ and $\vb{M}$,
+and that $\vb{M}$ has the opposite sign of $\vb{P}$.
+
+The polarization $\vb{P}$ is a function of $\vb{E}$.
+In addition to the inherent polarity
+of the material $\vb{P}_0$ (zero in most cases),
+there is a possibly nonlinear response
+to the applied $\vb{E}$-field:
+
+$$\begin{aligned}
+    \vb{P} =
+    \vb{P}_0 + \varepsilon_0 \chi_e^{(1)} \vb{E}
+    + \varepsilon_0 \chi_e^{(2)} |\vb{E}| \: \vb{E}
+    + \varepsilon_0 \chi_e^{(3)} |\vb{E}|^2 \: \vb{E} + ...
+\end{aligned}$$
+
+Where the $\chi_e^{(n)}$ are the **electric susceptibilities** of the medium.
+For simplicity, we often assume that only the $n\!=\!1$ term is nonzero,
+which is the linear response to $\vb{E}$.
+In that case, we define
+the **relative permittivity** $\varepsilon_r \equiv 1 + \chi_e^{(1)}$
+and the **absolute permittivity** $\varepsilon \equiv \varepsilon_r \varepsilon_0$,
+so that:
+
+$$\begin{aligned}
+    \vb{D}
+    = \varepsilon_0 \vb{E} + \vb{P}
+    = \varepsilon_0 \vb{E} + \varepsilon_0 \chi_e^{(1)} \vb{E}
+    = \varepsilon_0 \varepsilon_r \vb{E}
+    = \varepsilon \vb{E}
+\end{aligned}$$
+
+In reality, a material cannot respond instantly to $\vb{E}$,
+meaning that $\chi_e^{(1)}$ is a function of time,
+and that $\vb{P}$ is the convolution of $\chi_e^{(1)}(t)$ and $\vb{E}(t)$:
+
+$$\begin{aligned}
+    \vb{P}(t)
+    = (\chi_e^{(1)} * \vb{E})(t)
+    = \int_{-\infty}^\infty \chi_e^{(1)}(t - \tau) \: \vb{E}(\tau) \:d\tau
+\end{aligned}$$
+
+Note that this definition requires $\chi_e^{(1)}(t) = 0$ for $t < 0$
+in order to ensure causality,
+which leads to the [Kramers-Kronig relations](/know/concept/kramers-kronig-relations/).
diff --git a/content/know/concept/fermi-dirac-distribution/index.pdc b/content/know/concept/fermi-dirac-distribution/index.pdc
new file mode 100644
index 0000000..8820cbb
--- /dev/null
+++ b/content/know/concept/fermi-dirac-distribution/index.pdc
@@ -0,0 +1,86 @@
+---
+title: "Fermi-Dirac distribution"
+firstLetter: "F"
+publishDate: 2021-07-11
+categories:
+- Physics
+- Statistics
+- Quantum mechanics
+
+date: 2021-07-11T18:22:37+02:00
+draft: false
+markup: pandoc
+---
+
+# Fermi-Dirac distribution
+
+**Fermi-Dirac statistics** describe how identical **fermions**,
+which obey the [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/),
+will distribute themselves across the available states in a system at equilibrium.
+
+Consider one single-particle state $s$,
+which can contain $0$ or $1$ fermions.
+Because the occupation number $N_s$ is variable,
+we turn to the [grand canonical ensemble](/know/concept/grand-canonical-ensemble/),
+whose grand partition function $\mathcal{Z}_s$ is as follows,
+where we sum over all microstates of $s$:
+
+$$\begin{aligned}
+    \mathcal{Z}_s
+    = \sum_{N_s = 0}^1 \exp\!(- \beta N_s (\varepsilon_s - \mu))
+    = 1 + \exp\!(- \beta (\varepsilon_s - \mu))
+\end{aligned}$$
+
+Where $\mu$ is the chemical potential,
+and $\varepsilon_s$ is the energy contribution per particle in $s$,
+i.e. the total energy of all particles $E_s = \varepsilon_s N_s$.
