From 075683cdf4588fe16f41d9f7b46b9720b42b2553 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Wed, 17 Jul 2024 10:01:43 +0200 Subject: Improve knowledge base --- .../concept/bernstein-vazirani-algorithm/index.md | 4 +- .../know/concept/deutsch-jozsa-algorithm/index.md | 22 ++-- source/know/concept/dirac-notation/index.md | 16 ++- source/know/concept/ito-integral/index.md | 57 ++++----- .../concept/korteweg-de-vries-equation/index.md | 31 ++--- .../know/concept/kramers-kronig-relations/index.md | 133 ++++++++++++--------- source/know/concept/lagrange-multiplier/index.md | 2 +- source/know/concept/laser-rate-equations/index.md | 10 +- .../concept/lyddane-sachs-teller-relation/index.md | 4 +- source/know/concept/magnetohydrodynamics/index.md | 122 +++++++++---------- source/know/concept/material-derivative/index.md | 10 +- .../know/concept/maxwell-bloch-equations/index.md | 22 ++-- .../concept/rotating-wave-approximation/index.md | 32 ++--- source/know/concept/salt-equation/index.md | 4 +- source/know/concept/simons-algorithm/index.md | 46 ++++--- source/know/concept/two-fluid-equations/index.md | 47 +++++--- 16 files changed, 296 insertions(+), 266 deletions(-) (limited to 'source') diff --git a/source/know/concept/bernstein-vazirani-algorithm/index.md b/source/know/concept/bernstein-vazirani-algorithm/index.md index 884cca3..4f36d3c 100644 --- a/source/know/concept/bernstein-vazirani-algorithm/index.md +++ b/source/know/concept/bernstein-vazirani-algorithm/index.md @@ -24,8 +24,8 @@ of $$x$$ with an unknown $$N$$-bit string $$s$$: $$\begin{aligned} f(x) - = s \cdot x \:\:(\bmod \: 2) - = (s_1 x_1 + s_2 x_2 + \:...\: + s_N x_N) \:\:(\bmod \: 2) + \equiv s \cdot x \:\bmod 2 + = (s_1 x_1 + s_2 x_2 + \:...\: + s_N x_N) \:\bmod 2 \end{aligned}$$ The goal is to find $$s$$. diff --git a/source/know/concept/deutsch-jozsa-algorithm/index.md b/source/know/concept/deutsch-jozsa-algorithm/index.md index 44b06ad..223877a 100644 --- a/source/know/concept/deutsch-jozsa-algorithm/index.md +++ b/source/know/concept/deutsch-jozsa-algorithm/index.md @@ -72,8 +72,8 @@ $$\begin{aligned} + \frac{1}{2} \Ket{1} \Big( \Ket{0 \oplus f(1)} - \Ket{1 \oplus f(1)} \Big) \end{aligned}$$ -The parenthesized superpositions can be reduced. -Assuming that $$f(b) = 0$$, we notice: +The parenthesized superpositions can be reduced: +let us suppose that $$f(b) = 0$$, then: $$\begin{aligned} \Ket{0 \oplus f(b)} - \Ket{1 \oplus f(b)} @@ -91,7 +91,7 @@ $$\begin{aligned} \end{aligned}$$ We can thus combine both cases, $$f(b) = 0$$ or $$f(b) = 1$$, -into the following single expression: +into the following expression: $$\begin{aligned} \Ket{0 \oplus f(b)} - \Ket{1 \oplus f(b)} @@ -106,8 +106,8 @@ $$\begin{aligned} \frac{1}{2} \Big( (-1)^{f(0)} \Ket{0} + (-1)^{f(1)} \Ket{1} \Big) \Big( \Ket{0} - \Ket{1} \Big) \end{aligned}$$ -The second qubit in state $$\Ket{-}$$ is garbage; it is no longer of interest. -The first qubit is given by: +The second qubit in state $$\Ket{-}$$ is garbage (i.e. no longer of interest). +The first qubit is: $$\begin{aligned} \frac{1}{\sqrt{2}} \Big( (-1)^{f(0)} \Ket{0} + (-1)^{f(1)} \Ket{1} \Big) @@ -126,8 +126,8 @@ $$\begin{aligned} \end{aligned}$$ Depending on whether $$f$$ is constant or balanced, -the mearurement outcome of this state will be $$\Ket{0}$$ or $$\Ket{1}$$ -with 100\% probability. We have solved the problem! +the measurement outcome of this state will be $$\Ket{0}$$ or $$\Ket{1}$$ +with 100% probability. We have solved the problem! Note that we only consulted the oracle (i.e. applied $$U_f$$) once. A classical computer would need to query it twice, @@ -146,7 +146,7 @@ This algorithm is then implemented by the following quantum circuit: alt="Deutsch-Jozsa circuit" %} There are $$N$$ qubits in initial state $$\Ket{0}$$, and one in $$\Ket{1}$$. -For clarity, the oracle $$U_f$$ works like so: +The oracle $$U_f$$ performs this action: $$\begin{aligned} \Ket{x_1} \Ket{x_2} \cdots \Ket{x_N} \Ket{y} @@ -167,7 +167,7 @@ $$\begin{aligned} Where $$\Ket{x} = \Ket{x_1} \cdots \Ket{x_N}$$ denotes a classical binary state. For example, if $$x = 5 = 2^0 + 2^2$$ in the summation, then $$\Ket{x} = \Ket{1} \Ket{0} \Ket{1} \Ket{0}^{\otimes N-3}$$ -(from least to most significant). +(from least to most significant digit). We give this state to the oracle, and, by the same logic as for the Deutsch algorithm, @@ -217,8 +217,8 @@ we only need to measure the $$N$$ qubits once; $$f$$ is constant if and only if all are zero. The Deutsch-Jozsa algorithm needs only one oracle query to give an error-free result, -whereas a classical computer needs $$2^{N-1} + 1$$ queries in the worst case; -a revolutionary discovery. +whereas a classical computer needs $$2^{N-1} + 1$$ queries in the worst case. +A revolutionary discovery! ## References diff --git a/source/know/concept/dirac-notation/index.md b/source/know/concept/dirac-notation/index.md index 2830a33..bbf31e5 100644 --- a/source/know/concept/dirac-notation/index.md +++ b/source/know/concept/dirac-notation/index.md @@ -27,7 +27,8 @@ that maps kets $$\ket{V}$$ to other kets $$\ket{V'}$$. Recall that by definition the Hilbert inner product must satisfy: $$\begin{aligned} - \inprod{V}{W} = \inprod{W}{V}^* + \inprod{V}{W} + = \inprod{W}{V}^* \end{aligned}$$ So far, nothing has been said about the actual representation of bras or kets. @@ -36,12 +37,14 @@ the corresponding bras are given by the kets' adjoints, i.e. their transpose conjugates: $$\begin{aligned} - \ket{V} = + \ket{V} + = \begin{bmatrix} v_1 \\ \vdots \\ v_N \end{bmatrix} - \quad \implies \quad - \bra{V} = + \qquad \implies \qquad + \bra{V} + = \begin{bmatrix} v_1^* & \cdots & v_N^* \end{bmatrix} @@ -88,8 +91,9 @@ then the bras are *functionals* $$F[u(x)]$$ that take an arbitrary function $$u(x)$$ as an argument and return a scalar: $$\begin{aligned} - \ket{f} = f(x) - \quad \implies \quad + \ket{f} + = f(x) + \qquad \implies \qquad \bra{f} = F[u(x)] = \int_a^b f^*(x) \: u(x) \dd{x} diff --git a/source/know/concept/ito-integral/index.md b/source/know/concept/ito-integral/index.md index 4a725e1..9b092d6 100644 --- a/source/know/concept/ito-integral/index.md +++ b/source/know/concept/ito-integral/index.md @@ -10,8 +10,7 @@ layout: "concept" The **Itō integral** offers a way to integrate a given [stochastic process](/know/concept/stochastic-process/) $$G_t$$ -with respect to a [Wiener process](/know/concept/wiener-process/) $$B_t$$, -which is also a stochastic process. +with respect to a [Wiener process](/know/concept/wiener-process/) $$B_t$$. The Itō integral $$I_t$$ of $$G_t$$ is defined as follows: $$\begin{aligned} @@ -47,21 +46,21 @@ which can