diff options
16 files changed, 296 insertions, 266 deletions
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} |