+
+The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/)
+is the Landau potential $\Omega_s$, given by:
+
+$$\begin{aligned}
+    \Omega_s
+    = - k T \ln{\mathcal{Z}_s}
+    = - k T \ln\!\Big( 1 + \exp\!(- \beta (\varepsilon_s - \mu)) \Big)
+\end{aligned}$$
+
+The average number of particles $\expval{N_s}$
+in state $s$ is then found to be as follows:
+
+$$\begin{aligned}
+    \expval{N_s}
+    = - \pdv{\Omega_s}{\mu}
+    = k T \pdv{\ln{\mathcal{Z}_s}}{\mu}
+    = \frac{\exp\!(- \beta (\varepsilon_s - \mu))}{1 + \exp\!(- \beta (\varepsilon_s - \mu))}
+\end{aligned}$$
+
+By multiplying both the numerator and the denominator by $\exp\!(\beta (\varepsilon_s \!-\! \mu))$,
+we arrive at the standard form of
+the **Fermi-Dirac distribution** or **Fermi function** $f_F$:
+
+$$\begin{aligned}
+    \boxed{
+        \expval{N_s}
+        = f_F(\varepsilon_s)
+        = \frac{1}{\exp\!(\beta (\varepsilon_s - \mu)) + 1}
+    }
+\end{aligned}$$
+
+This tells the expected occupation number $\expval{N_s}$ of state $s$,
+given a temperature $T$ and chemical potential $\mu$.
+The corresponding variance $\sigma_s^2$ of $N_s$ is found to be:
+
+$$\begin{aligned}
+    \boxed{
+        \sigma_s^2
+        = k T \pdv{\expval{N_s}}{\mu}
+        = \expval{N_s} \big(1 - \expval{N_s}\big)
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  H. Gould, J. Tobochnik,
+    *Statistical and thermal physics*, 2nd edition,
+    Princeton.
diff --git a/content/know/concept/landau-quantization/index.pdc b/content/know/concept/landau-quantization/index.pdc
index 4212078..60b1331 100644
--- a/content/know/concept/landau-quantization/index.pdc
+++ b/content/know/concept/landau-quantization/index.pdc
@@ -13,7 +13,8 @@ markup: pandoc
 
 # Landau quantization
 
-When a particle with charge $q$ is moving in a homogeneous magnetic field,
+When a particle with charge $q$ is moving in a homogeneous
+[magnetic field](/know/concept/magnetic-field/),
 quantum mechanics decrees that its allowed energies split
 into degenerate discrete **Landau levels**,
 a phenomenon known as **Landau quantization**.
diff --git a/content/know/concept/larmor-precession/index.pdc b/content/know/concept/larmor-precession/index.pdc
new file mode 100644
index 0000000..3affdee
--- /dev/null
+++ b/content/know/concept/larmor-precession/index.pdc
@@ -0,0 +1,108 @@
+---
+title: "Larmor precession"
+firstLetter: "L"
+publishDate: 2021-07-02
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-07-02T15:48:41+02:00
+draft: false
+markup: pandoc
+---
+
+# Larmor precession
+
+Consider a stationary spin-1/2 particle,
+placed in a [magnetic field](/know/concept/magnetic-field/)
+with magnitude $B$ pointing in the $z$-direction.
+In that case, its Hamiltonian $\hat{H}$ is given by:
+
+$$\begin{aligned}
+    \hat{H} = - \gamma B \hat{S}_z = - \frac{\hbar}{2} \gamma B \hat{\sigma_z}
+\end{aligned}$$
+
+Where $\gamma = - q / m$ is the gyromagnetic ratio,
+and $\hat{\sigma}_z$ is the Pauli spin matrix for the $z$-direction.
+Since $\hat{H}$ is proportional to $\hat{\sigma}_z$,
+they share eigenstates $\ket{\downarrow}$ and $\ket{\uparrow}$.
