From 6e70f28ccbd5afc1506f71f013278a9d157ef03a Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 27 Oct 2022 20:40:09 +0200 Subject: Optimize last images, add proof template, improve CSS --- source/know/concept/binomial-distribution/index.md | 33 ++++-------- source/know/concept/boltzmann-equation/index.md | 31 ++++------- source/know/concept/convolution-theorem/index.md | 23 ++++---- .../know/concept/curvilinear-coordinates/index.md | 35 +++++------- source/know/concept/detailed-balance/index.md | 10 ++-- source/know/concept/dirac-delta-function/index.md | 20 +++---- source/know/concept/dynkins-formula/index.md | 23 +++----- .../concept/equation-of-motion-theory/index.md | 10 ++-- source/know/concept/euler-bernoulli-law/index.md | 11 ++-- source/know/concept/fourier-transform/index.md | 23 +++----- source/know/concept/fundamental-solution/index.md | 22 +++----- source/know/concept/greens-functions/index.md | 22 +++----- .../concept/gronwall-bellman-inequality/index.md | 33 ++++-------- source/know/concept/guiding-center-theory/index.md | 22 +++----- source/know/concept/hamiltonian-mechanics/index.md | 33 +++++------- .../know/concept/heaviside-step-function/index.md | 10 ++-- source/know/concept/holomorphic-function/index.md | 32 ++++------- source/know/concept/impulse-response/index.md | 10 ++-- source/know/concept/ito-integral/index.md | 24 ++++----- source/know/concept/ito-process/index.md | 34 +++++------- source/know/concept/laplace-transform/index.md | 21 +++----- source/know/concept/lindhard-function/index.md | 23 ++++---- .../concept/matsubara-greens-function/index.md | 41 +++++--------- .../know/concept/maxwell-bloch-equations/index.md | 10 ++-- .../know/concept/multi-photon-absorption/index.md | 13 +++-- source/know/concept/parsevals-theorem/index.md | 10 ++-- source/know/concept/repetition-code/index.md | 63 +++------------------- source/know/concept/residue-theorem/index.md | 10 ++-- source/know/concept/selection-rules/index.md | 46 +++++++--------- source/know/concept/superdense-coding/index.md | 63 +++------------------- 30 files changed, 234 insertions(+), 527 deletions(-) (limited to 'source/know') diff --git a/source/know/concept/binomial-distribution/index.md b/source/know/concept/binomial-distribution/index.md index 1193a93..dc75221 100644 --- a/source/know/concept/binomial-distribution/index.md +++ b/source/know/concept/binomial-distribution/index.md @@ -44,11 +44,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-mean" %} + Meanwhile, we find the following variance $$\sigma^2$$, with $$\sigma$$ being the standard deviation: @@ -74,12 +71,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-var" %} + As $$N \to \infty$$, the binomial distribution turns into the continuous normal distribution, @@ -119,11 +112,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-normal" %} ## References diff --git a/source/know/concept/boltzmann-equation/index.md b/source/know/concept/boltzmann-equation/index.md index 9ed2fd2..d2631b2 100644 --- a/source/know/concept/boltzmann-equation/index.md +++ b/source/know/concept/boltzmann-equation/index.md @@ -145,11 +145,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
If we instead choose the momentum $$Q = m \vb{v}$$, we find that the **first moment** of the BTE describes conservation of momentum, @@ -174,11 +170,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
Finally, if we choose the kinetic energy $$Q = m |\vb{v}|^2 / 2$$, we find that the **second moment** gives conservation of energy, @@ -237,11 +229,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-moment2" %} diff --git a/source/know/concept/convolution-theorem/index.md b/source/know/concept/convolution-theorem/index.md index 742c8ff..510417a 100644 --- a/source/know/concept/convolution-theorem/index.md +++ b/source/know/concept/convolution-theorem/index.md @@ -12,6 +12,8 @@ is equal to a product in the frequency domain. This is especially useful for computation, replacing an $$\mathcal{O}(n^2)$$ convolution with an $$\mathcal{O}(n \log(n))$$ transform and product. + + ## Fourier transform The convolution theorem is usually expressed as follows, where @@ -27,11 +29,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-fourier" %} + ## Laplace transform @@ -79,11 +78,8 @@ $$\begin{aligned} \boxed{\hat{\mathcal{L}}\{(f * g)(t)\} = \tilde{f}(s) \: \tilde{g}(s)} \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-laplace" %} diff --git a/source/know/concept/curvilinear-coordinates/index.md b/source/know/concept/curvilinear-coordinates/index.md index cb22e43..48a5a72 100644 --- a/source/know/concept/curvilinear-coordinates/index.md +++ b/source/know/concept/curvilinear-coordinates/index.md @@ -48,6 +48,7 @@ we derive general formulae to convert expressions from Cartesian coordinates to the new orthogonal system $$(x_1, x_2, x_3)$$. + ## Basis vectors Consider the the vector form of the line element $$\dd{\ell}$$, @@ -86,6 +87,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Gradient In an orthogonal coordinate system, @@ -102,11 +104,8 @@ $$\begin{gathered} } \end{gathered}$$ -
