From 7cb1bd307e6d3f1279731bebadbc6f994ed1105a Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 3 Jul 2026 17:18:50 +0200 Subject: Improve knowledge base --- .../concept/bose-einstein-distribution/index.md | 36 ++++----- .../know/concept/boussinesq-wave-theory/index.md | 2 +- source/know/concept/coupled-mode-theory/index.md | 4 +- source/know/concept/debye-length/index.md | 30 ++++---- .../know/concept/fermi-dirac-distribution/index.md | 38 +++++----- .../concept/korteweg-de-vries-equation/index.md | 2 +- source/know/concept/kubo-formula/index.md | 23 +++--- source/know/concept/larmor-precession/index.md | 24 +++--- source/know/concept/lindhard-function/index.md | 86 +++++++++++----------- .../concept/matsubara-greens-function/index.md | 10 +-- source/know/concept/matsubara-sum/index.md | 8 +- .../concept/pauli-exclusion-principle/index.md | 2 +- source/know/concept/salt-equation/index.md | 2 +- 13 files changed, 137 insertions(+), 130 deletions(-) (limited to 'source') diff --git a/source/know/concept/bose-einstein-distribution/index.md b/source/know/concept/bose-einstein-distribution/index.md index 5640e69..ea5ca68 100644 --- a/source/know/concept/bose-einstein-distribution/index.md +++ b/source/know/concept/bose-einstein-distribution/index.md @@ -14,19 +14,19 @@ which do not obey the [Pauli exclusion principle](/know/concept/pauli-exclusion- distribute themselves across the available states in a system at equilibrium. -Consider a single-particle state $$s$$, +Consider a single-particle state $$\ket{i}$$, which can contain any number of bosons. -Since the occupation number $$N$$ is variable, +Since the occupation number $$n_i$$ is variable, we use the [grand canonical ensemble](/know/concept/grand-canonical-ensemble/), whose grand partition function $$\mathcal{Z}$$ is as shown below, -where $$\varepsilon$$ is the energy per particle, +where $$\varepsilon_i$$ is the energy per particle, and $$\mu$$ is the chemical potential. We evaluate the sum in $$\mathcal{Z}$$ as a geometric series: $$\begin{aligned} \mathcal{Z} - = \sum_{N = 0}^\infty \Big( e^{-\beta (\varepsilon - \mu)} \Big)^{N} - = \frac{1}{1 - e^{-\beta (\varepsilon - \mu)}} + = \sum_{m = 0}^\infty \Big( e^{-\beta (\varepsilon_i - \mu)} \Big)^{m} + = \frac{1}{1 - e^{-\beta (\varepsilon_i - \mu)}} \end{aligned}$$ The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/) @@ -35,42 +35,42 @@ is the Landau potential $$\Omega$$, given by: $$\begin{aligned} \Omega = - k T \ln{\mathcal{Z}} - = k T \ln\!\big( 1 - e^{-\beta (\varepsilon - \mu)} \big) + = k T \ln\!\big( 1 - e^{-\beta (\varepsilon_i - \mu)} \big) \end{aligned}$$ -The average number of particles $$\expval{N}$$ in $$s$$ +The average number of particles $$\expval{n_i}$$ in $$\ket{i}$$ is then found by taking a derivative of $$\Omega$$: $$\begin{aligned} - \expval{N} + \expval{n_i} = - \pdv{\Omega}{\mu} = k T \pdv{\ln{\mathcal{Z}}}{\mu} - = \frac{e^{-\beta (\varepsilon - \mu)}}{1 - e^{-\beta (\varepsilon - \mu)}} + = \frac{e^{-\beta (\varepsilon_i - \mu)}}{1 - e^{-\beta (\varepsilon_i - \mu)}} \end{aligned}$$ -By multiplying both the numerator and the denominator by $$e^{\beta(\varepsilon \!-\! \mu)}$$, +By multiplying both the numerator and the denominator by $$e^{\beta(\varepsilon_i - \mu)}$$, we arrive at the standard form of the **Bose-Einstein distribution** $$f_B$$: $$\begin{aligned} \boxed{ - \expval{N} - = f_B(\varepsilon) - = \frac{1}{e^{\beta (\varepsilon - \mu)} - 1} + \expval{n_i} + = f_B(\varepsilon_i) + = \frac{1}{e^{\beta (\varepsilon_i - \mu)} - 1} } \end{aligned}$$ -This gives the expected occupation number $$\expval{N}$$ -of state $$s$$ with energy $$\varepsilon$$, +This gives the expected occupation number $$\expval{n_i}$$ +of state $$\ket{i}$$ with energy $$\varepsilon_i$$, given a temperature $$T$$ and chemical potential $$\mu$$. {% comment %} -The corresponding variance $$\sigma^2 \equiv \expval{N^2} - \expval{N}^2$$ is found to be: +The corresponding variance $$\sigma^2 \equiv \expval{n_i^2} - \expval{n_i}^2$$ is found to be: $$\begin{aligned} \boxed{ \sigma^2 - = k T \pdv{\expval{N}}{\mu} - = \expval{N} \big(1 + \expval{N}\!