be applied recursively, leading to: $$\begin{aligned} X_{t+h} \approx X_{t} + f(X_t) \: h - \quad \implies \quad + \qquad \implies \qquad X_t \approx X_0 + \sum_{s = 0}^{s = t} f(X_s) \: h \end{aligned}$$ -In the limit $$h \to 0$$, this leads to the following unsurprising integral for $$X_t$$: +In the limit $$h \to 0$$, this unsurprisingly leads to the following integral for $$X_t$$: $$\begin{aligned} - \int_0^t f(X_s) \dd{s} - = \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h + \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h + = \int_0^t f(X_s) \dd{s} \end{aligned}$$ In contrast, consider the *stochastic differential equation* below, where $$\xi_t$$ represents white noise, -which is informally the $$t$$-derivative +which is informally defined as the $$t$$-derivative of the Wiener process $$\xi_t = \idv{B_t}{t}$$: $$\begin{aligned} @@ -89,9 +88,9 @@ $$\begin{aligned} = X_0 + \int_0^t g(X_s) \dd{B_s} \end{aligned}$$ -This integral is *defined* as below, -analogously to the first, but with $$h$$ replaced by -the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process. +The meaning of such an integral is *defined* below. +It is analogous to the deterministic case, +but $$h$$ is replaced by the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process. This is an Itō integral: $$\begin{aligned} @@ -100,7 +99,7 @@ $$\begin{aligned} \end{aligned}$$ For more information about applying the Itō integral in this way, -see the [Itō calculus](/know/concept/ito-process/). +see [Itō calculus](/know/concept/ito-process/). @@ -131,7 +130,7 @@ $$\begin{aligned} A more interesting property is the **Itō isometry**, which expresses the expectation of the square of an Itō integral of $$G_t$$ as a simpler "ordinary" integral of the expectation of $$G_t^2$$ -(which exists by the definition of Itō-integrability): +(which exists due to the definition of Itō-integrability): $$\begin{aligned} \boxed{ @@ -172,24 +171,16 @@ $$\begin{aligned} However, $$\mathcal{F}_t$$ says nothing about the increment $$(B_{t + h} \!-\! B_t) \sim \mathcal{N}(0, h)$$, -meaning that the conditional expectation is zero: +meaning that the conditional expectation is zero for $$t \ge s + h$$: $$\begin{aligned} \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big] = 0 - \qquad \mathrm{for}\; t \ge s + h \end{aligned}$$ -By swapping $$s$$ and $$t$$, the exact same result can be obtained for $$s \ge t \!+\! h$$: - -$$\begin{aligned} - \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big] - = 0 - \qquad \mathrm{for}\; s \ge t + h -\end{aligned}$$ - -This leaves only one case which can be nonzero: $$[t, t\!+\!h] = [s, s\!+\!h]$$. -Applying the law of total expectation again yields: +By swapping $$s$$ and $$t$$, the exact same result can be obtained for $$s \ge t \!+\! h$$. +This leaves only one possibly nonzero case: $$[t, t\!+\!h] = [s, s\!+\!h]$$. +Applying the law of total expectation again: $$\begin{aligned} \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2 @@ -198,15 +189,15 @@ $$\begin{aligned} &= \sum_{t = a}^{t = b} \mathbf{E} \bigg[ \mathbf{E} \Big[ G_t^2 (B_{t + h} \!-\! B_t)^2 \Big| \mathcal{F}_t \Big] \bigg] \end{aligned}$$ -We know $$G_t$$, and the expectation value of $$(B_{t+h} \!-\! B_t)^2$$, -since the increment is normally distributed, is simply the variance $$h$$: +We know $$G_t$$, +and the expectation value of $$(B_{t+h} \!-\! B_t)^2$$ is simply the variance $$h$$: $$\begin{aligned} \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2 &= \sum_{t = a}^{t = b} \mathbf{E} \big[ G_t^2 \big] h - \longrightarrow - \int_a^b \mathbf{E} \big[ G_t^2 \big] \dd{t} \end{aligned}$$ + +Taking the limit $$h \to 0$$ then yields the desired result. {% include proof/end.html id="proof-isometry" %} @@ -239,7 +230,7 @@ $$\begin{aligned} \end{aligned}$$ We now have everything we need to calculate $$\mathbf{E} [ I_t | \mathcal{F_s} ]$$, -giving the martingale property: +leading to the martingale property: $$\begin{aligned} \mathbf{E} \big[ I_t | \mathcal{F}_s \big] @@ -250,10 +241,10 @@ $$\begin{aligned} For the existence of $$I_t$$, we need $$\mathbf{E}[G_t^2]$$ to be integrable over the target interval, -so from the Itō isometry we have $$\mathbf{E}[I]^2 < \infty$$, -and therefore $$\mathbf{E}[I] < \infty$$, -so $$I_t$$ has all the properties of a Martingale, -since it is trivially $$\mathcal{F}_t$$-adapted. +which implies via the Itō isometry that $$\mathbf{E}[I]^2$$ is finite. +Therefore $$\mathbf{E}[I]$$ is also finite, +so $$I_t$$ has all the properties of a Martingale +(since it is trivially $$\mathcal{F}_t$$-adapted). {% include proof/end.html id="proof-martingale" %} diff --git a/source/know/concept/korteweg-de-vries-equation/index.md b/source/know/concept/korteweg-de-vries-equation/index.md index 2857e23..e8035d1 100644 --- a/source/know/concept/korteweg-de-vries-equation/index.md +++ b/source/know/concept/korteweg-de-vries-equation/index.md @@ -162,11 +162,11 @@ $$\begin{aligned} = q_0 - \frac{g}{q_0} \Big( \eta(x, t) + \alpha + \gamma(x, t) \Big) \end{aligned}$$ -Where $$\alpha$$ is a constant parameter -(which we will use to handle velocity discrepancies -between the linear and nonlinear theories). +Where $$\alpha$$ is a constant parameter, +which we will use to handle velocity discrepancies +between the linear and nonlinear theories. The correction represented by $$\gamma$$ is much smaller, -i.e. $$\eta \sim \alpha \gg \gamma$$. +i.e. $$\eta \gg \alpha \gg \gamma$$. We insert this ansatz into the above equations, yielding: $$\begin{aligned} @@ -265,14 +265,15 @@ $$\begin{aligned} \equiv \frac{h^3}{3} - \frac{h T}{g \rho} \end{aligned}$$ -What about $$\alpha$$? +But what about $$\alpha$$? Looking at the ansatz for $$f$$, we see that -the body of water is already assumed to be moving at $$q_0$$, -minus $$g \alpha / q_0$$, so by varying $$\alpha$$ -we are modifying the water's velocity. -The term in the KdV equation simply corrects for our chosen value of $$\alpha$$. -It has no deeper meaning than that: for any value of $$\alpha$$, -the full range of KdV solutions can still be obtained. +the body of water is assumed to be moving at $$q_0 - g \alpha / q_0$$, +and $$q_0$$ is set to $$\pm \sqrt{g h}$$ by almost all authors, +so $$\alpha$$ controls the velocity of our reference frame. +Nonlinear waves do not travel at the same speed as linear waves, +so we can choose $$\alpha$$ to make the wave stationary +without breaking the $$q_0$$ "tradition". +That term in the KdV equation simply corrects for our chosen value of $$\alpha$$. @@ -383,14 +384,16 @@ These are the final scale parameter values, leading to the desired dimensionless form: $$\begin{aligned} - 0 - &= \tilde{\eta}_{\tilde{t}} - 6 \tilde{\eta} \tilde{\eta}_{\tilde{x}} + \tilde{\eta}_{\tilde{x} \tilde{x} \tilde{x}} + \boxed{ + 0 + = \tilde{\eta}_{\tilde{t}} - 6 \tilde{\eta} \tilde{\eta}_{\tilde{x}} + \tilde{\eta}_{\tilde{x} \tilde{x} \tilde{x}} + } \end{aligned}$$ Recall that $$\alpha$$ sets the background fluid velocity, and $$v_c$$ controls the coordinate system's motion: our choice of $$v_c$$ simply cancels out the effect of $$\alpha$$. -This reveals the point of $$\alpha$$: +This demonstrates the purpose of $$\alpha$$: the KdV equation has solutions moving at various speeds, so, for a given $$\eta$$, we can always choose $$\alpha$$ (and hence $$v_c$$) such that the wave appears stationary. diff --git a/source/know/concept/kramers-kronig-relations/index.md b/source/know/concept/kramers-kronig-relations/index.md index 711023e..68e27dc 100644 --- a/source/know/concept/kramers-kronig-relations/index.md +++ b/source/know/concept/kramers-kronig-relations/index.md @@ -10,124 +10,145 @@ categories: layout: "concept" --- -Let $$\chi(t)$$ be a complex function describing -the response of a system to an impulse $$f(t)$$ starting at $$t = 0$$. -The **Kramers-Kronig relations** connect the real and imaginary parts of $$\chi(t)$$, -such that one can be reconstructed from the other. -Suppose we can only measure $$\chi_r(t)$$ or $$\chi_i(t)$$: +Let $$\chi(t)$$ be the response function of a system +to an external impulse $$f(t)$$, which starts at $$t = 0$$. +Assuming initial equilibrium, the principle of causality +states that there is no response before the impulse, +so $$\chi(t) = 0$$ for $$t < 0$$. +To enforce this, we demand that $$\chi(t)$$ satisfies a **causality test**, +where $$\Theta(t)$$ is the [Heaviside step function](/know/concept/heaviside-step-function/): $$\begin{aligned} - \chi(t) = \chi_r(t) + i \chi_i(t) + \chi(t) + = \chi(t) \: \Theta(t) \end{aligned}$$ -Assuming that the system was at rest until $$t = 0$$, -the response $$\chi(t)$$ cannot depend on anything from $$t < 0$$, -since the known impulse $$f(t)$$ had not started yet, -This principle is called **causality**, and to enforce it, -we use the [Heaviside step function](/know/concept/heaviside-step-function/) -$$\Theta(t)$$ to create a **causality test** for $$\chi(t)$$: - -$$\begin{aligned} - \chi(t) = \chi(t) \: \Theta(t) -\end{aligned}$$ - -If we [Fourier transform](/know/concept/fourier-transform/) this equation, -then it will become a convolution in the frequency domain +If we take the [Fourier transform](/know/concept/fourier-transform/) (FT) +$$\chi(t) \!\to\! \tilde{\chi}(\omega)$$ of this equation, +the right-hand side becomes a convolution in the frequency domain thanks to the [convolution theorem](/know/concept/convolution-theorem/), -where $$A$$, $$B$$ and $$s$$ are constants from the FT definition: +where $$A$$, $$B$$ and $$s$$ are constants determined by +how we choose to define our FT: $$\begin{aligned} \tilde{\chi}(\omega) - = (\tilde{\chi} * \tilde{\Theta})(\omega) - = B \int_{-\infty}^\infty \tilde{\chi}(\omega') \: \tilde{\Theta}(\omega - \omega') \dd{\omega'} + &= (\tilde{\chi} * \tilde{\Theta})(\omega) + \\ + &= B \int_{-\infty}^\infty \tilde{\chi}(\omega') \: \tilde{\Theta}(\omega - \omega') \dd{\omega'} \end{aligned}$$ -We look up the FT of the step function $$\tilde{\Theta}(\omega)$$, +We look up the full expression for $$\tilde{\Theta}(\omega)$$, which involves the signum function $$\mathrm{sgn}(t)$$, the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta$$, -and the Cauchy principal value $$\pv{}$$. -We arrive at: +and the [Cauchy principal value](/know/concept/cauchy-principal-value/) $$\pv{}$$. +Inserting that, we arrive at: $$\begin{aligned} \tilde{\chi}(\omega) &= \frac{A B}{|s|} \pv{\int_{-\infty}^\infty \tilde{\chi}(\omega') - \Big( \pi \delta(\omega - \omega') + i \:\mathrm{sgn} \frac{1}{\omega - \omega'} \Big) \dd{\omega'}} + \bigg( \pi \delta(\omega - \omega') + i \frac{\mathrm{sgn}(s)}{\omega - \omega'} \bigg) \dd{\omega'}} \\ - &= \Big( \frac{1}{2} \frac{2 \pi A B}{|s|} \Big) \tilde{\chi}(\omega) - + i \Big( \frac{\mathrm{sgn}(s)}{2 \pi} \frac{2 \pi A B}{|s|} \Big) + &= \bigg( \frac{2}{2} \frac{\pi A B}{|s|} \bigg) \tilde{\chi}(\omega) + + i \: \mathrm{sgn}(s) \bigg( \frac{2 \pi}{2 \pi} \frac{A B}{|s|} \bigg) \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} \end{aligned}$$ -From the definition of the Fourier transform we know that -$$2 \pi A B / |s| = 1$$: +From the definition of the FT we know that +$$2 \pi A B / |s| = 1$$, so this reduces to: $$\begin{aligned} \tilde{\chi}(\omega) &= \frac{1}{2} \tilde{\chi}(\omega) - + \mathrm{sgn}(s) \frac{i}{2 \pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} + + i \: \mathrm{sgn}(s) \frac{1}{2 \pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} \end{aligned}$$ -We isolate this equation for $$\tilde{\chi}(\omega)$$ -to get the final version of the causality test: +We rearrange this equation a bit to get the final version of the causality test: $$\begin{aligned} \boxed{ \tilde{\chi}(\omega) - = - \mathrm{sgn}(s) \frac{i}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} + = i \: \mathrm{sgn}(s) \frac{1}{\pi} + \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} } \end{aligned}$$ -By inserting $$\tilde{\chi}(\omega) = \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega)$$ -and splitting the equation into real and imaginary parts, -we get the Kramers-Kronig relations: +Next, we split $$\tilde{\chi}(\omega)$$ +into its real and imaginary parts, +i.e. $$\tilde{\chi}(\omega) = \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega)$$: + +$$\begin{aligned} + \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega) + = i \: \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{\omega - \omega'} \dd{\omega'}} + - \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{\omega - \omega'} \dd{\omega'}} +\end{aligned}$$ + +This equation can likewise be split into real and imaginary parts, +leading to the **Kramers-Kronig relations**, +which enable us to reconstruct $$\tilde{\chi}_r(\omega)$$ +from $$\tilde{\chi}_i(\omega)$$ and vice versa: $$\begin{aligned} \boxed{ \begin{aligned} \tilde{\chi}_r(\omega) - &= \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{\omega' - \omega} \dd{\omega'}} + &= - \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{\omega - \omega'} \dd{\omega'}} \\ \tilde{\chi}_i(\omega) - &= - \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{\omega' - \omega} \dd{\omega'}} + &= \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{\omega - \omega'} \dd{\omega'}} \end{aligned} } \end{aligned}$$ -If the