- - - -
## Divergence @@ -145,11 +143,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-div" %} + ## Laplacian @@ -246,6 +241,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Curl The curl of a vector $$\vb{V}$$ is as follows @@ -264,11 +260,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-curl" %} + ## Differential elements diff --git a/source/know/concept/detailed-balance/index.md b/source/know/concept/detailed-balance/index.md index b89d5da..98f9bd3 100644 --- a/source/know/concept/detailed-balance/index.md +++ b/source/know/concept/detailed-balance/index.md @@ -103,11 +103,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-reversibility" %} diff --git a/source/know/concept/dirac-delta-function/index.md b/source/know/concept/dirac-delta-function/index.md index 518eba1..0185b78 100644 --- a/source/know/concept/dirac-delta-function/index.md +++ b/source/know/concept/dirac-delta-function/index.md @@ -65,11 +65,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
An even more impressive property is the behaviour of the derivative of $$\delta(x)$$: @@ -89,11 +85,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
This property also generalizes nicely for the higher-order derivatives: diff --git a/source/know/concept/dynkins-formula/index.md b/source/know/concept/dynkins-formula/index.md index c0d20c5..307f098 100644 --- a/source/know/concept/dynkins-formula/index.md +++ b/source/know/concept/dynkins-formula/index.md @@ -39,11 +39,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
The general definition of resembles that of a classical derivative, and indeed, the generator $$\hat{A}$$ can be thought of as a differential operator. @@ -104,11 +100,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+Isolating this equation for $$\mathbf{E}[Y_\tau \!\mid\! X_0]$$ then gives Dynkin's formula. +{% include proof/end.html id="proof-dynkin" %} + A common application of Dynkin's formula is predicting when the stopping time $$\tau$$ occurs, and in what state $$X_\tau$$ this happens. diff --git a/source/know/concept/equation-of-motion-theory/index.md b/source/know/concept/equation-of-motion-theory/index.md index 02ed856..c1ed8da 100644 --- a/source/know/concept/equation-of-motion-theory/index.md +++ b/source/know/concept/equation-of-motion-theory/index.md @@ -63,11 +63,8 @@ $$\begin{aligned} = - \sum_{\nu''} u_{\nu \nu''} \hat{c}_{\nu''} \end{aligned}$$ -
- - - -
Substituting this into $$G_{\nu \nu'}^R$$'s equation of motion, we recognize another Green's function $$G_{\nu'' \nu'}^R$$: diff --git a/source/know/concept/euler-bernoulli-law/index.md b/source/know/concept/euler-bernoulli-law/index.md index dad67ca..5a6c38d 100644 --- a/source/know/concept/euler-bernoulli-law/index.md +++ b/source/know/concept/euler-bernoulli-law/index.md @@ -81,11 +81,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-field" %} + In any case, the beam experiences a bending torque with an $$x$$-component $$T_x$$ given by: diff --git a/source/know/concept/fourier-transform/index.md b/source/know/concept/fourier-transform/index.md index 0bc849b..c86d997 100644 --- a/source/know/concept/fourier-transform/index.md +++ b/source/know/concept/fourier-transform/index.md @@ -67,6 +67,7 @@ on whether the analysis is for forward ($$s > 0$$) or backward-propagating ($$s < 0$$) waves. + ## Derivatives The FT of a derivative has a very useful property. @@ -113,6 +114,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Multiple dimensions The Fourier transform is straightforward to generalize to $$N$$ dimensions. @@ -150,11 +152,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