\big) + = k T \pdv{\expval{n_i}}{\mu} + = \expval{n_i} \big(1 + \expval{n_i}\!\big) } \end{aligned}$$ {% endcomment %} diff --git a/source/know/concept/boussinesq-wave-theory/index.md b/source/know/concept/boussinesq-wave-theory/index.md index e5fd433..b5f91e1 100644 --- a/source/know/concept/boussinesq-wave-theory/index.md +++ b/source/know/concept/boussinesq-wave-theory/index.md @@ -275,7 +275,7 @@ $$\begin{aligned} \end{aligned}$$ The smallest term we will include is $$a h^2 / \lambda^3$$; -anything smaller (specifically containing $$a^2 / \lambda^2$$) will be discarded. +anything smaller (i.e. containing a factor of $$a^2 / \lambda^2$$) will be discarded. Of course, this decision is arbitrary: higher-order approximations exist for deeper water and/or taller waves, but we stick with Boussinesq's original choice, leaving: diff --git a/source/know/concept/coupled-mode-theory/index.md b/source/know/concept/coupled-mode-theory/index.md index 6a5ec1b..23b6470 100644 --- a/source/know/concept/coupled-mode-theory/index.md +++ b/source/know/concept/coupled-mode-theory/index.md @@ -10,8 +10,8 @@ layout: "concept" Given an optical resonator (e.g. a photonic crystal cavity), consider one of its quasinormal modes -with frequency $$\omega_0$$ and decay rate $$1 / \tau_0$$. -Its complex amplitude $$A$$ is governed by: +with frequency $$\omega_0$$ and decay rate $$1 / \tau_0$$ in isolation. +Its complex amplitude $$A$$ then obeys: $$\begin{aligned} \dv{A}{t} diff --git a/source/know/concept/debye-length/index.md b/source/know/concept/debye-length/index.md index 5961c4f..063e308 100644 --- a/source/know/concept/debye-length/index.md +++ b/source/know/concept/debye-length/index.md @@ -12,8 +12,7 @@ If a charged object is put in a plasma, it repels like charges and attracts opposite charges, leading to a **Debye sheath** around the object's surface with a net opposite charge. -This has the effect of **shielding** the object's presence -from the rest of the plasma. +This has the effect of **shielding** the rest of the plasma from the object's presence. We start from [Gauss' law](/know/concept/maxwells-equations/) for the [electric field](/know/concept/electric-field/) $$\vb{E}$$, @@ -23,12 +22,12 @@ and splitting the charge density into ions $$n_i$$ and electrons $$n_e$$: $$\begin{aligned} \nabla^2 \phi(\vb{r}) - = - \frac{1}{\varepsilon_0} \Big( q_i n_i(\vb{r}) + q_e n_e(\vb{r}) + q_t \delta(\vb{r}) \Big) + = - \frac{1}{\varepsilon_0} \Big( q_i n_i(\vb{r}) + q_e n_e(\vb{r}) + Q \delta(\vb{r}) \Big) \end{aligned}$$ The last term represents a *test particle*, which will be shielded. -This particle is a point charge $$q_t$$, +This particle is a point charge $$Q$$, whose density is simply a [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(\vb{r})$$, and is not included in $$n_i$$ or $$n_e$$. @@ -63,10 +62,10 @@ where we have assumed quasi-neutrality such that $$q_i n_{i0} = q_e n_{e0}$$: $$\begin{aligned} \nabla^2 \phi &= - \frac{1}{\varepsilon_0} - \bigg( q_i n_{i0} - n_{i0} \frac{q_i^2 \phi}{k_B T_i} + q_e n_{e0} - n_{e0} \frac{q_e^2 \phi}{k_B T_e} + q_t \delta(\vb{r}) \bigg) + \bigg( q_i n_{i0} - n_{i0} \frac{q_i^2 \phi}{k_B T_i} + q_e n_{e0} - n_{e0} \frac{q_e^2 \phi}{k_B T_e} + Q \delta(\vb{r}) \bigg) \\ &= \bigg( \frac{n_{i0} q_i^2}{\varepsilon_0 