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Diffstat (limited to 'source/know')
69 files changed, 3695 insertions, 1191 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/canonical-ensemble/index.md b/source/know/concept/canonical-ensemble/index.md index 8a96e91..da7d436 100644 --- a/source/know/concept/canonical-ensemble/index.md +++ b/source/know/concept/canonical-ensemble/index.md @@ -178,7 +178,7 @@ $$\begin{aligned} \end{aligned}$$ Rearranging and substituting -the [fundamental thermodynamic relation](/know/concept/fundamental-thermodynamic-relation/) +the [fundamental thermodynamic relation](/know/concept/fundamental-relation-of-thermodynamics/) then gives: $$\begin{aligned} diff --git a/source/know/concept/clausius-mossotti-relation/index.md b/source/know/concept/clausius-mossotti-relation/index.md index a0f4916..03bdcac 100644 --- a/source/know/concept/clausius-mossotti-relation/index.md +++ b/source/know/concept/clausius-mossotti-relation/index.md @@ -55,7 +55,8 @@ the dipole term will be dominant in that case, given by: $$\begin{aligned} V_i(\vb{r}) - \approx \frac{1}{4 \pi \varepsilon_0} \frac{1}{|\vb{r}|^2} \int \rho_i(\vb{r}') \: |\vb{r}'| \cos{\theta} \dd{\vb{r}'} + \approx \frac{1}{4 \pi \varepsilon_0} \frac{1}{|\vb{r}|^2} + \int_{-\infty}^\infty \rho_i(\vb{r}') \: |\vb{r}'| \cos{\theta} \dd{\vb{r}'} \end{aligned}$$ Where $$\theta$$ is the angle between $$\vb{r}$$ and $$\vb{r}'$$, @@ -64,7 +65,8 @@ with the unit vector $$\vu{r}$$, normalized from $$\vb{r}$$: $$\begin{aligned} V_i(\vb{r}) - = \frac{1}{4 \pi \varepsilon_0} \frac{1}{|\vb{r}|^2} \: \vu{r} \cdot \!\!\int \vb{r}' \rho_i(\vb{r}') \dd{\vb{r}'} + = \frac{1}{4 \pi \varepsilon_0} \frac{1}{|\vb{r}|^2} + \: \vu{r} \cdot \!\!\int_{-\infty}^\infty \vb{r}' \rho_i(\vb{r}') \dd{\vb{r}'} \end{aligned}$$ The integral is a more general definition of the dipole moment $$\vb{p}_i$$. diff --git a/source/know/concept/convolution-theorem/index.md b/source/know/concept/convolution-theorem/index.md index 3f9eafb..8462fcc 100644 --- a/source/know/concept/convolution-theorem/index.md +++ b/source/know/concept/convolution-theorem/index.md @@ -24,10 +24,10 @@ and $$A$$ and $$B$$ are the constants from its definition: $$\begin{aligned} \boxed{ \begin{aligned} - A \cdot (f * g)(x) + A \: (f * g)(x) &= \hat{\mathcal{F}}{}^{-1}\Big\{ \tilde{f}(k) \: \tilde{g}(k) \Big\} \\ - B \cdot (\tilde{f} * \tilde{g})(k) + B \: (\tilde{f} * \tilde{g})(k) &= \hat{\mathcal{F}}\Big\{ f(x) \: g(x) \Big\} \end{aligned} } @@ -45,7 +45,7 @@ $$\begin{aligned} \\ &= A \int_{-\infty}^\infty g(x') \: f(x - x') \dd{x'} \\ - &= A \cdot (f * g)(x) + &= A \: (f * g)(x) \end{aligned}$$ Then we do the same again, @@ -59,7 +59,7 @@ $$\begin{aligned} \\ &= B \int_{-\infty}^\infty \tilde{g}(k') \: \tilde{f}(k - k') \dd{k'} \\ - &= B \cdot (\tilde{f} * \tilde{g})(k) + &= B \: (\tilde{f} * \tilde{g})(k) \end{aligned}$$ {% include proof/end.html id="proof-fourier" %} 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: $$\begin{aligned} - \ket{f} = f(x) - \quad \implies \quad + \ket{f} + = f(x) + \qquad \implies \qquad \bra{f} = F[u(x)] = \int_a^b f^*(x) \: u(x) \dd{x} diff --git a/source/know/concept/electromagnetic-wave-equation/index.md b/source/know/concept/electromagnetic-wave-equation/index.md index a27fe6f..559d943 100644 --- a/source/know/concept/electromagnetic-wave-equation/index.md +++ b/source/know/concept/electromagnetic-wave-equation/index.md @@ -1,7 +1,7 @@ --- title: "Electromagnetic wave equation" sort_title: "Electromagnetic