Differentiation is more complicated for $$N > 1$$, but the FT is still useful, @@ -197,11 +195,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-laplacian" %} diff --git a/source/know/concept/fundamental-solution/index.md b/source/know/concept/fundamental-solution/index.md index 312cc2e..947aada 100644 --- a/source/know/concept/fundamental-solution/index.md +++ b/source/know/concept/fundamental-solution/index.md @@ -42,11 +42,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-solution" %} + While the impulse response is typically used for initial value problems, the fundamental solution $$G$$ is used for boundary value problems. @@ -117,11 +114,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-reciprocity" %} diff --git a/source/know/concept/greens-functions/index.md b/source/know/concept/greens-functions/index.md index ddba2cd..eda5671 100644 --- a/source/know/concept/greens-functions/index.md +++ b/source/know/concept/greens-functions/index.md @@ -21,6 +21,7 @@ but in general they are not the same, except in a special case, see below. + ## Single-particle functions If the two operators are single-particle creation/annihilation operators, @@ -146,11 +147,8 @@ $$\begin{gathered} G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t') \end{gathered}$$ -
- - - -
+{% include proof/end.html id="proof-time-delta" %} If the Hamiltonian is both time-independent and non-interacting, then the time-dependence of $$\hat{c}_\nu$$ @@ -214,6 +211,7 @@ $$\begin{aligned} \end{aligned}$$ + ## As fundamental solutions In the absence of interactions, @@ -237,11 +235,8 @@ $$\begin{aligned} = \frac{\hbar^2}{2 m} \nabla^2 \hat{\Psi}(\vb{r}) \end{aligned}$$ -
- - - -
After substituting this into the equation of motion, we recognize $$G^R(\vb{r}, t; \vb{r}', t')$$ itself: diff --git a/source/know/concept/gronwall-bellman-inequality/index.md b/source/know/concept/gronwall-bellman-inequality/index.md index 8096aaf..da1bcad 100644 --- a/source/know/concept/gronwall-bellman-inequality/index.md +++ b/source/know/concept/gronwall-bellman-inequality/index.md @@ -26,11 +26,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-original" %} + Grönwall's inequality can be generalized to non-differentiable functions. Suppose we know: @@ -84,11 +81,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-integral" %} + In the special case where $$\alpha(t)$$ is non-decreasing with $$t$$, the inequality reduces to: @@ -151,11 +145,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-special" %} diff --git a/source/know/concept/guiding-center-theory/index.md b/source/know/concept/guiding-center-theory/index.md index 5368966..412c88b 100644 --- a/source/know/concept/guiding-center-theory/index.md +++ b/source/know/concept/guiding-center-theory/index.md @@ -72,6 +72,7 @@ we can use this average to approximately remove the finer dynamics, and focus only on the guiding center. + ## Uniform electric and magnetic field Consider the case where $$\vb{E}$$ and $$\vb{B}$$ are both uniform, @@ -149,6 +150,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Non-uniform magnetic field Next, consider a more general case, where $$\vb{B}$$ is non-uniform, @@ -193,11 +195,8 @@ $$\begin{aligned} \approx - \frac{u_L^2}{2 \omega_c} \nabla B \end{aligned}$$ -
- - - -
With this, the guiding center's equation of motion is reduced to the following: @@ -332,11 +330,8 @@ $$\begin{aligned} \approx - u_{gc\parallel} \frac{\vb{R}_c}{R_c^2} \end{aligned}$$ -
- - - -
With this, we arrive at the following equation of motion for the guiding center: diff --git a/source/know/concept/hamiltonian-mechanics/index.md b/source/know/concept/hamiltonian-mechanics/index.md index 19e55b0..03ff2dd 100644 --- a/source/know/concept/hamiltonian-mechanics/index.md +++ b/source/know/concept/hamiltonian-mechanics/index.md @@ -15,6 +15,7 @@ It is built on the shoulders of [Lagrangian mechanics](/know/concept/lagrangian- which is in turn built on [variational calculus](/know/concept/calculus-of-variations/). + ## Definitions In Lagrangian mechanics, use a Lagrangian $$L$$, @@ -90,6 +91,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Canonical equations Lagrangian mechanics has a single Euler-Lagrange equation per object, @@ -105,11 +107,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
Just like in Lagrangian mechanics, if $$H$$ does not explicitly contain $$q_n$$, then $$q_n$$ is called a **cyclic coordinate**, and leads to the conservation of $$p_n$$: @@ -175,11 +173,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