k_B T_i} + \frac{n_{e0} q_e^2}{\varepsilon_0 k_B T_e} \bigg) \phi - - \frac{q_t}{\varepsilon_0} \delta(\vb{r}) + - \frac{Q}{\varepsilon_0} \delta(\vb{r}) \end{aligned}$$ We now define the **ion** and **electron Debye lengths** @@ -101,24 +100,27 @@ suggesting exponential decay: $$\begin{aligned} \nabla^2 \phi(\vb{r}) &= \frac{1}{\lambda_D^2} \phi(\vb{r}) - - \frac{q_t}{\varepsilon_0} \delta(\vb{r}) + - \frac{Q}{\varepsilon_0} \delta(\vb{r}) \end{aligned}$$ -This has the following solution, -known as the **Yukawa potential**, -which decays exponentially, -representing the plasma's **self-shielding** -over a characteristic distance $$\lambda_D$$: +This has the solution below, known as the **Yukawa potential**, +which looks like Coulomb's law but with an extra exponential factor, +representing the plasma's **self-shielding**: $$\begin{aligned} \boxed{ \phi(r) - = \frac{q_t}{4 \pi \varepsilon_0 r} \exp\!\Big( \!-\!\frac{r}{\lambda_D} \Big) + = \frac{Q}{4 \pi \varepsilon_0 r} \exp\!\Big( \!-\!\frac{r}{\lambda_D} \Big) } \end{aligned}$$ +We call it *self*-shielding because in reality +$$Q$$ is simply an electron or ion of the plasma. +This explains why plasmas are macroscopically neutral, +despite consisting of charged particles. + Note that $$r$$ is a scalar, -i.e. the potential depends only on the radial distance to $$q_t$$. +i.e. the potential depends only on the radial distance to $$Q$$. This treatment only makes sense if the plasma is sufficiently dense, such that there is a large number of particles diff --git a/source/know/concept/fermi-dirac-distribution/index.md b/source/know/concept/fermi-dirac-distribution/index.md index 2a38eb3..7554e5a 100644 --- a/source/know/concept/fermi-dirac-distribution/index.md +++ b/source/know/concept/fermi-dirac-distribution/index.md @@ -13,18 +13,18 @@ layout: "concept" which obey the [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/), distribute themselves across the available states in a system at equilibrium. -Consider one single-particle state $$s$$, +Consider one single-particle state $$\ket{i}$$, which can contain $$0$$ or $$1$$ fermions. -Because the occupation number $$N$$ is variable, +Because the occupation number $$n_i$$ is variable, we turn to the [grand canonical ensemble](/know/concept/grand-canonical-ensemble/), whose grand partition function $$\mathcal{Z}$$ is as follows, -where $$\varepsilon$$ is the energy of $$s$$ +where $$\varepsilon_i$$ is the energy of $$\ket{i}$$ and $$\mu$$ is the chemical potential: $$\begin{aligned} \mathcal{Z} - = \sum_{N = 0}^1 \Big( e^{-\beta (\varepsilon - \mu)} \Big)^N - = 1 + e^{-\beta (\varepsilon - \mu)} + = \sum_{m = 0}^1 \Big( e^{-\beta (\varepsilon_i - \mu)} \Big)^m + = 1 + e^{-\beta (\varepsilon_i - \mu)} \end{aligned}$$ The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/) @@ -33,43 +33,43 @@ is the Landau potential $$\Omega$$, given by: $$\begin{aligned} \Omega = - k T \ln{\mathcal{Z}} - = - k T \ln\!\Big( 1 + e^{-\beta (\varepsilon - \mu)} \Big) + = - k T \ln\!\Big( 1 + e^{-\beta (\varepsilon_i - \mu)} \Big) \end{aligned}$$ -The average number of particles $$\expval{N}$$ -in $$s$$ is then found by taking a derivative of $$\Omega$$: +The average number of particles $$\expval{n_i}$$ +in $$\ket{i}$$ is then found by taking a derivative of $$\Omega$$: $$\begin{aligned} - \expval{N} + \expval{n_i} = - \pdv{\Omega}{\mu} = k T \pdv{\ln{\mathcal{Z}}}{\mu} - = \frac{e^{-\beta (\varepsilon - \mu)}}{1 + e^{-\beta (\varepsilon - \mu)}} + = \frac{e^{-\beta (\varepsilon_i - \mu)}}{1 + e^{-\beta (\varepsilon_i - \mu)}} \end{aligned}$$ -By multiplying both the numerator and the denominator by $$e^{\beta (\varepsilon \!