+The respective eigenenergies $E_{\downarrow}$ and $E_{\uparrow}$ are as follows:
+
+$$\begin{aligned}
+    E_{\downarrow} = \frac{\hbar}{2} \gamma B
+    \qquad
+    E_{\uparrow} = - \frac{\hbar}{2} \gamma B
+\end{aligned}$$
+
+Because $\hat{H}$ is time-independent,
+the general time-dependent solution $\ket{\chi(t)}$ is of the following form,
+where $a$ and $b$ are constants,
+and the exponentials are "twiddle factors":
+
+$$\begin{aligned}
+    \ket{\chi(t)}
+    = a \exp\!(- i E_{\downarrow} t / \hbar) \: \ket{\downarrow}
+    \:+\: b \exp\!(- i E_{\uparrow} t / \hbar) \: \ket{\uparrow}
+\end{aligned}$$
+
+For our purposes, we can safely assume that $a$ and $b$ are real,
+and then say that there exists an angle $\theta$
+satisfying $a = \sin\!(\theta / 2)$ and $b = \cos\!(\theta / 2)$, such that:
+
+$$\begin{aligned}
+    \ket{\chi(t)} = \sin\!(\theta / 2) \exp\!(- i E_{\downarrow} t / \hbar) \: \ket{\downarrow}
+    \:+\: \cos\!(\theta / 2) \exp\!(- i E_{\uparrow} t / \hbar) \: \ket{\uparrow}
+\end{aligned}$$
+
+Now, we find the expectation values of the spin operators
+$\expval*{\hat{S}_x}$, $\expval*{\hat{S}_y}$, and $\expval*{\hat{S}_z}$.
+The first is:
+
+$$\begin{aligned}
+    \matrixel{\chi}{\hat{S}_x}{\chi}
+    &= \frac{\hbar}{2}
+    \begin{bmatrix} a \exp\!(i E_{\downarrow} t / \hbar) \\ b \exp\!(i E_{\uparrow} t / \hbar) \end{bmatrix}^{\mathrm{T}}
+    \cdot
+    \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
+    \cdot
+    \begin{bmatrix} a \exp\!(- i E_{\downarrow} t / \hbar) \\ b \exp\!(- i E_{\uparrow} t / \hbar) \end{bmatrix}
+    \\
+    &= \frac{\hbar}{2}
+    \begin{bmatrix} a \exp\!(i E_{\downarrow} t / \hbar) \\ b \exp\!(i E_{\uparrow} t / \hbar) \end{bmatrix}^{\mathrm{T}}
+    \cdot
+    \begin{bmatrix} b \exp\!(- i E_{\uparrow} t / \hbar) \\ a \exp\!(- i E_{\downarrow} t / \hbar) \end{bmatrix}
+    \\
+    &= \frac{\hbar}{2} \Big( a b \exp\!(i (E_{\downarrow} \!-\! E_{\uparrow}) t / \hbar)
+    + b a \exp\!(i (E_{\uparrow} \!-\! E_{\downarrow}) t / \hbar) \Big)
+    \\
+    &= \frac{\hbar}{2} \cos\!(\theta/2) \sin\!(\theta/2) \Big( \exp\!(i \gamma B t) + \exp\!(- i \gamma B t) \Big)
+    \\
+    &= \frac{\hbar}{2} \cos\!(\gamma B t) \Big( \cos\!(\theta/2) \sin\!(\theta/2) + \cos\!(\theta/2) \sin\!(\theta/2) \Big)
+    \\
+    &= \frac{\hbar}{2} \sin\!(\theta) \cos\!(\gamma B t)
+\end{aligned}$$
+
+The other two are calculated in the same way,
+with the following results:
+
+$$\begin{aligned}
+    \matrixel{\chi}{\hat{S}_y}{\chi} = - \frac{\hbar}{2} \sin\!(\theta) \sin\!(\gamma B t)
+    \qquad
+    \matrixel{\chi}{\hat{S}_z}{\chi} = \frac{\hbar}{2} \cos\!(\theta)
+\end{aligned}$$
+
+The result is that the spin axis is off by $\theta$ from the $z$-direction,
+and is rotating (or **precessing**) around the $z$-axis at the **Larmor frequency** $\omega$:
+
+$$\begin{aligned}
+    \boxed{
+        \omega = \gamma B
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  D.J. Griffiths, D.F. Schroeter,
+    *Introduction to quantum mechanics*, 3rd edition,
+    Cambridge.