Assuming that $$H$$ does not explicitly depend on $$t$$, the above property naturally leads us to an alternative @@ -247,11 +241,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-transformation" %} + If you have experience with quantum mechanics, the latter equation should look suspiciously similar diff --git a/source/know/concept/heaviside-step-function/index.md b/source/know/concept/heaviside-step-function/index.md index 15d1729..9f5d4ec 100644 --- a/source/know/concept/heaviside-step-function/index.md +++ b/source/know/concept/heaviside-step-function/index.md @@ -57,11 +57,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
The use of $$\pv{}$$ without an integral is an abuse of notation, and means that this result only makes sense when wrapped in an integral. diff --git a/source/know/concept/holomorphic-function/index.md b/source/know/concept/holomorphic-function/index.md index 5dde240..cf252c0 100644 --- a/source/know/concept/holomorphic-function/index.md +++ b/source/know/concept/holomorphic-function/index.md @@ -61,6 +61,7 @@ and imaginary parts satisfy these equations. This gives an idea of how strict the criteria are to qualify as holomorphic. + ## Integration formulas Holomorphic functions satisfy **Cauchy's integral theorem**, which states @@ -73,11 +74,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-int-theorem" %} + An interesting consequence is **Cauchy's integral formula**, which states that the value of $$f(z)$$ at an arbitrary point $$z_0$$ is @@ -110,11 +108,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
Similarly, **Cauchy's differentiation formula**, or **Cauchy's integral formula for derivatives** @@ -149,11 +143,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-dv-formula" %} diff --git a/source/know/concept/impulse-response/index.md b/source/know/concept/impulse-response/index.md index 397ac2d..661ed3f 100644 --- a/source/know/concept/impulse-response/index.md +++ b/source/know/concept/impulse-response/index.md @@ -30,11 +30,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
This is useful for solving initial value problems, because any initial condition can be satisfied diff --git a/source/know/concept/ito-integral/index.md b/source/know/concept/ito-integral/index.md index f087f97..4a725e1 100644 --- a/source/know/concept/ito-integral/index.md +++ b/source/know/concept/ito-integral/index.md @@ -29,6 +29,7 @@ and $$\mathbf{E}[G_t^2]$$ is integrable for $$t \in [a, b]$$. If $$I_t$$ exists, $$G_t$$ is said to be **Itō-integrable** with respect to $$B_t$$. + ## Motivation Consider the following simple first-order differential equation for $$X_t$$, @@ -99,7 +100,8 @@ $$\begin{aligned} \end{aligned}$$ For more information about applying the Itō integral in this way, -see the [Itō calculus](/know/concept/ito-calculus/). +see the [Itō calculus](/know/concept/ito-process/). + ## Properties @@ -138,11 +140,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
Furthermore, Itō integrals are [martingales](/know/concept/martingale/), meaning that the average noise contribution is zero, which makes intuitive sense, since true white noise cannot be biased. -
- - - -
+{% include proof/end.html id="proof-martingale" %} diff --git a/source/know/concept/ito-process/index.md b/source/know/concept/ito-process/index.md index f192e28..2756e33 100644 --- a/source/know/concept/ito-process/index.md +++ b/source/know/concept/ito-process/index.md @@ -61,6 +61,7 @@ since only the current value of $$X_t$$ determines the future, and $$B_t$$ is also a Markov process. + ## Itō's lemma Classically, given $$y \equiv h(x(t), t)$$, @@ -83,11 +84,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-lemma" %} + The most important application of Itō's lemma is to perform coordinate transformations, to make the solution of a given Itō SDE easier. + ## Coordinate transformations The simplest coordinate transformation is a scaling of the time axis. @@ -208,6 +207,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Existence and uniqueness It is worth knowing under what condition a solution to a given SDE exists, @@ -232,11 +232,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
If a solution exists, it is also worth knowing whether it is unique. Suppose that $$f$$ and $$g$$ satisfy the following inequalities, @@ -301,11 +297,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