-\! \mu)}$$, +By multiplying both the numerator and the denominator by $$e^{\beta (\varepsilon_i - \mu)}$$, we arrive at the standard form of the **Fermi-Dirac distribution** or **Fermi function** $$f_F$$: $$\begin{aligned} \boxed{ - \expval{N} - = f_F(\varepsilon) - = \frac{1}{e^{\beta (\varepsilon - \mu)} + 1} + \expval{n_i} + = f_F(\varepsilon_i) + = \frac{1}{e^{\beta (\varepsilon_i - \mu)} + 1} } \end{aligned}$$ -This gives the expected occupation number $$\expval{N}$$ -of state $$s$$ with energy $$\varepsilon$$, +This gives the expected occupation number $$\expval{n_i}$$ +of state $$\ket{i}$$ with energy $$\varepsilon_i$$, given a temperature $$T$$ and chemical potential $$\mu$$. {% comment %} -The corresponding variance $$\sigma^2 \equiv \expval{N^2} - \expval{N}^2$$ is found to be: +The corresponding variance $$\sigma^2 \equiv \expval{n_i^2} - \expval{n_i}^2$$ is found to be: $$\begin{aligned} \boxed{ \sigma^2 - = k T \pdv{\expval{N}}{\mu} - = \expval{N} \big(1 - \expval{N}\big) + = k T \pdv{\expval{n_i}}{\mu} + = \expval{n_i} \big(1 - \expval{n_i}\big) } \end{aligned}$$ {% endcomment %} diff --git a/source/know/concept/korteweg-de-vries-equation/index.md b/source/know/concept/korteweg-de-vries-equation/index.md index e8035d1..13b1ee2 100644 --- a/source/know/concept/korteweg-de-vries-equation/index.md +++ b/source/know/concept/korteweg-de-vries-equation/index.md @@ -152,7 +152,7 @@ rather than transform the coordinate system, the velocity is incorporated into his ansatz for $$f$$; in other words, he assumed that the entire liquid is moving at $$q_0$$. For a wave going in the positive $$x$$-direction, -the linearized problem then predicts a profile $$\eta(x \!-\! (\sqrt{g h} \!+\! q_0))$$, +the linearized problem then predicts a profile $$\eta(x \!-\! (\sqrt{g h} \!+\! q_0) t)$$, so de Vries chose $$q_0 = -\sqrt{g h}$$ to make it stationary. Analogously, $$q_0 = \sqrt{g h}$$ for a backward-moving wave. With this in mind, the ansatz is: diff --git a/source/know/concept/kubo-formula/index.md b/source/know/concept/kubo-formula/index.md index 4cb39ac..41fda3c 100644 --- a/source/know/concept/kubo-formula/index.md +++ b/source/know/concept/kubo-formula/index.md @@ -27,14 +27,15 @@ respectively refer to the Schrödinger and [interaction pictures](/know/concept/interaction-picture/): $$\begin{aligned} - \expval{\hat{A}}(t) + \expval{\hat{A}(t)} = \matrixel{\psi_S(t)}{\hat{A}_S}{\psi_S(t)} &= \matrixel{\psi_I(t)}{\hat{A}_I(t)}{\psi_I(t)} \\ &= \matrixel{\psi_I(t_0)\,}{\,\hat{K}_I^\dagger(t, t_0) \hat{A}_I(t) \hat{K}_I(t, t_0)\,}{\,\psi_I(t_0)} \end{aligned}$$ -Where the time evolution operator $$\hat{K}_I(t, t_0)$$ is as follows, +Where the [time evolution operator](/know/concept/time-evolution-operator/) +$$\hat{K}_I(t, t_0)$$ is as follows, which we Taylor-expand: $$\begin{aligned} @@ -71,18 +72,18 @@ where $$\Expval{}$$ is the expectation value for $$\Ket{\psi(t)}$$, and $$\Expval{}_0$$ is the expectation value for $$\Ket{\psi_I(t_0)}$$: $$\begin{aligned} - \expval{\hat{A}}(t) + \expval{\hat{A}(t)} = \expval{\hat{K}_I^\dagger \hat{A}_I \hat{K}_I}_0 = \expval{\hat{A}_I(t)}_0 - \frac{i}{\hbar} \int_{t_0}^t \Expval{\Comm{\hat{A}_I(t)}{\hat{H}_{1,I}(t')}}_0 \dd{t'} \end{aligned}$$ -Now we define $$\delta\!\expval{\hat{A}}\!(t)$$ +Now we define $$\delta\!\expval{\hat{A}(t)}$$ as the change of $$\expval{\hat{A}}$$ due to the perturbation $$\hat{H}_1$$, -and insert $$\expval{\hat{A}}(t)$$: +and insert $$\expval{\hat{A}(t)}$$: $$\begin{aligned} - \delta\!\expval{\hat{A}}\!(t) - \equiv \expval{\hat{A}}(t) - \expval{\hat{A}_I}_0 + \delta\!\expval{\hat{A}(t)} + \equiv \expval{\hat{A}(t)} - \expval{\hat{A}_I(t)}_0 = - \frac{i}{\hbar} \int_{t_0}^t \Expval{\Comm{\hat{A}_I(t)}{\hat{H}_{1,I}(t')}}_0 \dd{t'} \end{aligned}$$ @@ -94,7 +95,7 @@ describing the response of $$\expval{\hat{A}}$$ to first order in $$\hat{H}_1$$: $$\begin{aligned} \boxed{ - \delta\!