wave equation" -date: 2021-09-09 +date: 2024-09-08 # Originally 2021-09-09, major rewrite categories: - Physics - Electromagnetism @@ -9,236 +9,281 @@ categories: layout: "concept" --- -The electromagnetic wave equation describes -the propagation of light through various media. -Since an electromagnetic (light) wave consists of +Light, i.e. **electromagnetic waves**, consist of an [electric field](/know/concept/electric-field/) and a [magnetic field](/know/concept/magnetic-field/), -we need [Maxwell's equations](/know/concept/maxwells-equations/) -in order to derive the wave equation. +one inducing the other and vice versa. +The existence and classical behavior of such waves +can be derived using only [Maxwell's equations](/know/concept/maxwells-equations/), +as we will demonstrate here. - -## Uniform medium - -We will use all of Maxwell's equations, -but we start with Ampère's circuital law for the "free" fields $$\vb{H}$$ and $$\vb{D}$$, -in the absence of a free current $$\vb{J}_\mathrm{free} = 0$$: - -$$\begin{aligned} - \nabla \cross \vb{H} - = \pdv{\vb{D}}{t} -\end{aligned}$$ - -We assume that the medium is isotropic, linear, -and uniform in all of space, such that: +We start from Faraday's law of induction, +where we assume that the system consists of materials +with well-known (linear) relative magnetic permeabilities $$\mu_r(\vb{r})$$, +such that $$\vb{B} = \mu_0 \mu_r \vb{H}$$: $$\begin{aligned} - \vb{D} = \varepsilon_0 \varepsilon_r \vb{E} - \qquad \quad - \vb{H} = \frac{1}{\mu_0 \mu_r} \vb{B} + \nabla \cross \vb{E} + = - \pdv{\vb{B}}{t} + = - \mu_0 \mu_r \pdv{\vb{H}}{t} \end{aligned}$$ -Which, upon insertion into Ampère's law, -yields an equation relating $$\vb{B}$$ and $$\vb{E}$$. -This may seem to contradict Ampère's "total" law, -but keep in mind that $$\vb{J}_\mathrm{bound} \neq 0$$ here: +We move $$\mu_r(\vb{r})$$ to the other side, +take the curl, and insert Ampère's circuital law: $$\begin{aligned} - \nabla \cross \vb{B} - = \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{\vb{E}}{t} + \nabla \cross \bigg( \frac{1}{\mu_r} \nabla \cross \vb{E} \bigg) + &= - \mu_0 \pdv{}{t} \big( \nabla \cross \vb{H} \big) + \\ + &= - \mu_0 \bigg( \pdv{\vb{J}_\mathrm{free}}{t} + \pdvn{2}{\vb{D}}{t} \bigg) \end{aligned}$$ -Now we take the curl, rearrange, -and substitute $$\nabla \cross \vb{E}$$ according to Faraday's law: +For simplicity, we only consider insulating materials, +since light propagation in conductors is a complex beast. +We thus assume that there are no free currents $$\vb{J}_\mathrm{free} = 0$$, leaving: $$\begin{aligned} - \nabla \cross (\nabla \cross \vb{B}) - = \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{}{t}(\nabla \cross \vb{E}) - = - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{B}}{t} + \nabla \cross \bigg( \frac{1}{\mu_r} \nabla \cross \vb{E} \bigg) + &= - \mu_0 \pdvn{2}{\vb{D}}{t} \end{aligned}$$ -Using a vector identity, we rewrite the leftmost expression, -which can then be reduced thanks to Gauss' law for magnetism $$\nabla \cdot \vb{B} = 0$$: +Having $$\vb{E}$$ and $$\vb{D}$$ in the same equation is not ideal, +so we should make a choice: +do we restrict ourselves to linear media +(so $$\vb{D} = \varepsilon_0 \varepsilon_r \vb{E}$$), +or do we allow materials with more complicated responses +(so $$\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$$, with $$\vb{P}$$ unspecified)? +The former is usually sufficient: $$\begin{aligned} - - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{B}}{t} - &= \nabla (\nabla \cdot \vb{B}) - \nabla^2 \vb{B} - = - \nabla^2 \vb{B} + \boxed{ + \nabla \cross \bigg( \frac{1}{\mu_r} \nabla \cross \vb{E} \bigg) + = - \mu_0 \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{E}}{t} + } \end{aligned}$$ -This describes $$\vb{B}$$. -Next, we repeat the process for $$\vb{E}$$: -taking the curl of Faraday's law yields: +This is the general linear form of the **electromagnetic wave equation**, +where $$\mu_r$$ and $$\varepsilon_r$$ +both depend on $$\vb{r}$$ in order to describe the structure of the system. +We can obtain a similar equation for $$\vb{H}$$, +by starting from Ampère's law under the same assumptions: $$\begin{aligned} - \nabla \cross (\nabla \cross \vb{E}) - = - \pdv{}{t}(\nabla \cross \vb{B}) - = - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{E}}{t} + \nabla \cross \vb{H} + = \pdv{\vb{D}}{t} + = \varepsilon_0 \varepsilon_r \pdv{\vb{E}}{t} \end{aligned}$$ -Which can be rewritten using same vector identity as before, -and then reduced by assuming that there is no net charge density $$\rho = 0$$ -in Gauss' law, such that $$\nabla \cdot \vb{E} = 0$$: +Taking the curl and substituting Faraday's law on the right yields: $$\begin{aligned} - - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{E}}{t} - &= \nabla (\nabla \cdot \vb{E}) - \nabla^2 \vb{E} - = - \nabla^2 \vb{E} + \nabla \cross \bigg( \frac{1}{\varepsilon_r} \nabla \cross \vb{H} \bigg) + &= \varepsilon_0 \pdv{}{t} \big( \nabla \cross \vb{E} \big) + = - \varepsilon_0 \pdvn{2}{\vb{B}}{t} \end{aligned}$$ -We thus arrive at the following two (implicitly coupled) -wave equations for $$\vb{E}$$ and $$\vb{B}$$, -where we have defined the phase velocity $$v \equiv 1 / \sqrt{\mu_0 \mu_r \varepsilon_0 \varepsilon_r}$$: +And then we insert $$\vb{B} = \mu_0 \mu_r \vb{H}$$ to get the analogous +electromagnetic wave equation for $$\vb{H}$$: $$\begin{aligned} \boxed{ - \pdvn{2}{\vb{E}}{t} - \frac{1}{v^2} \nabla^2 \vb{E} - = 0 - } - \qquad \quad - \boxed{ - \pdvn{2}{\vb{B}}{t} - \frac{1}{v^2} \nabla^2 \vb{B} - = 0 + \nabla \cross \bigg( \frac{1}{\varepsilon_r} \nabla \cross \vb{H} \bigg) + = - \mu_0 \varepsilon_0 \mu_r \pdvn{2}{\vb{H}}{t} } \end{aligned}$$ -Traditionally, it is said that the solutions are as follows, -where the wavenumber $$|\vb{k}| = \omega / v$$: - -$$\begin{aligned} - \vb{E}(\vb{r}, t) - &= \vb{E}_0 \exp(i \vb{k} \cdot \vb{r} - i \omega t) - \\ - \vb{B}(\vb{r}, t) - &= \vb{B}_0 \exp(i \vb{k} \cdot \vb{r} - i \omega t) -\end{aligned}$$ - -In fact, thanks to linearity, these **plane waves** can be treated as -terms in a Fourier series, meaning that virtually -*any* function $$f(\vb{k} \cdot \vb{r} - \omega t)$$ is a valid solution. +This is equivalent to the problem for $$\vb{E}$$, +since they are coupled by Maxwell's equations. +By solving either, subject to Gauss's laws +$$\nabla \cdot (\varepsilon_r \vb{E}) = 0$$ and $$\nabla \cdot (\mu_r \vb{H}) = 0$$, +the behavior of light in a given system can be deduced. +Note that Gauss's laws enforce that the wave's fields are transverse, +i.e. they must be perpendicular to the propagation direction. -Keep in mind that in reality $$\vb{E}$$ and $$\vb{B}$$ are real, -so although it is mathematically convenient to use plane waves, -in the end you will need to take the real part. -## Non-uniform medium +## Homogeneous linear media -A useful generalization is to allow spatial change -in the relative permittivity $$\varepsilon_r(\vb{r})$$ -and the relative permeability $$\mu_r(\vb{r})$$. -We still assume that the medium is linear and