time-domain response function $$\chi(t)$$ is real -(so far we have assumed it to be complex), -then we can take advantage of the fact that -the FT of a real function satisfies -$$\tilde{\chi}(-\omega) = \tilde{\chi}^*(\omega)$$, i.e. $$\tilde{\chi}_r(\omega)$$ -is even and $$\tilde{\chi}_i(\omega)$$ is odd. We multiply the fractions by -$$(\omega' + \omega)$$ above and below: +The sign of these expressions deserves special attention: +it depends on an author's choice of FT definition via $$\mathrm{sgn}(s)$$, +and, to make matters even more confusing, +many also choose to use the opposite sign in the denominator, +i.e. they write $$\omega' - \omega$$ instead of $$\omega - \omega'$$. + +In the special case where $$\chi(t)$$ is real, +we can take advantage of the property that +the FT of a real function always satisfies +$$\tilde{\chi}(-\omega) = \tilde{\chi}^*(\omega)$$. +Here, this means that $$\tilde{\chi}_r(\omega)$$ is even +and $$\tilde{\chi}_i(\omega)$$ is odd. +To use this fact, we simultaneously +multiply and divide the integrands by $$\omega + \omega'$$: $$\begin{aligned} \tilde{\chi}_r(\omega) - &= \mathrm{sgn}(s) \bigg( \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} - + \frac{\omega}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} \bigg) + &= - \mathrm{sgn}(s) \frac{1}{\pi} + \bigg( \!\pv{\int_{-\infty}^\infty \frac{\omega \tilde{\chi}_i(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} + + \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} \bigg) \\ \tilde{\chi}_i(\omega) - &= - \mathrm{sgn}(s) \bigg( \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} - + \frac{\omega}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} \bigg) + &= \mathrm{sgn}(s) \frac{1}{\pi} + \bigg( \!\pv{\int_{-\infty}^\infty \frac{\omega \tilde{\chi}_r(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} + + \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_r(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} \bigg) \end{aligned}$$ -For $$\tilde{\chi}_r(\omega)$$, the second integrand is odd, so we can drop it. -Similarly, for $$\tilde{\chi}_i(\omega)$$, the first integrand is odd. -We therefore find the following variant of the Kramers-Kronig relations: +In $$\tilde{\chi}_r(\omega)$$'s equation, the first integrand is odd, +so the integral's value is zero. +Similarly, for $$\tilde{\chi}_i(\omega)$$, the second integrand is odd, so we drop it too. +We thus arrive at the following common variant of the Kramers-Kronig relations, +only valid for real $$\chi(t)$$: $$\begin{aligned} \boxed{ \begin{aligned} \tilde{\chi}_r(\omega) - &= \mathrm{sgn}(s) \frac{2}{\pi} \pv{\int_0^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} + &= - \mathrm{sgn}(s) \frac{2}{\pi} + \pv{\int_0^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} \\ \tilde{\chi}_i(\omega) - &= - \mathrm{sgn}(s) \frac{2 \omega}{\pi} \pv{\int_0^\infty \frac{\tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} + &= \mathrm{sgn}(s) \frac{2}{\pi} + \pv{\int_0^\infty \frac{\omega \tilde{\chi}_r(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} \end{aligned} } \end{aligned}$$ -To reiterate: this version is only valid if $$\chi(t)$$ is real in the time domain. +Note that we have modified the integration limits +using the fact that the integrands are even, +leading to an extra factor of $$2$$. diff --git a/source/know/concept/lagrange-multiplier/index.md b/source/know/concept/lagrange-multiplier/index.md index 6b5e3fc..4c2e957 100644 --- a/source/know/concept/lagrange-multiplier/index.md +++ b/source/know/concept/lagrange-multiplier/index.md @@ -117,7 +117,7 @@ We often assign $$\lambda$$ an algebraic expression rather than a value, usually without even bothering to calculate its final actual value. In fact, in some cases, $$\lambda$$'s only function is to help us reason about the interdependence of a system of equations -(see [example 3](https://en.wikipedia.org/wiki/Lagrange_multiplier#Example_3:_Entropy) on Wikipedia); +(see Wikipedia's [entropy example](https://en.wikipedia.org/wiki/Lagrange_multiplier#Examples)); then $$\lambda$$ is not even given an expression! Hence it is sometimes also called an *undetermined multiplier*. diff --git a/source/know/concept/laser-rate-equations/index.md b/source/know/concept/laser-rate-equations/index.md index c81f02b..feec168 100644 --- a/source/know/concept/laser-rate-equations/index.md +++ b/source/know/concept/laser-rate-equations/index.md @@ -30,7 +30,7 @@ $$\begin{aligned} Where $$n$$ is the background medium's refractive index, $$\omega_0$$ the two-level system's gap resonance frequency, -$$|g| \equiv |\matrixel{e}{\vu{x}}{g}|$$ the transition dipole moment, +$$|g| \equiv |\!\matrixel{e}{\vu{x}}{g}\!|$$ the transition dipole moment, $$\gamma_\perp$$ and $$\gamma_\parallel$$ empirical decay rates, and $$D_0$$ the equilibrium inversion. Note that $$\vb{E}^{-} = (\vb{E}^{+})^*$$. @@ -110,7 +110,7 @@ $$\begin{aligned} Where the Lorentzian gain curve $$\gamma(\omega)$$ (which also appears in the [SALT equation](/know/concept/salt-equation/)) -represents a laser's preferred spectrum for amplification, +represents the laser's preferred spectrum for amplification, and is defined like so: $$\begin{aligned} @@ -139,7 +139,7 @@ $$\begin{aligned} Next, we insert our ansatz for $$\vb{E}^{+}$$ and $$\vb{P}^{+}$$ into the third MBE, and rewrite $$\vb{P}_0^{+}$$ as above. -Using our identity for $$\gamma(\omega)$$, +Using the aforementioned identity for $$\gamma(\omega)$$ and the fact that $$\vb{E}_0^{+} \cdot \vb{E}_0^{-} = |\vb{E}|^2$$, we find: $$\begin{aligned} @@ -218,8 +218,8 @@ $$\begin{aligned} \end{aligned}$$ Where $$\gamma_e$$ is a redefinition of $$\gamma_\parallel$$ -depending on the electron decay processes, -and the photon loss rate $$\gamma_p$$, the gain $$G$$, +depending on the electron decay processes. +The photon loss rate $$\gamma_p$$, the gain $$G$$, and the carrier supply rate $$R_\mathrm{pump}$$ are defined like so: diff --git a/source/know/concept/lyddane-sachs-teller-relation/index.md b/source/know/concept/lyddane-sachs-teller-relation/index.md index e80bf00..9cec9dc 100644 --- a/source/know/concept/lyddane-sachs-teller-relation/index.md +++ b/source/know/concept/lyddane-sachs-teller-relation/index.md @@ -219,8 +219,8 @@ i.e. the material becomes a perfect reflector: $$\begin{aligned} R - = \bigg| \frac{i \sqrt{|\varepsilon_r|} - 1}{i \sqrt{|\varepsilon_r|} + 1} \bigg|^2 - = \frac{|\varepsilon_r|^2 + 1^2}{|\varepsilon_r|^2 + 1^2} + = \bigg| \frac{i \sqrt{-\varepsilon_r} - 1}{i \sqrt{-\varepsilon_r} + 1} \bigg|^2 + = \frac{\varepsilon_r^2 + 1^2}{\varepsilon_r^2 + 1^2} = 1 \end{aligned}$$ diff --git a/source/know/concept/magnetohydrodynamics/index.md b/source/know/concept/magnetohydrodynamics/index.md index bcc23f3..4431dfa 100644 --- a/source/know/concept/magnetohydrodynamics/index.md +++ b/source/know/concept/magnetohydrodynamics/index.md @@ -24,24 +24,23 @@ and electric current density $$\vb{J}$$ are: $$\begin{aligned} p - = p_i + p_e - \qquad \quad + &= p_i + p_e + \\ \vb{J} - = q_i n_i \vb{u}_i + q_e n_e \vb{u}_e + &= q_i n_i \vb{u}_i + q_e n_e \vb{u}_e \end{aligned}$$ Meanwhile, the macroscopic mass density $$\rho$$ -and center-of-mass flow velocity $$\vb{u}$$ -are as follows, although the ions dominate due to their large mass: +and center-of-mass flow velocity $$\vb{u}$$ are as follows, +although the ions dominate both due to their large mass, +so $$\rho \approx m_i n_i$$ and $$\vb{u} \approx \vb{u}_i$$: $$\begin{aligned} \rho - = m_i n_i + m_e n_e - \approx m_i n_i - \qquad \quad + &= m_i n_i + m_e n_e + \\ \vb{u} - = \frac{1}{\rho} \Big( m_i n_i \vb{u}_i + m_e n_e \vb{u}_e \Big) - \approx \vb{u}_i + &= \frac{1}{\rho} \Big( m_i n_i \vb{u}_i + m_e n_e \vb{u}_e \Big) \end{aligned}$$ With these quantities in mind, @@ -75,9 +74,9 @@ $$\begin{aligned} \end{aligned}$$ We will assume that electrons' inertia -is negligible compared to the [Lorentz force](/know/concept/lorentz-force/). -Let $$\tau_\mathrm{char}$$ be the characteristic timescale of the plasma's dynamics, -i.e. nothing noticable happens in times shorter than $$\tau_\mathrm{char}$$, +is negligible compared to the Lorentz force. +Let $$\tau_\mathrm{char}$$ be the characteristic timescale of the plasma's dynamics +(i.e. nothing notable happens in times shorter than $$\tau_\mathrm{char}$$), then this assumption can be written as: $$\begin{aligned} @@ -86,15 +85,14 @@ $$\begin{aligned} \sim \frac{m_e n_e |\vb{u}_e| / \tau_\mathrm{char}}{q_e n_e |\vb{u}_e| |\vb{B}|} = \frac{m_e}{q_e |\vb{B}| \tau_\mathrm{char}} = \frac{1}{\omega_{ce} \tau_\mathrm{char}} - \ll 1 \end{aligned}$$ -Where we have recognized the cyclotron frequency $$\omega_c$$ (see Lorentz force article). +Where we have recognized the cyclotron frequency $$\omega_c$$ +(see [Lorentz force](/know/concept/lorentz-force/)). In other words, our assumption is equivalent to the electron gyration period $$2 \pi / \omega_{ce}$$ -being small compared to the macroscopic dynamics' timescale $$\tau_\mathrm{char}$$. -By construction, we can thus ignore the left-hand side -of the electron momentum equation, leaving: +being small compared to the macroscopic timescale $$\tau_\mathrm{char}$$. +We can thus ignore the left-hand side of the electron momentum equation, leaving: $$\begin{aligned} m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} @@ -138,8 +136,8 @@ $$\begin{aligned} However, we found this by combining two equations into one, so some information was implicitly lost; -we need a second momentum equation. -Therefore, we return to the electrons' momentum equation, +we need a second one to keep our system of equations complete. +Therefore we return to the electrons' momentum equation, after a bit of rearranging: $$\begin{aligned} @@ -154,14 +152,14 @@ so: $$\begin{aligned} \vb{E} + \vb{u}_e \cross \vb{B} - \frac{\nabla p_e}{q_e n_e} = \eta \vb{J} - \qquad \quad + \qquad \qquad \eta \equiv \frac{f_{ei} m_e}{n_e q_e^2} \end{aligned}$$ Where $$\eta$$ is the electrical resistivity of the plasma, see [Spitzer resistivity](/know/concept/spitzer-resistivity/) -for more information, and a rough estimate of this quantity for a plasma. +for more information and a rough estimate of its value in a plasma. Now, using that $$\vb{u} \approx \vb{u}_i$$, we add $$(\vb{u} \!-\! \vb{u}_i) \cross \vb{B} \approx 0$$ to the equation, @@ -183,34 +181,37 @@ $$\begin{aligned} - \nabla \cross \frac{\nabla p_e}{q_e n_e} \end{aligned}$$ -Where we have used Faraday's law. +Where we have used [Faraday's law](/know/concept/maxwells-equations/). This is the **induction equation**, and is used to compute $$\vb{B}$$. The pressure term can be rewritten using the ideal gas law $$p_e = k_B T_e n_e$$: $$\begin{aligned} \nabla \cross \frac{\nabla p_e}{q_e n_e} - = \frac{k_B}{q_e} \nabla \cross \frac{\nabla (n_e T_e)}{n_e} - = \frac{k_B}{q_e} \nabla \cross \Big( \nabla T_e + T_e \frac{\nabla n_e}{n_e} \Big) + &= \frac{k_B}{q_e} \nabla \cross \frac{\nabla (n_e T_e)}{n_e} + \\ + &= \frac{k_B}{q_e} \nabla \cross \Big( \nabla T_e + T_e \frac{\nabla n_e}{n_e} \Big) \end{aligned}$$ The curl of a gradient is always zero, and we notice that $$\nabla n_e / n_e = \nabla\! \ln(n_e)$$. -Then we use the vector identity $$\nabla \cross (f \nabla g) = \nabla f \cross \nabla g$$, -leading to: +Then we use the vector identity $$\nabla \cross (f \nabla g) = \nabla f \cross \nabla g$$ to get: $$\begin{aligned} \nabla \cross \frac{\nabla p_e}{q_e n_e} - = \frac{k_B}{q_e} \nabla \cross \big( T_e \: \nabla\! \ln(n_e) \big) - = \frac{k_B}{q_e} \big( \nabla T_e \cross \nabla\! \ln(n_e) \big) - = \frac{k_B}{q_e n_e} \big( \nabla T_e \cross \nabla n_e \big) + &= \frac{k_B}{q_e} \nabla \cross \big( T_e \: \nabla\! \ln(n_e) \big) + \\ + &= \frac{k_B}{q_e} \big( \nabla T_e \cross \nabla\! \ln(n_e) \big) + \\ + &= \frac{k_B}{q_e n_e} \big( \nabla T_e \cross \nabla n_e \big) \end{aligned}$$ It is reasonable to assume that $$\nabla T_e$$ and $$\nabla n_e$$ point in roughly the same direction, in which case the pressure term can be neglected. Consequently, $$p_e$$ has no effect on the dynamics of $$\vb{B}$$, -so we argue that it can be dropped from the original (non-curled) equation too, leaving: +so we argue that it can also be dropped +from the original equation (before taking the curl): $$\begin{aligned} \boxed{ @@ -232,20 +233,18 @@ $$\begin{aligned} From Faraday's law, we can obtain a scale estimate for $$\vb{E}$$. Recall that $$\tau_\mathrm{char}$$ is the characteristic timescale of the plasma, -and let $$\lambda_\mathrm{char} \gg \lambda_D$$ be its characteristic lengthscale: +and let $$\lambda_\mathrm{char} \gg \lambda_D$$ be its characteristic length scale: $$\begin{aligned} \nabla \cross \vb{E} = - \pdv{\vb{B}}{t} - \quad \implies \quad + \qquad \implies \qquad |\vb{E}| \sim \frac{\lambda_\mathrm{char}}{\tau_\mathrm{char}} |\vb{B}| \end{aligned}$$ -From this, we find when we can neglect -the last term in Ampère's law: -the characteristic velocity $$v_\mathrm{char}$$ -must be tiny compared to $$c$$, +From this, we find that we can neglect the last term in Ampère's law +as long as the characteristic velocity $$v_\mathrm{char}$$ is tiny compared to $$c$$, i.e. the plasma must be non-relativistic: $$\begin{aligned} @@ -254,7 +253,6 @@ $$\begin{aligned} \sim \frac{|\vb{E}| / \tau_\mathrm{char}}{|\vb{B}| c^2 / \lambda_\mathrm{char}} \sim \frac{|\vb{B}| \lambda_\mathrm{char}^2 / \tau_\mathrm{char}^2}{|\vb{B}| c^2} = \frac{v_\mathrm{char}^2}{c^2} - \ll 1 \end{aligned}$$ We thus have the following reduced form of Ampère's law, @@ -265,7 +263,7 @@ $$\begin{aligned} \nabla \cross \vb{B} = \mu_0 \vb{J} } - \qquad \quad + \qquad \qquad \boxed{ \nabla \cross \vb{E} = - \pdv{\vb{B}}{t} @@ -287,10 +285,12 @@ the [material derivative](/know/concept/material-derivative/) $$\mathrm{D} \rho / \mathrm{D} t$$ as follows: $$\begin{aligned} - \pdv{\rho}{t} + \nabla \cdot (\rho \vb{u}) - = \pdv{\rho}{t} + \rho \nabla \cdot \vb{u} + \vb{u} \cdot \nabla \rho - = \rho \nabla \cdot \vb{u} + \frac{\mathrm{D} \rho}{\mathrm{D} t} - = 0 + 0 + &= \pdv{\rho}{t} + \nabla \cdot (\rho \vb{u}) + \\ + &= \pdv{\rho}{t} + \rho \nabla \cdot \vb{u} + \vb{u} \cdot \nabla \rho + \\ + &= \rho \nabla \cdot \vb{u} + \frac{\mathrm{D} \rho}{\mathrm{D} t} \end{aligned}$$ Inserting this into the equation of state @@ -311,6 +311,7 @@ but we have merged $$n_i$$ and $$n_e$$ into $$\rho$$, and $$p_i$$ and $$p_i$$ into $$p$$. + ## Ohm's law variants It is worth discussing the generalized Ohm's law in more detail. @@ -321,29 +322,27 @@ $$\begin{aligned} = \eta \vb{J} \end{aligned}$$ -However, most authors neglect some of its terms: -this form is used for **Hall MHD**, -where $$\vb{J} \cross \vb{B}$$ is called the *Hall term*. -This term can be dropped in any of the following cases: +However, most authors neglect some terms: +the full form is used for **Hall MHD**, +where $$\vb{J} \cross \vb{B}$$ is called the **Hall term**. +It can be dropped in any of the following cases: -$$\begin{gathered} +$$\begin{aligned} 1 - \gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \vb{u} \cross \vb{B} \big|} + &\gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \vb{u} \cross \vb{B} \big|} \sim \frac{\rho v_\mathrm{char} / \tau_\mathrm{char}}{v_\mathrm{char} |\vb{B}| q_i n_i} \approx \frac{m_i n_i}{|\vb{B}| q_i n_i \tau_\mathrm{char}} = \frac{1}{\omega_{ci} \tau_\mathrm{char}} - \ll 1 \\ 1 - \gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \eta \vb{J} \big|} + &\gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \eta \vb{J} \big|} \sim \frac{|\vb{J}| |\vb{B}| q_e^2 n_e}{f_{ei} m_e |\vb{J}| q_e n_e} = \frac{|\vb{B}| q_e}{f_{ei} m_e} = \frac{\omega_{ce}}{f_{ei}} - \ll 1 -\end{gathered}$$ +\end{aligned}$$ Where we have used the MHD momentum equation with $$\nabla p \approx 0$$ -to obtain the scale estimate $$\vb{J} \cross \vb{B} \sim \rho v_\mathrm{char} / \tau_\mathrm{char}$$. +to obtain the scale estimate $$|\vb{J} \cross \vb{B}| \sim \rho v_\mathrm{char} / \tau_\mathrm{char}$$. In other words, if the ion gyration period is short $$\tau_\mathrm{char} \gg \omega_{ci}$$, and/or if the electron gyration period is long compared to the electron-ion collision period $$\omega_{ce} \ll f_{ei}$$, @@ -354,18 +353,17 @@ $$\begin{aligned} = \eta \vb{J} \end{aligned}$$ -Finally, we can neglect the resisitive term $$\eta \vb{J}$$ +Finally, we can neglect the resistive term $$\eta \vb{J}$$ if the Lorentz force is much larger. We formalize this condition as follows, -where we have used Ampère's law to find $$\vb{J} \sim \vb{B} / \mu_0 \lambda_\mathrm{char}$$: +where we have used Ampère's law to find $$|\vb{J}| \sim |\vb{B}| / \mu_0 \lambda_\mathrm{char}$$: $$\begin{aligned} 1 \ll \frac{\big| \vb{u} \cross \vb{B} \big|}{\big| \eta \vb{J} \big|} - \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta \vb{J}} + \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta |\vb{J}|} \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta |\vb{B}| / \mu_0 \lambda_\mathrm{char}} = \mathrm{R_m} - \gg 1 \end{aligned}$$ Where we have defined the **magnetic Reynolds number** $$\mathrm{R_m}$$ as follows, @@ -379,13 +377,15 @@ $$\begin{aligned} \end{aligned}$$ If $$\mathrm{R_m} \ll 1$$, the plasma is "electrically viscous", -such that resistivity needs to be accounted for, +meaning resistivity needs to be accounted for, whereas if $$\mathrm{R_m} \gg 1$$, the resistivity is negligible, in which case we have **ideal MHD**: $$\begin{aligned} - \vb{E} + \vb{u} \cross \vb{B} - = 0 + \boxed{ + \vb{E} + \vb{u} \cross \vb{B} + = 0 + } \end{aligned}$$ diff --git a/source/know/concept/material-derivative/index.md b/source/know/concept/material-derivative/index.md index 6bb83c5..4eb43e9 100644 --- a/source/know/concept/material-derivative/index.md +++ b/source/know/concept/material-derivative/index.md @@ -36,7 +36,7 @@ $$\begin{aligned} In effect, we have simply made the coordinate $$\va{r}$$ dependent on time, and have specifically chosen the time-dependence to track the parcel. -The net evolution of $$f$$ is then its "true" (i.e. non-partial) derivative with respect to $$t$$, +The evolution of $$f$$ is then its derivative with respect to $$t$$, allowing us to apply the chain rule: $$\begin{aligned} @@ -58,11 +58,7 @@ $$\begin{aligned} Note that $$\va{v} = \va{v}(\va{r}, t)$$, that is, the velocity can change with time ($$t$$-dependence), and depends on which parcel we track ($$\va{r}$$-dependence). - -Of course, the parcel is in our imagination: -$$\va{r}$$ does not really depend on $$t$$; -after all, we are dealing with a continuum. -Nevertheless, the right-hand side of the equation is very useful, +This result is very useful for fluid dynamics, and is known as the **material derivative** or **comoving derivative**: $$\begin{aligned} @@ -76,7 +72,7 @@ The first term is called the **local rate of change**, and the second is the **advective rate of change**. In effect, the latter moves the frame of reference along with the material, so that we can find the evolution of $$f$$ -without needing to worry about the continuum's motion. +without needing to explicitly account for the continuum's motion. That was for a scalar field $$f(\va{r}, t)$$, but in fact the definition also works for vector fields $$\va{U}(\va{r}, t)$$: diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md index 1214703..28885af 100644 --- a/source/know/concept/maxwell-bloch-equations/index.md +++ b/source/know/concept/maxwell-bloch-equations/index.md @@ -17,8 +17,8 @@ where $$\varepsilon_g$$ and $$\varepsilon_e$$ are the time-independent eigenener and the weights $$c_g$$ and $$c_g$$ are functions of $$t$$: $$\begin{aligned} - \ket{\Psi} - &= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar} + \ket{\Psi(t)} + &= c_g(t) \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e(t) \ket{e} e^{-i \varepsilon_e t / \hbar} \end{aligned}$$ This system is being perturbed by an electromagnetic wave @@ -32,8 +32,8 @@ $$\begin{aligned} Where the forward-propagating component $$\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^{-})^*$$. +and similarly $$\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}$$ as defined above, the pure [density operator](/know/concept/density-operator/) @@ -92,7 +92,7 @@ $$\begin{aligned} \end{aligned}$$ However, the light wave affects the electron, -so the actual electromagnetic dipole moment $$\vb{p}$$ is as follows, +so the true electromagnetic dipole moment $$\vb{p}$$ is as follows, using [Laporte's selection rule](/know/concept/selection-rules/) to remove diagonal terms by assuming that the electron's orbitals are spatially odd or even: @@ -106,9 +106,9 @@ $$\begin{aligned} \\ &= 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) + &= \vb{p}_0^{-} \rho_{ge} + \vb{p}_0^{+} \rho_{eg} \\ - &\equiv \vb{p}^{-}(t) + \vb{p}^{+}(t) + &\equiv \vb{p}^{-} + \vb{p}^{+} \end{aligned}$$ Where we have split $$\vb{p}$$ analogously to $$\vb{E}$$ @@ -117,8 +117,9 @@ Its equation of motion can then be found from the optical Bloch equations: $$\begin{aligned} \dv{\vb{p}^{+}}{t} - = \vb{p}_0^{+} \dv{\rho_{eg}}{t} - = - \vb{p}_0^{+} \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg} + &= \vb{p}_0^{+} \dv{\rho_{eg}}{t} + \\ + &= - \vb{p}_0^{+} \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg} + \frac{i}{\hbar} \vb{p}_0^{+} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \Big( \rho_{gg} - \rho_{ee} \Big) \end{aligned}$$ @@ -147,7 +148,8 @@ we find its equation of motion to be: $$\begin{aligned} \dv{d}{t} &= \dv{\rho_{ee}}{t} - \dv{\rho_{gg}}{t} - = 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} + \\ + &= 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} + \frac{i 2}{\hbar} \Big( \vb{p}^{-} \cdot \vb{E}^{+} - \vb{p}^{+} \cdot \vb{E}^{-} \Big) \end{aligned}$$ diff --git a/source/know/concept/rotating-wave-approximation/index.md b/source/know/concept/rotating-wave-approximation/index.md index edb13e9..54e0675 100644 --- a/source/know/concept/rotating-wave-approximation/index.md +++ b/source/know/concept/rotating-wave-approximation/index.md @@ -16,7 +16,7 @@ in the [electric dipole approximation](/know/concept/electric-dipole-approximati $$\begin{aligned} \hat{H}_1(t) - = \hat{V} \cos(\omega t) + \equiv \hat{V} \cos(\omega t) = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big) \end{aligned}$$ @@ -26,17 +26,17 @@ of the system that is getting perturbed by $$\hat{H}_1$$. As an example, consider a two-level system consisting of states $$\ket{g}$$ and $$\ket{e}$$, -with a resonance frequency $$\omega_0 = (E_e \!-\! E_g) / \hbar$$. +with a resonance frequency $$\omega_0 \equiv (E_e \!-\! E_g) / \hbar$$. From the [amplitude rate equations](/know/concept/amplitude-rate-equations/), we know that the general superposition state $$\ket{\Psi} = c_g \ket{g} + c_e \ket{e}$$ evolves as: $$\begin{aligned} i \hbar \dv{c_g}{t} - &= \matrixel{g}{\hat{H}_1(t)}{g} \: c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} \: c_e(t) \: e^{- i \omega_0 t} + &= \matrixel{g}{\hat{H}_1(t)}{g} c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} c_e(t) \: e^{- i \omega_0 t} \\ i \hbar \dv{c_e}{t} - &= \matrixel{e}{\hat{H}_1(t)}{g} \: c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} \: c_e(t) + &= \matrixel{e}{\hat{H}_1(t)}{g} c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} c_e(t) \end{aligned}$$ Typically, $$\hat{V}$$ has odd spatial parity, in which case @@ -66,15 +66,10 @@ $$\begin{aligned} At last, here we make the **rotating wave approximation**: since $$\omega$$ is assumed to be close to $$\omega_0$$, -we argue that $$\omega \!+\! \omega_0$$ is so much larger than $$\omega \!-\! \omega_0$$ -that those oscillations turn out negligible -if the system is observed over a reasonable time interval. - -Specifically, since both exponentials have the same weight, -the fast ($$\omega \!+\! \omega_0$$) oscillations -have a tiny amplitude compared to the slow ($$\omega \!-\! \omega_0$$) ones. -Furthermore, since they average out to zero over most realistic time intervals, -the fast terms can be dropped, leaving: +we argue that $$\omega \!+\! \omega_0$$ is much larger than $$\omega \!-\! \omega_0$$, +so that those oscillations average out to zero +when the system is observed over a realistic time interval. +Hence we drop those terms: $$\begin{aligned} \boxed{ @@ -103,13 +98,12 @@ $$\begin{aligned} This approximation's name is a bit confusing: the idea is that going from the Schrödinger to the [interaction picture](/know/concept/interaction-picture/) -has the effect of removing the exponentials of $$\omega_0$$ from the above equations, -i.e. multiplying them by $$e^{i \omega_0 t}$$ and $$e^{- i \omega_0 t}$$ +involves removing the exponentials of $$\omega_0$$ from the above equations, +i.e. they are multiplied by $$e^{i \omega_0 t}$$ and $$e^{- i \omega_0 t}$$ respectively, which can be regarded as a rotation. - -Relative to this rotation, when we split the wave $$\cos(\omega t)$$ -into two exponentials, one co-rotates, and the other counter-rotates. -We keep only the co-rotating waves, hence the name. +When we split the wave $$\cos(\omega t)$$ into two exponentials, +one co-rotates relative to this rotation, and the other counter-rotates. +We keep only the co-rotating terms, hence the name. The rotating wave approximation is usually used in the context of the two-level quantum system for light-matter interactions, diff --git a/source/know/concept/salt-equation/index.md b/source/know/concept/salt-equation/index.md index d7f8ef3..d47383f 100644 --- a/source/know/concept/salt-equation/index.md +++ b/source/know/concept/salt-equation/index.md @@ -275,10 +275,10 @@ so there are multiple active modes competing for charge carriers. Below threshold (i.e. before any mode is lasing), the problem is linear in $$\Psi_n$$, but above threshold it is nonlinear via $$h(\vb{x})$$. -Then the amplitude of $$\Psi_n$$ gets adjusted +Then the amplitude of $$\Psi_n$$ adjusts itself such that its respective $$k_n$$ never leaves the real axis. Once a mode is lasing, hole burning makes it harder for any other modes to activate, -since they modes must compete for the carrier supply $$D_0$$. +since they must compete for the carrier supply $$D_0$$. diff --git a/source/know/concept/simons-algorithm/index.md b/source/know/concept/simons-algorithm/index.md index 63bb808..6404ab0 100644 --- a/source/know/concept/simons-algorithm/index.md +++ b/source/know/concept/simons-algorithm/index.md @@ -16,7 +16,7 @@ the [Deutsch-Jozsa algorithm](/know/concept/deutsch-jozsa-algorithm/) and the [Bernstein-Vazirani