\expval{\hat{A}}\!(t) + \delta\!\expval{\hat{A}(t)} = \int_{t_0}^\infty C^R_{A H_1}(t, t') \dd{t'} } \end{aligned}$$ @@ -142,7 +143,7 @@ With this, the Kubo formula can be written as follows, where we have set $$t_0 = - \infty$$: $$\begin{aligned} - \delta\!\expval{A}\!(t) + \delta\!\expval{\hat{A}(t)} = \int_{-\infty}^\infty C^R_{A B}(t - t') f(t') \dd{t'} = (C^R_{A B} * f)(t) \end{aligned}$$ @@ -150,12 +151,12 @@ $$\begin{aligned} This is a convolution, so the [convolution theorem](/know/concept/convolution-theorem/) states that the [Fourier transform](/know/concept/fourier-transform/) -of $$\delta\!\expval{\hat{A}}\!(t)$$ is simply the product +of $$\delta\!\expval{\hat{A}(t)}$$ is simply the product of the transforms of $$C^R_{AB}$$ and $$f$$: $$\begin{aligned} \boxed{ - \delta\!\expval{\hat{A}}\!(\omega) + \delta\!\expval{\hat{A}(\omega)} = \tilde{C}{}^R_{A B}(\omega) \: \tilde{f}(\omega) } \end{aligned}$$ diff --git a/source/know/concept/larmor-precession/index.md b/source/know/concept/larmor-precession/index.md index 774af7b..601dae7 100644 --- a/source/know/concept/larmor-precession/index.md +++ b/source/know/concept/larmor-precession/index.md @@ -36,8 +36,8 @@ and the exponentials are "twiddle factors": $$\begin{aligned} \Ket{\chi(t)} - = a \exp(- i E_{\downarrow} t / \hbar) \: \Ket{\downarrow} - \:+\: b \exp(- i E_{\uparrow} t / \hbar) \: \Ket{\uparrow} + = a e^{- i E_{\downarrow} t / \hbar} \Ket{\downarrow} + \:+\: b e^{- i E_{\uparrow} t / \hbar} \Ket{\uparrow} \end{aligned}$$ For our purposes, we can safely assume that $$a$$ and $$b$$ are real, @@ -45,8 +45,8 @@ and then say that there exists an angle $$\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} + \Ket{\chi(t)} = \sin(\theta / 2) \: e^{- i E_{\downarrow} t / \hbar} \Ket{\downarrow} + \:+\: \cos(\theta / 2) \: e^{- i E_{\uparrow} t / \hbar} \Ket{\uparrow} \end{aligned}$$ Now, we find the expectation values of the spin operators @@ -56,23 +56,23 @@ The first is: $$\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}} + \begin{bmatrix} a e^{i E_{\downarrow} t / \hbar} \\ b e^{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} + \begin{bmatrix} a e^{- i E_{\downarrow} t / \hbar} \\ b e^{- 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}} + \begin{bmatrix} a e^{i E_{\downarrow} t / \hbar} \\ b e^{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} + \begin{bmatrix} b e^{- i E_{\uparrow} t / \hbar} \\ a e^{- 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} \Big( a b e^{i (E_{\downarrow} \!-\! E_{\uparrow}) t / \hbar} + + b a e^{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(\theta/2) \sin(\theta/2) \Big( e^{i \gamma B t} + e^{- 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} \cos(\gamma B t) \cdot 2 \cos(\theta/2) \sin(\theta/2) \\ &= \frac{\hbar}{2} \sin(\theta) \cos(\gamma B t) \end{aligned}$$ diff --git a/source/know/concept/lindhard-function/index.md b/source/know/concept/lindhard-function/index.md index fd620df..5f11d36 100644 --- a/source/know/concept/lindhard-function/index.md +++ b/source/know/concept/lindhard-function/index.md @@ -19,7 +19,7 @@ which describes the change in $$\Expval{\hat{n}}$$ due to a time-dependent perturbation $$\hat{H}_1$$: $$\begin{aligned} - \delta\!\Expval{ {\hat{n}}}\!(\vb{r}, t) + \delta\!\Expval{\hat{n}(\vb{r}, t)} = -\frac{i}{\hbar} \int_{-\infty}^\infty \Theta(t - t') \Expval{\Comm{\hat{n}_I(\vb{r}, t)}{\hat{H}_{1,I}(t')}}_0 \dd{t'} \end{aligned}$$ @@ -39,7 +39,7 @@ and $$U(\vb{r})$$ is an arbitrary potential function. The Kubo formula becomes: $$\begin{aligned} - \delta\!