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cheat-sheet-example.Rmd
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---
title: ""
output:
pdf_document:
template: template.tex
---
# Some Simple Examples
## One-line equation
Euler's constant: $e = \mathop {\lim }\limits_{n \to \infty } \left( {1 + \frac{1}{n}} \right)^n$
## Multi-line equation
This renders as a "pop-up" in Rstudio if you've selected "Inline" with the "Show equation and image previews" option at Tools > Global Options... > Rmarkdown
$$
\begin{array}{lcl}
I &=& \int (x + 1) dx \\
&=& \frac{x^2}{2} + x + C.
\end{array}
$$
## Comments with `\iffalse` (and closing `\fi`)
You shouldn't see anything in the output PDF here (although there is an equation in the Rmarkdown document).
\iffalse
Integration by parts: $\int {u\frac{{dv}}{{dx}}} dx = uv - \int {\frac{{du}}{{dx}}} vdx$
\fi
# A "Big" Test of Equations
**Source:** [Griffith Quantum Mechanics Time Dependent Perturbation theory CheatSheet (UCB 137B final) on www.overleaf.com](https://www.overleaf.com/articles/griffith-quantum-mechanics-time-dependent-perturbation-theory-cheatsheet-ucb-137b-final/jwynrzctvqgp)
\section{TIPT}
\begin{align*}
H = H_0 + H' \\
E_n = E_n^0 + E_n^1 \\
|\psi_n\rangle = |\psi_n^0\rangle + |\psi_n^1\rangle \\
E_n^1 = \langle \psi_n^0 | H' | \psi_n^0 \rangle \\
|\psi_n^1\rangle = \sum_{m\neq n} \frac{\langle \psi_m^0 | H' | \psi_n^0 \rangle}{E_n^0-E_m^0} | \psi_m^0\rangle
\end{align*}
\subsection{Degenerate Case}
\begin{align*}
\text{Degenerate space: } \{|i\rangle\} \to E\\
\quad W_{ab} = \langle a | H' | b \rangle \text{ Non-Diagonal} \\
\text{Eigenvalue and Eivenvectors} \to E_n^1, |\hat{i}\rangle
\end{align*}
\section{Variational Method}
\begin{align*}
\langle H \rangle(\lambda) &=
\frac{\langle \psi(x,\lambda)|H|\psi(x,\lambda) \rangle}{\langle \psi(x,\lambda)|\psi(x,\lambda) \rangle} \\
\langle H \rangle(\lambda) &\geq E_{.g.s} \\
\frac{d}{d\lambda} \langle H \rangle(\lambda_0) = 0 &\Rightarrow
\langle H \rangle(\lambda_0) \approx E_{.g.s}
\end{align*}
\section{WKB Method}
\begin{align*}
\frac{d^2 \psi(x)}{dx^2} &= -k^2(x) \psi(x)& \\ k(x)&=\frac1\hbar \sqrt{2m(E-V(x)} \\
\phi(x) &= \int^x k(x) dx \\
\psi(x) &= \frac1{\sqrt{k(x)}} (C_+ e^{i \phi(x)} + C_- e^{-i \phi(x)})\\
&= \frac1{\sqrt{k(x)}} (C_1 \sin{\phi(x)} + C_2 \cos{\phi(x)})
\end{align*}
\subsection{Energy Level}
\begin{align*}
\int_{R_{classical}} k(x) dx = n\pi \\
\text{one $\infty$ wall} \quad n \to n-1/4 \\
\text{No $\infty$ wall} \quad n \to n-1/2
\end{align*}
\subsection{Tunneling}
\begin{align*}
T = e^{-2\gamma} \\
\gamma = \int_{R_{forbidden}} k(x) dx
\end{align*}
\section{TDPT}
\begin{align*}
H = H_0 + V(t) \\
\text{Eigenstate of $H_0$: } |n\rangle, E_n\\
\text{transition: } |i\rangle \to | f\rangle \\
V_{f i}(t) = \langle f | V(t) | i \rangle \\
\omega_{f i} = (E_f-E_i)/\hbar \\
c_f(T) = \frac{-i}{\hbar} \int_0^T V_{f i}(t) e^{-i \omega_{f i} t} dt
\end{align*}
\subsection{Constant Perturbation}
\begin{align*}
V(t) = \begin{cases}
V, & 0 \leq t \leq T \\
0, & \text{otherwise}
\end{cases} \\
V_{fi}(t)= constant \\
P_{i\to f}(t)= |c_f(t)|^2 = 4 \frac{|V_{fi}|^2}{\hbar^2} \frac{\sin^2(\omega_{fi})t/2}{\omega_{fi}^2} \\
\omega_{fi} \to 0 \quad \text{(degenerate states):} \\
|c_f(t)|^2 = \frac{|V_{fi}|^2}{\hbar^2} t^2
\end{align*}
\subsection{Absorption}