algorithm](/know/concept/bernstein-vazirani-algorithm/), the problem it solves, known as **Simon's problem**, is of no practical use, -but nevertheless Simon's algorithm is an important landmark. +but nevertheless Simon's algorithm is an important milestone. Simon's problem is this: we are given a "black box" function $$f(x)$$ @@ -27,8 +27,9 @@ We are promised that there exists an $$s$$ such that for all $$x_1$$ and $$x_2$$ $$\begin{aligned} f(x_1) = f(x_2) - \quad \Leftrightarrow \quad - x_2 = s \oplus x_1 + \qquad \Leftrightarrow \qquad + x_2 + = s \oplus x_1 \end{aligned}$$ In other words, regardless of what $$f(x)$$ does behind the scenes, @@ -94,7 +95,8 @@ where $$x \cdot y$$ is the bitwise dot product: $$\begin{aligned} \frac{1}{\sqrt{2^n}} \sum_{x = 0}^{2^n - 1} \Ket{x} \Ket{f(x)} \quad \to \boxed{H^{\otimes n}} \to \quad - &\frac{1}{2^n} \sum_{x = 0}^{2^n - 1} \bigg( \sum_{y = 0}^{2^n - 1} (-1)^{x \cdot y} \Ket{y} \bigg) \Ket{f(x)} + &\frac{1}{\sqrt{2^n}} \sum_{x = 0}^{2^n - 1} + \bigg( \frac{1}{\sqrt{2^n}} \sum_{y = 0}^{2^n - 1} (-1)^{x \cdot y} \Ket{y} \bigg) \Ket{f(x)} \end{aligned}$$ Next, we measure all qubits. @@ -106,42 +108,47 @@ where $$f(x_1) = f(x_2)$$ and $$x_2 = s \oplus x_1$$: $$\begin{alignedat}{2} &\mathrm{if} \: s = 0: \qquad - &&\frac{1}{\sqrt{2^{n}}} \sum_{y = 0}^{2^n - 1} (-1)^{x_1 \cdot y} \Ket{y} \Ket{f(x_1)} + &&\bigg( \frac{1}{\sqrt{2^{n}}} \sum_{y = 0}^{2^n - 1} (-1)^{x_1 \cdot y} \Ket{y} \bigg) \Ket{f(x_1)} \\ &\mathrm{if} \: s \neq 0: \qquad - &&\frac{1}{\sqrt{2^{n+1}}} \sum_{y = 0}^{2^n - 1} \Big( (-1)^{x_1 \cdot y} + (-1)^{x_2 \cdot y} \Big) \Ket{y} \Ket{f(x_1)} + &&\bigg( \frac{1}{\sqrt{2^n}} \sum_{y = 0}^{2^n - 1} \frac{1}{\sqrt{2}} \Big( (-1)^{x_1 \cdot y} + (-1)^{x_2 \cdot y} \Big) \Ket{y} \bigg) \Ket{f(x_1)} \end{alignedat}$$ -If $$s = 0$$, we get an equiprobable superposition of all $$y$$. -So, when we measure the first $$n$$ qubits, the result is a uniformly random number, +If $$s = 0$$, we get an equal superposition of all $$y$$, +so when we measure the first $$n$$ qubits, +the result is a uniformly random number, regardless of the phase $$(-1)^{x_1 \cdot y}$$. -If $$s \neq 0$$, the situation is more interesting, +If $$s \neq 0$$, we get an "extra superposition", +since $$x_1 \neq x_2$$ but both are candidate inputs. +This is a more interesting situation, because we can only measure $$y$$-values where: $$\begin{aligned} - (-1)^{x_1 \cdot y} + (-1)^{x_2 \cdot y} \neq 0 + 0 + \neq (-1)^{x_1 \cdot y} + (-1)^{x_2 \cdot y} \end{aligned}$$ Since $$x_2 = s \oplus x_1$$ by definition, we can rewrite this as follows: $$\begin{aligned} - (-1)^{x_1 \cdot y} + (-1)^{x_1 \cdot y \oplus s \cdot y} + 0 + \neq (-1)^{x_1 \cdot y} + (-1)^{x_1 \cdot y \oplus s \cdot y} = (-1)^{x_1 \cdot y} + (-1)^{x_1 \cdot y} (-1)^{s \cdot y} - \neq 0 \end{aligned}$$ -Clearly, the expression can only be nonzero if $$s \cdot y$$ is even. +Clearly, this expression can only be nonzero if $$s \cdot y$$ is even. In other words, when we measure the first $$n$$ qubits, we get a random $$y$$-value, for which $$s \cdot y$$ is guaranteed to be even. In both cases $$s = 0$$ and $$s \neq 0$$, -we measure a $$y$$-value that satisfies the equation: +measuring the first $$n$$ qubits gives a $$y$$-value satisfying: $$\begin{aligned} - s \cdot y = 0 \:\:(\bmod 2) + s \cdot y + = 0 \:\bmod 2 \end{aligned}$$ This tells us something about $$s$$, albeit not much. @@ -150,13 +157,16 @@ we get various $$y$$-values $$y_1, ..., y_N$$, from which we can build a system of linear equations: $$\begin{aligned} - s \cdot y_1 &= 0 \:\:(\bmod 2) + s \cdot y_1 + &= 0 \:\bmod 2 \\ - s \cdot y_2 &= 0 \:\:(\bmod 2) + s \cdot y_2 + &= 0 \:\bmod 2 \\ &\:\:\vdots \\ - s \cdot y_N &= 0 \:\:(\bmod 2) + s \cdot y_N + &= 0 \:\bmod 2 \end{aligned}$$ This can be solved efficiently by a classical computer. diff --git a/source/know/concept/two-fluid-equations/index.md b/source/know/concept/two-fluid-equations/index.md index e224e3e..a00a2f9 100644 --- a/source/know/concept/two-fluid-equations/index.md +++ b/source/know/concept/two-fluid-equations/index.md @@ -98,15 +98,17 @@ leading to the following **continuity equations**: $$\begin{aligned} \boxed{ - \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i) - = 0 - \qquad \quad - \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) - = 0 + \begin{aligned} + 0 + &= \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i) + \\ + 0 + &= \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) + \end{aligned} } \end{aligned}$$ -These are 8 equations (2 scalar continuity, 2 vector momentum), +These are 8 equations (2 scalars for continuity, 2 vectors for momentum), but 16 unknowns $$\vb{u}_i$$, $$\vb{u}_e$$, $$\vb{E}$$, $$\vb{B}$$, $$n_i$$, $$n_e$$, $$p_i$$ and $$p_e$$. We would like to close this system, so we need 8 more. An obvious choice is [Maxwell's equations](/know/concept/maxwells-equations/), @@ -115,9 +117,13 @@ in particular Faraday's and Ampère's law $$\begin{aligned} \boxed{ - \nabla \cross \vb{E} = - \pdv{\vb{B}}{t} - \qquad \quad - \nabla \cross \vb{B} = \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big) + \begin{aligned} + \nabla \cross \vb{E} + &= - \pdv{\vb{B}}{t} + \\ + \nabla \cross \vb{B} + &= \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big) + \end{aligned} } \end{aligned}$$ @@ -129,7 +135,7 @@ it turns out that: $$\begin{aligned} \frac{\mathrm{D}}{\mathrm{D} t} \big( p V^\gamma \big) = 0 - \qquad \quad + \qquad \qquad \gamma \equiv \frac{C_P}{C_V} = \frac{N + 2}{N} @@ -146,7 +152,7 @@ for some constant $$C$$: $$\begin{aligned} \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p}{n^\gamma} \Big) = 0 - \quad \implies \quad + \qquad \implies \qquad p = C n^\gamma \end{aligned}$$ @@ -155,11 +161,13 @@ giving us a set of 16 equations for 16 unknowns: $$\begin{aligned} \boxed{ - \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big) - = 0 - \qquad \quad - \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big) - = 0 + \begin{aligned} + 0 + &= \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big) + \\ + 0 + &= \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big) + \end{aligned} } \end{aligned}$$ @@ -169,15 +177,16 @@ using simple differentiation and the ideal gas law: $$\begin{aligned} p = C n^\gamma - \quad \implies \quad + \qquad \implies \qquad \nabla p = \gamma \frac{C n^{\gamma}}{n} \nabla n = \gamma p \frac{\nabla n}{n} = \gamma k_B T \nabla n \end{aligned}$$ -Note that the ideal gas law was not used immediately, -to allow for $$\gamma \neq 1$$. +Note that we waited until now to use the ideal gas law, +in order to include the case $$\gamma \neq 1$$. + ## Fluid drifts -- cgit v1.2.3