\Expval{ {\hat{n}}}\!(\vb{r}, t) + \delta\!\Expval{\hat{n}(\vb{r}, t)} = \iint_{-\infty}^\infty \chi(\vb{r}, \vb{r}'; t, t') \: U(\vb{r}') \: e^{i (\omega + i \eta) t'} \dd{t'} \dd{\vb{r}'} \end{aligned}$$ @@ -95,8 +95,9 @@ $$\begin{aligned} \: e^{i (\vb{q}_2 + \vb{q}) \cdot \vb{r}'} \dd{\vb{q}_2} \dd{\vb{r}'} \end{aligned}$$ -For $$V \to \infty$$ we get a Dirac delta function, -but in fact the conclusion holds for finite $$V$$ too: +This gives a Dirac delta function for $$V \to \infty$$ +(a limit that we will take properly later, +but beware that some authors set $$V = 1$$ until then): $$\begin{aligned} \chi(\vb{q}; t, t') @@ -107,8 +108,9 @@ $$\begin{aligned} \end{aligned}$$ Similarly, if the unperturbed Hamiltonian $$\hat{H}_0$$ is time-independent, -$$\chi$$ only depends on the time difference $$t - t'$$. -Note that $$\delta{\Expval{\hat{n}}}$$ already has the form of a Fourier transform, +$$\chi$$ only depends on the time difference $$t\!-\!t'$$. +Note that $$\delta{\Expval{\hat{n}}}$$ already has the form of a Fourier transform +$$t\!-\!t' \to \omega\!+\!i \eta$$, which gives us an opportunity to rewrite $$\chi$$ in the [Lehmann representation](/know/concept/lehmann-representation/): @@ -119,12 +121,12 @@ $$\begin{aligned} \Big( e^{-\beta E_\nu} - e^{- \beta E_{\nu'}} \Big) \end{aligned}$$ -Where $$\Ket{\nu}$$ and $$\Ket{\nu'}$$ are many-electron eigenstates of $$\hat{H}_0$$, +Where $$\Ket{\nu}$$ and $$\Ket{\nu'}$$ are many-particle eigenstates of $$\hat{H}_0$$, and $$Z$$ is the [grand partition function](/know/concept/grand-canonical-ensemble/). -According to the [convolution theorem](/know/concept/convolution-theorem/) -$$\delta{\Expval{\hat{n}}}(\vb{q}, \omega) = \chi(\vb{q}, \omega) \: U(\vb{q})$$. -In anticipation, we swap $$\nu$$ and $$\nu''$$ in the second term, -so the general response function is written as: +To get ready for the calculations ahead, +we swap $$\nu$$ and $$\nu'$$ in the second term, +so the response function is as shown below. +All operators are in the Schrödinger picture from now on: $$\begin{aligned} \chi(\vb{q}, \omega) @@ -135,7 +137,6 @@ $$\begin{aligned} {\hbar (\omega + i \eta) + E_{\nu'} - E_\nu} \bigg) e^{-\beta E_\nu} \end{aligned}$$ -All operators are in the Schrödinger picture from now on, hence we dropped the subscript $$S$$. To proceed, we need to rewrite $$\hat{n}(\vb{q})$$ somehow. If we neglect electron-electron interactions, @@ -180,9 +181,8 @@ with per-value spacing $$2 \pi / V^{1/D}$$ along each axis. Consequently, each orbital $$\psi_\vb{k}$$ uniquely occupies a volume $$(2 \pi)^D / V$$ in $$\vb{k}$$-space, so we make the approximation $$\sum_{\vb{k}} \approx V / (2 \pi)^D \int_{-\infty}^\infty \dd{\vb{k}}$$. -This becomes exact for $$V \to \infty$$, -in which case $$\vb{k}$$ also becomes continuous again, -which is what we want for jellium. +This is exact in the limit $$V \to \infty$$, +in which case $$\vb{k}$$ also becomes a continuous variable again. We apply this standard trick from condensed matter physics to $$\hat{n}$$, and $$V$$ cancels out: @@ -341,45 +341,49 @@ $$\begin{aligned} } \end{aligned}$$ -From this, we would like to get the -[dielectric function](/know/concept/dielectric-function/) $$\varepsilon_r$$. -Recall its definition, where $$U_\mathrm{tot}$$, $$U_\mathrm{ext}$$, and $$U_\mathrm{ind}$$ -are the total, external and induced potentials, respectively: +This is its most general form, but for practical calculations +we need to formally take the limit $$V \to \infty$$ +and then use $$\sum_{\vb{k}} = V / (2 \pi)^{D} \int_{-\infty}^{\infty} \dd{\vb{k}}$$. +Furthermore, electrons are spin-1/2 particles, +so each orbital contains two, meaning +$$\sum_{\sigma}$$ simply gives a constant