Using these properties, it can then be shown that if all of the above conditions are satisfied, diff --git a/source/know/concept/laplace-transform/index.md b/source/know/concept/laplace-transform/index.md index c7f352a..94c3742 100644 --- a/source/know/concept/laplace-transform/index.md +++ b/source/know/concept/laplace-transform/index.md @@ -35,6 +35,7 @@ using [partial fraction decomposition](/know/concept/partial-fraction-decomposit and then looking up the individual terms. + ## Derivatives The derivative of a transformed function is the transform @@ -55,11 +56,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
The Laplace transform of a derivative introduces the initial conditions into the result. Notice that $$f(0)$$ is the initial value in the original $$t$$-domain: @@ -98,11 +95,8 @@ and $$f^{(0)}(t) = f(t)$$. As an example, $$\hat{\mathcal{L}}\{f'''(t)\}$$ becomes $$- f''(0) - s f'(0) - s^2 f(0) + s^3 \tilde{f}(s)$$. -
- - - -
+{% include proof/end.html id="proof-dv-t" %} diff --git a/source/know/concept/lindhard-function/index.md b/source/know/concept/lindhard-function/index.md index 4033148..fd620df 100644 --- a/source/know/concept/lindhard-function/index.md +++ b/source/know/concept/lindhard-function/index.md @@ -149,11 +149,8 @@ $$\begin{aligned} = \hat{n}^\dagger(\vb{q}) \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-density" %} + When neglecting interactions, it is tradition to rename $$\chi$$ to $$\chi_0$$. We insert $$\hat{n}$$, suppressing spin: @@ -290,12 +287,10 @@ $$\begin{aligned} = \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k}} - \hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k} + \vb{q}} \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-commutator" %} + We substitute this result into $$\chi_0$$, and reintroduce the spin index $$\sigma$$ associated with $$\vb{k}$$: diff --git a/source/know/concept/matsubara-greens-function/index.md b/source/know/concept/matsubara-greens-function/index.md index fdcadb3..fd46abf 100644 --- a/source/know/concept/matsubara-greens-function/index.md +++ b/source/know/concept/matsubara-greens-function/index.md @@ -83,11 +83,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
Due to this limited domain $$\tau \in [-\hbar \beta, \hbar \beta]$$, the [Fourier transform](/know/concept/fourier-transform/) @@ -157,11 +153,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
Let us now define the **Matsubara frequencies** $$\omega_n$$ as a species-dependent subset of $$k_n$$: @@ -228,11 +220,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
If we actually evaluate this, we obtain the following form of $$C_{AB}$$, @@ -283,11 +271,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-lehmann" %} + This gives us the primary use of the Matsubara Green's function $$C_{AB}$$: calculating the retarded $$C_{AB}^R$$ and advanced $$C_{AB}^A$$. diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md index b306c7d..ba8a677 100644 --- a/source/know/concept/maxwell-bloch-equations/index.md +++ b/source/know/concept/maxwell-bloch-equations/index.md @@ -296,11 +296,8 @@ $$\begin{aligned} \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} \end{aligned}$$ -
- - - -
With this, the equation for the population inversion $$d$$ takes the following final form: diff --git a/source/know/concept/multi-photon-absorption/index.md b/source/know/concept/multi-photon-absorption/index.md index 5dd9887..80dbc9b 100644 --- a/source/know/concept/multi-photon-absorption/index.md +++ b/source/know/concept/multi-photon-absorption/index.md @@ -73,11 +73,8 @@ $$\begin{aligned} = 2 \pi \: \delta(x) \: t \end{aligned}$$ -
- - - -
## One-photon absorption @@ -187,6 +183,7 @@ Note that this transition is only possible when $$\matrixel{u}{\vu{p}}{0} \neq 0 i.e. for any odd-numbered final state $$\Ket{u}$$. + ## Two-photon absorption Next, we go to second-order perturbation theory. @@ -255,6 +252,7 @@ Notice that the rate is proportional to $$|\vb{E}|^4$$, so this effect is only noticeable at high light intensities. + ## Three-photon absorption For third-order perturbation theory, @@ -327,6 +325,7 @@ The rate is proportional to $$|\vb{E}|^6$$, so this effect only appears at extremely high light intensities. + ## N-photon absorption A pattern has appeared in these calculations: diff --git a/source/know/concept/parsevals-theorem/index.md b/source/know/concept/parsevals-theorem/index.md index df90244..377f3a1 100644 --- a/source/know/concept/parsevals-theorem/index.md +++ b/source/know/concept/parsevals-theorem/index.md @@ -24,11 +24,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
For this reason, physicists like to define the Fourier transform with $$A\!=\!B\!=\!1 / \sqrt{2\pi}$$ and $$|s|\!=\!1$$, because then it nicely diff --git a/source/know/concept/repetition-code/index.md b/source/know/concept/repetition-code/index.md index 678211e..89e6f4d 100644 --- a/source/know/concept/repetition-code/index.md +++ b/source/know/concept/repetition-code/index.md @@ -164,63 +164,12 @@ while $$ZZI$$ cannot protect the 3rd qubit. But by using both, we know exactly which qubit was flipped thanks to the eigenvalues: - - - - - - - - - - - - - - - - - - - - - - - - - - -