isotropic, so: +In the special case where the medium is completely uniform, +$$\mu_r$$ and $$\varepsilon_r$$ no longer depend on $$\vb{r}$$, +so they can be moved to the other side: $$\begin{aligned} - \vb{D} - = \varepsilon_0 \varepsilon_r(\vb{r}) \vb{E} - \qquad \quad - \vb{B} - = \mu_0 \mu_r(\vb{r}) \vb{H} + \nabla \cross \big( \nabla \cross \vb{E} \big) + &= - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{E}}{t} + \\ + \nabla \cross \big( \nabla \cross \vb{H} \big) + &= - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{H}}{t} \end{aligned}$$ -Inserting these expressions into Faraday's and Ampère's laws -respectively yields: +This can be rewritten using the vector identity +$$\nabla \cross (\nabla \cross \vb{V}) = \nabla (\nabla \cdot \vb{V}) - \nabla^2 \vb{V}$$: $$\begin{aligned} - \nabla \cross \vb{E} - = - \mu_0 \mu_r(\vb{r}) \pdv{\vb{H}}{t} - \qquad \quad - \nabla \cross \vb{H} - = \varepsilon_0 \varepsilon_r(\vb{r}) \pdv{\vb{E}}{t} + \nabla (\nabla \cdot \vb{E}) - \nabla^2 \vb{E} + &= - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{E}}{t} + \\ + \nabla (\nabla \cdot \vb{H}) - \nabla^2 \vb{H} + &= - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{H}}{t} \end{aligned}$$ -We then divide Ampère's law by $$\varepsilon_r(\vb{r})$$, -take the curl, and substitute Faraday's law, giving: +Which can be reduced using Gauss's laws +$$\nabla \cdot \vb{E} = 0$$ and $$\nabla \cdot \vb{H} = 0$$ +thanks to the fact that $$\varepsilon_r$$ and $$\mu_r$$ are constants in this case. +We therefore arrive at: $$\begin{aligned} - \nabla \cross \Big( \frac{1}{\varepsilon_r} \nabla \cross \vb{H} \Big) - = \varepsilon_0 \pdv{}{t}(\nabla \cross \vb{E}) - = - \mu_0 \mu_r \varepsilon_0 \pdvn{2}{\vb{H}}{t} + \boxed{ + \nabla^2 \vb{E} - \frac{n^2}{c^2} \pdvn{2}{\vb{E}}{t} + = 0 + } \end{aligned}$$ -Next, we exploit linearity by decomposing $$\vb{H}$$ and $$\vb{E}$$ -into Fourier series, with terms given by: - $$\begin{aligned} - \vb{H}(\vb{r}, t) - = \vb{H}(\vb{r}) \exp(- i \omega t) - \qquad \quad - \vb{E}(\vb{r}, t) - = \vb{E}(\vb{r}) \exp(- i \omega t) + \boxed{ + \nabla^2 \vb{H} - \frac{n^2}{c^2} \pdvn{2}{\vb{H}}{t} + = 0 + } \end{aligned}$$ -By inserting this ansatz into the equation, -we can remove the explicit time dependence: +Where $$c = 1 / \sqrt{\mu_0 \varepsilon_0}$$ is the speed of light in a vacuum, +and $$n = \sqrt{\mu_0 \varepsilon_0}$$ is the refractive index of the medium. +Note that most authors write the magnetic equation with $$\vb{B}$$ instead of $$\vb{H}$$; +both are correct thanks to linearity. + +In a vacuum, where $$n = 1$$, these equations are sometimes written as +$$\square \vb{E} = 0$$ and $$\square \vb{H} = 0$$, +where $$\square$$ is the **d'Alembert operator**, defined as follows: $$\begin{aligned} - \nabla \cross \Big( \frac{1}{\varepsilon_r} \nabla \cross \vb{H} \Big) \exp(- i \omega t) - = \mu_0 \varepsilon_0 \omega^2 \mu_r \vb{H} \exp(- i \omega t) + \boxed{ + \square + \equiv \nabla^2 - \frac{1}{c^2} \pdvn{2}{}{t} + } \end{aligned}$$ -Dividing out $$\exp(- i \omega t)$$, -we arrive at an eigenvalue problem for $$\omega^2$$, -with $$c = 1 / \sqrt{\mu_0 \varepsilon_0}$$: +Note that some authors define it with the opposite sign. +In any case, the d'Alembert operator is important for special relativity. + +The solution to the homogeneous electromagnetic wave equation +are traditionally said to be the so-called **plane waves** given by: $$\begin{aligned} - \boxed{ - \nabla \cross \Big( \frac{1}{\varepsilon_r(\vb{r})} \nabla \cross \vb{H |