factor: $$\begin{aligned} - U_\mathrm{tot} - = U_\mathrm{ext} + U_\mathrm{ind} - = \frac{U_\mathrm{ext}}{\varepsilon_r} + \boxed{ + \chi_0(\vb{q}, \omega) + = \frac{2}{(2 \pi)^{D}} \int_{-\infty}^{\infty} + \frac{n_F(\xi_{\vb{k}}) - n_F(\xi_{\vb{k} + \vb{q}})} + {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}} \dd{\vb{k}} + } \end{aligned}$$ -Note that these are all *energy* potentials: -this choice is justified because all energy potentials -are caused by electric fields in this case. -The *electric* potential is recoverable as -$$\Phi_\mathrm{tot} = q_e U_\mathrm{tot}$$, +From this, we would like to get the +[dielectric function](/know/concept/dielectric-function/) $$\varepsilon_r$$. +When an external [electric field](/know/concept/electric-field/) is applied, +the electrons respond and thereby modify the net field inside the material. +We include this effect in our *energy* potential $$U$$, +such that the net *electric* potential +$$\Phi_\mathrm{tot} = U / q_e$$, where $$q_e < 0$$ is the charge of an electron. +This is not the same as including direct electron-electron interactions! -From the Lindhard response function $$\chi_0$$, -we get the induced particle density offset $$\delta{\Expval{\hat{n}}}$$ -caused by a potential $$U$$. -The density $$\delta{\Expval{\hat{n}}}$$ should be self-consistent, -implying $$U = U_\mathrm{tot}$$. -In other words, we have a linear relation -$$\delta{\Expval{\hat{n}}} = \chi_0 U_\mathrm{tot}$$, -so the standard formula for $$\varepsilon_r$$ gives: +We thus have a linear relation for the induced *particle* density +$$\delta\!\Expval{\hat{n}(\vb{q}, \omega)} = \chi_0(\vb{q}, \omega) \: U(\vb{q})$$ +thanks to the [convolution theorem](/know/concept/convolution-theorem/). +The corresponding induced *charge* density is given by +$$\rho_\mathrm{ind} = q_e^2 \chi_0 \Phi_\mathrm{tot}$$, +so the standard formula for $$\varepsilon_r$$ yields: $$\begin{aligned} \boxed{ \varepsilon_r(\vb{q}, \omega) - = 1 - \frac{U_{ee}(\vb{q})}{V} - \sum_{\sigma \vb{k}} \frac{n_F(\xi_{\vb{k}}) - n_F(\xi_{\vb{k} + \vb{q}})}{\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}} + = 1 - U_{ee}(\vb{q}) \frac{2}{(2 \pi)^{D}} + \int_{-\infty}^{\infty} \frac{n_F(\xi_{\vb{k}}) - n_F(\xi_{\vb{k} + \vb{q}})}{\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}} \dd{\vb{k}} } \end{aligned}$$ -Where $$U_{ee}(\vb{q}) = q_e^2 / (\varepsilon_0 |\vb{q}|^2)$$ -is Coulomb repulsion. -This is the **Lindhard dielectric function** of a free -non-interacting electron gas, +Where $$U_{ee}(\vb{q}) = q_e^2 / (\varepsilon_0 |\vb{q}|^2)$$ is Coulomb repulsion. +This is the **Lindhard dielectric function** of a free non-interacting electron gas, at any temperature and for any dimensionality. diff --git a/source/know/concept/matsubara-greens-function/index.md b/source/know/concept/matsubara-greens-function/index.md index 5e753db..6f60edf 100644 --- a/source/know/concept/matsubara-greens-function/index.md +++ b/source/know/concept/matsubara-greens-function/index.md @@ -64,7 +64,7 @@ $$\begin{aligned} With $$-$$ for bosons, and $$+$$ for fermions, due to the time-ordered product for $$\tau > \tau'$$. -On this domain $$[-\hbar \beta, \hbar \beta]$$, +On this domain $$]\!-\!\hbar \beta, \hbar \beta[$$, the Matsubara Green's function $$C_{AB}$$ obeys a useful shift relation: it is $$\hbar \beta$$-periodic for bosons, @@ -133,7 +133,7 @@ $$\begin{aligned} {% include proof/end.html id="proof-period" %} -Due to this limited domain $$\tau \in [-\hbar \beta, \hbar \beta]$$, +Due to this limited domain $$\tau \in \:]\!-\!