- Error - - $$ZZI$$ - - $$IZZ$$ -
- $$I$$ - - $$+1$$ - - $$+1$$ -
- $$X_1$$ - - $$-1$$ - - $$+1$$ -
- $$X_2$$ - - $$-1$$ - - $$-1$$ -
- $$X_1$$ - - $$+1$$ - - $$-1$$ -
+| **Error** | $$ZZI$$ | $$IZZ$$ | +| :-: | :-: | :-: | +| $$I$$ | $$+1$$ | $$+1$$ | +| $$X_1$$ | $$-1$$ | $$+1$$ | +| $$X_2$$ | $$-1$$ | $$-1$$ | +| $$X_1$$ | $$+1$$ | $$-1$$ | Where e.g. $$X_3$$ denotes that the 3rd qubit was flipped. The measurement outcomes on the last three rows are called **error syndromes**, diff --git a/source/know/concept/residue-theorem/index.md b/source/know/concept/residue-theorem/index.md index b58e3c2..a0f515e 100644 --- a/source/know/concept/residue-theorem/index.md +++ b/source/know/concept/residue-theorem/index.md @@ -41,11 +41,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
This theorem might not seem very useful, but in fact, by cleverly choosing the contour $$C$$, diff --git a/source/know/concept/selection-rules/index.md b/source/know/concept/selection-rules/index.md index 373486e..620e345 100644 --- a/source/know/concept/selection-rules/index.md +++ b/source/know/concept/selection-rules/index.md @@ -25,6 +25,7 @@ between $$\ell_i$$, $$\ell_f$$, $$m_i$$ and $$m_f$$, which, if not met, guarantee that the above matrix element is zero. + ## Parity rules Let $$\hat{O}$$ denote any operator which is odd under spatial inversion @@ -73,6 +74,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Dipole rules Arguably the most common operator found in such matrix elements @@ -87,11 +89,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-dipole-m" %} + Meanwhile, for the total angular momentum $$\ell$$ we have the following: @@ -177,11 +176,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
## Rotational rules @@ -502,11 +494,8 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
+{% include proof/end.html id="proof-rotation-scalar" %} + Similarly, given a general (pseudo)vector operator $$\vu{V}$$, which, by nature, must satisfy the following commutation relations, @@ -631,6 +620,7 @@ $$\begin{gathered} \end{gathered}$$ + ## Superselection rule Selection rules are not always about atomic electron transitions, or angular momenta even. diff --git a/source/know/concept/superdense-coding/index.md b/source/know/concept/superdense-coding/index.md index ba6e898..4338205 100644 --- a/source/know/concept/superdense-coding/index.md +++ b/source/know/concept/superdense-coding/index.md @@ -27,63 +27,12 @@ Based on the values of the two classical bits $$(a_1, a_2)$$, Alice performs the following operations on her side $$A$$ of the Bell state: - - - - - - - - - - - - - - - - - - - - - - - - - - -
- $$(a_1, a_2)$$ - - Operator - - Result -
- $$00$$ - - $$\hat{I}$$ - - $$\displaystyle \ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B + \Ket{1}_A \Ket{1}_B \Big)$$ -
- $$01$$ - - $$\hat{\sigma}_z$$ - - $$\displaystyle \ket{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B - \Ket{1}_A \Ket{1}_B \Big)$$ -
- $$10$$ - - $$\hat{\sigma}_x$$ - - $$\displaystyle \ket{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B + \Ket{1}_A \Ket{0}_B \Big)$$ -
- $$11$$ - - $$\hat{\sigma}_x \hat{\sigma}_z$$ - - $$\displaystyle \ket{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B - \Ket{1}_A \Ket{0}_B \Big)$$ -
+| $$(a_1, a_2)$$ | **Operator** | **Result** | +| :-: | :-: | :-: | +| $$00$$ | $$\hat{I}$$ | $$\displaystyle \ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B + \Ket{1}_A \Ket{1}_B \Big)$$ | +| $$01$$ | $$\hat{\sigma}_z$$ | $$\displaystyle \ket{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B - \Ket{1}_A \Ket{1}_B \Big)$$ | +| $$10$$ | $$\hat{\sigma}_x$$ | $$\displaystyle \ket{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B + \Ket{1}_A \Ket{0}_B \Big)$$ | +| $$11$$ | $$\hat{\sigma}_x \hat{\sigma}_z$$ | $$\displaystyle \ket{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B - \Ket{1}_A \Ket{0}_B \Big)$$ | Her actions affect the state on Bob's side $$B$$ due to entanglement. Alice then sends her qubit $$A$$ to Bob over the quantum channel, -- cgit v1.2.3