diff --git a/content/know/concept/magnetic-field/index.pdc b/content/know/concept/magnetic-field/index.pdc
new file mode 100644
index 0000000..2ad5fbf
--- /dev/null
+++ b/content/know/concept/magnetic-field/index.pdc
@@ -0,0 +1,110 @@
+---
+title: "Magnetic field"
+firstLetter: "M"
+publishDate: 2021-07-12
+categories:
+- Physics
+- Electromagnetism
+
+date: 2021-07-12T09:46:31+02:00
+draft: false
+markup: pandoc
+---
+
+## Magnetic field
+
+The **magnetic field** $\vb{B}$ is a vector field
+that describes magnetic effects,
+and is defined as the field
+that correctly predicts the Lorentz force
+on a particle with electric charge $q$:
+
+$$\begin{aligned}
+    \vb{F}
+    = q \vb{v} \cross \vb{B}
+\end{aligned}$$
+
+If an object is placed in a magnetic field $\vb{B}$,
+and wants to rotate to align itself with the field,
+then its **magnetic dipole moment** $\vb{m}$
+is defined from the aligning torque $\vb{\tau}$:
+
+$$\begin{aligned}
+    \vb{\tau} = \vb{m} \times \vb{B}
+\end{aligned}$$
+
+Where $\vb{m}$ has units of $\mathrm{J / T}$.
+From this, the **magnetization** $\vb{M}$ is defined as follows,
+and roughly represents the moments per unit volume:
+
+$$\begin{aligned}
+    \vb{M} \equiv \dv{\vb{m}}{V}
+    \:\:\iff\:\:
+    \vb{m} = \int_V \vb{M} \dd{V}
+\end{aligned}$$
+
+If $\vb{M}$ has the same magnitude and orientation throughout the body,
+then $\vb{m} = \vb{M} V$, where $V$ is the volume.
+Therefore, $\vb{M}$ has units of $\mathrm{A / m}$.
+
+A nonzero $\vb{M}$ complicates things,
+since it contributes to the field
+and hence modifies $\vb{B}$.
+We thus define
+the "free" **auxiliary field** $\vb{H}$
+from the "bound" field $\vb{M}$
+and the "net" field $\vb{B}$:
+
+$$\begin{aligned}
+    \vb{H} \equiv \frac{1}{\mu_0} \vb{B} - \vb{M}
+    \:\:\iff\:\:
+    \vb{B} = \mu_0 (\vb{H} + \vb{M})
+\end{aligned}$$
+
+Where the **magnetic permeability of free space** $\mu_0$ is a known constant.
+It is important to point out some inconsistencies here:
+$\vb{B}$ contains a factor of $\mu_0$, and thus measures **flux density**,
+while $\vb{H}$ and $\vb{M}$ do not contain $\mu_0$,
+and therefore measure **field intensity**.
+Note that this convention is the opposite of the analogous
+[electric fields](/know/concept/electric-field/)
+$\vb{E}$, $\vb{D}$ and $\vb{P}$.
+Also note that $\vb{P}$ has the opposite sign convention of $\vb{M}$.
+
+Some objects, called **ferromagnets** or **permanent magnets**,
+have an inherently nonzero $\vb{M}$.
+Others objects, when placed in a $\vb{B}$-field,
+may instead gain an induced $\vb{M}$.
+
+When $\vb{M}$ is induced,
+its magnitude is usually proportional
+to the applied field strength $\vb{H}$:
+
+$$\begin{aligned}
+    \vb{B}
+    = \mu_0(\vb{H} + \vb{M})
+    = \mu_0 (\vb{H} + \chi_m \vb{H})
+    = \mu_0 \mu_r \vb{H}
+    = \mu \vb{H}
+\end{aligned}$$
+
+Where $\chi_m$ is the **volume magnetic susceptibility**,
+and $\mu_r \equiv 1 + \chi_m$ and $\mu \equiv \mu_r \mu_0$ are
+the **relative permeability** and **absolute permeability**
+of the medium, respectively.
+Materials with intrinsic magnetization, i.e. ferromagnets,
+do not have a well-defined $\chi_m$.
+
+If $\chi_m > 0$, the medium is **paramagnetic**,
+meaning it strengthens the net field $\vb{B}$.
+Otherwise, if $\chi_m < 0$, the medium is **diamagnetic**,
+meaning it counteracts the applied field $\vb{H}$.
+
+For $|\chi_m| \ll 1$, as is often the case,
+the magnetization $\vb{M}$ can be approximated by:
+
+$$\begin{aligned}
+    \vb{M}
+    = \chi_m \vb{H}
+    \approx \chi_m \vb{B} / \mu_0
+\end{aligned}$$