\hbar \beta, \hbar \beta[$$, the [Fourier transform](/know/concept/fourier-transform/) of $$C_{AB}(\tau)$$ consists of discrete frequencies $$k_n \equiv n \pi / (\hbar \beta)$$. @@ -288,7 +288,7 @@ $$\begin{aligned} \matrixel{n'}{\hat{B}}{n} e^{(E_n - E_{n'})(\tau - \tau') / \hbar} \end{aligned}$$ -We take the Fourier transform by integrating over $$[0, \hbar \beta]$$: +We take the Fourier transform by integrating over $$]0, \hbar \beta[$$: $$\begin{aligned} C_{AB}(i \omega_m) @@ -324,7 +324,7 @@ $$\begin{aligned} \end{aligned}$$ Since $$\tau \!-\! \tau' < 0$$ this time, -we take the Fourier transform over $$[-\hbar \beta, 0]$$: +we take the Fourier transform over $$]\!-\!\hbar \beta, 0[$$: $$\begin{aligned} C_{AB}(i \omega_m) @@ -341,7 +341,7 @@ $$\begin{aligned} \Big( e^{-\beta E_n} - e^{-i \hbar \omega_m \beta} e^{-\beta E_{n'}} \Big) \\ &= \mp \frac{1}{Z} \sum_{n n'} \frac{\matrixel{n}{\hat{B}}{n'} \matrixel{n'}{\hat{A}}{n}}{i \hbar \omega_m - E_n + E_{n'}} - \Big( e^{- \beta E_n} \pm e^{-\beta E_{n'}} \Big) + \Big( e^{- \beta E_n} \mp e^{-\beta E_{n'}} \Big) \\ &= \frac{1}{Z} \sum_{n n'} \frac{\matrixel{n}{\hat{B}}{n'} \matrixel{n'}{\hat{A}}{n}}{i \hbar \omega_m - E_n + E_{n'}} \Big( e^{- \beta E_{n'}} \mp e^{-\beta E_n} \Big) diff --git a/source/know/concept/matsubara-sum/index.md b/source/know/concept/matsubara-sum/index.md index 0e04455..3347476 100644 --- a/source/know/concept/matsubara-sum/index.md +++ b/source/know/concept/matsubara-sum/index.md @@ -23,7 +23,7 @@ $$\begin{aligned} $$g(z)$$ is a *meromorphic* function on the complex frequency plane, i.e. it is [holomorphic](/know/concept/holomorphic-function/) except for a known set of simple poles, -and $$\tau \in [-\hbar \beta, \hbar \beta]$$ is a real parameter. +and $$\tau \in \:]\!-\!\hbar \beta, \hbar \beta[$$ is a real parameter. The Matsubara frequencies $$i \omega_n$$ are defined as follows for bosons (subscript $$B$$) or fermions (subscript $$F$$): @@ -77,9 +77,9 @@ $$\begin{aligned} h(z) \equiv \begin{cases} - n_{B,F}(z) & \mathrm{if}\; \tau \ge 0 + n_{B,F}(z) & \mathrm{if}\; 0 \le \tau < \hbar \beta \\ - -n_{B,F}(-z) & \mathrm{if}\; \tau \le 0 + -n_{B,F}(-z) & \mathrm{if}\; \!-\!\hbar \beta < \tau \le 0 \end{cases} \end{aligned}$$ @@ -107,7 +107,7 @@ $$\begin{aligned} &= \lim_{z \to i \omega_n}\!\bigg( \frac{z - i \omega_n}{e^{\hbar \beta z} + 1} \bigg) = \lim_{\eta \to 0}\!\bigg( \frac{i \omega_n + \eta - i \omega_n}{e^{i \hbar \beta \omega_n} e^{\hbar \beta \eta} + 1} \bigg) \\ - &= \lim_{\eta \to 0}\!\bigg( \frac{\eta}{e^{\hbar \beta \eta} + 1} \bigg) + &= \lim_{\eta \to 0}\!\bigg( \frac{\eta}{-e^{\hbar \beta \eta} + 1} \bigg) = \lim_{\eta \to 0}\!\bigg( \frac{\eta}{- 1 - \hbar \beta \eta + 1} \bigg) = - \frac{1}{\hbar \beta} \end{aligned}$$ diff --git a/source/know/concept/pauli-exclusion-principle/index.md b/source/know/concept/pauli-exclusion-principle/index.md index 9821718..5b83b69 100644 --- a/source/know/concept/pauli-exclusion-principle/index.md +++ b/source/know/concept/pauli-exclusion-principle/index.md @@ -106,7 +106,7 @@ $$\begin{aligned} = C \Ket{a}\Ket{a} \end{aligned}$$ -However, for fermions, the state is unnormalizable and thus unphysical: +However, for fermions, the state is unnormalizable and therefore unphysical: $$\begin{aligned} \Ket{\Psi(a, a)}_f diff --git a/source/know/concept/salt-equation/index.md b/source/know/concept/salt-equation/index.md index d47383f..e6ed5e5 100644 --- a/source/know/concept/salt-equation/index.md +++ b/source/know/concept/salt-equation/index.md @@ -80,7 +80,7 @@ $$\begin{aligned} + \frac{i}{\hbar} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \: D \end{aligned}$$ -With being $$\vb{p}_0^{+} \vb{p}_0^{-}$$ a dyadic product. +With $$\vb{p}_0^{+} \vb{p}_0^{-}$$ denoting a dyadic product. Isolating the latter equation for $$\vb{p}_n$$ gives us: $$\begin{aligned} -- cgit v1.3