\begin{align*}
V(t) = V \sin(\omega t) \\
P_{i\to f}(t) = \frac{|V_{fi}|^2}{\hbar^2} \frac{\sin^2((\omega_{fi}-\omega)t/2)}{(\omega_{fi}-\omega)^2}
\end{align*}
\subsection{Simulated Emission}
\begin{align*}
E_i > E_f,\quad \omega_{fi}<0 \\
P_{i\to f}(t) = \frac{|V_{fi}|^2}{\hbar^2} \frac{\sin^2((\omega_{fi}+\omega)t/2)}{(\omega_{fi}+\omega)^2}
\end{align*}
\subsection{Fermi Golden Rule}
\begin{align*}
E_i \to E_f\text{ (continuous states)} \\
P_{i\to f} =
\frac{2\pi}{\hbar} |\langle f | V| i\rangle |^2 \rho(E_f) t \\
\end{align*}
\subsection{Selection Rule}
For spherical symmetric potential:
\begin{align*}
\langle n',l',m'| \vec{r} | n,l,m \rangle &\neq 0 \text{ when:} \\
\Delta l &= \pm 1 \text{ and:}\\
\Delta l &= \pm 1 \text{ or } 0
\end{align*}
\section{Scattering}
\begin{align*}
\psi(r,\theta) &= e^{ikz} + f(\theta) \frac{e^{ikr}}{r}, \text{ for large r} \\
k &= \frac{\sqrt{2mE}}\hbar \\
\frac{d \sigma}{d \Omega} &= |f(\theta)|^2 \\
\sigma &= \int d\omega \frac{d \sigma}{d \Omega}
\end{align*}
\subsection{Born Approximation}
\begin{align*}
f(\theta) = -\frac{m}{2\pi \hbar^2} \int V(\vec{r}) e^{i(\vec{k}'-\vec{k})\cdot \vec{r}} d^3\vec{r}
\intertext{Low Energy:}
f(\theta) = -\frac{m}{2\pi \hbar^2} \int V(\vec{r}) d^3\vec{r}
\intertext{Spherical symmetric:}
f(\theta) = -\frac{2m}{\hbar^2 \kappa} \int_0^\infty r V(r) \sin(\kappa r) dr \\
\kappa = 2k\sin(\theta/2)
\end{align*}
\subsubsection{Yukawa Potential}
\begin{align*}
V(r) = V_0 \frac{e^{-r/R}}{r} \\
f(\theta) = -\frac{2mV_0 R^2}{\hbar^2} \frac{1}{1+4k^2R^2\sin^2(\theta/2)} \\
\sigma = (\frac{2mV_0R^2}{\hbar^2})^2 \frac{4\pi}{1+4k^2R^2}
\end{align*}
\subsubsection{Rutherford Scattering}
Let $V_0 = q_1q_2/4\pi \epsilon_0$, $R=\infty$:
\begin{align*}
f(\theta) = -\frac{2mq_1q_2}{4\pi\epsilon_0\hbar^2\kappa^2}
\end{align*}
\subsection{Partial Waves}
\begin{align*}
f(\theta) &= \frac1k \sum_{i=0}^\infty (2l+1)e^{i \delta_l} \sin(\delta_l) P_l(\cos(\theta)) \\
\sigma &= \frac{4\pi}{k^2}\sum_{l=0}^\infty (2l+1) \sin^2(\delta_l)
\end{align*}
\subsubsection{Optical Theorem}
\begin{equation*}
Im[f(0)] = \frac{k \sigma}{4\pi}
\end{equation*}
\subsubsection{Hard Ball}
\begin{align*}
\delta_l = \tan^{-1}(\frac{j_l(ka)}{\eta_l(ka)}) \\
ka << 1 \to \sigma = 4\pi a^2
\end{align*}
\section{Useful Models}
\subsection{Density of States}
\begin{align*}
E &= \hbar^2 k^2 /2m \\
dN &= \frac{L^3}{(2\pi)^3} d^3k = \frac{L^3}{(2\pi)^3} d\Omega dk \\
dN &= \frac{L^3}{(2\pi)^3} 4\pi \frac{m}{\hbar^2 k} dE \\
\rho(E) &= \frac{dN}{dE} = \frac{L^3}{2\pi^2} \frac{mk}{\hbar^2}
\end{align*}
\subsection{infinite square well}
\begin{align*}
H(x) = \frac{p^2}{2m} + \begin{cases}
0, & 0\leq x\leq a \\
\infty, & \text{otherwise}
\end{cases} \\
E_n = \frac1{2m} (\frac{n \pi \hbar}{a})^2 \\
\psi_n = \sqrt{\frac{2}{a}} \sin(\frac{n\pi x}{a}) e^{-iE_nt/\hbar}
\end{align*}
\subsection{Harmonic Oscillator}
\begin{align*}
H(x) &= \frac{p^2}{2m} + \frac12 m\omega^2x^2 \\
E_n &= (n+1/2)\hbar \omega \\
\psi_n(x) &= \frac1{\sqrt{2^n n!}} (\frac{m\omega}{\pi \hbar})^{1/4} e^{-\zeta^2/2} H_n(\zeta) \\
\zeta &= \sqrt{\frac{m\omega}{\hbar}x}
\end{align*}
\subsection{Virial Theorem}
\begin{align*}
2 \langle T \rangle = \langle \vec{r} \cdot \nabla V \rangle
\quad \text{(3D)}\\
2 \langle T \rangle = \langle x \frac{dV}{dx} \rangle \quad \text{(1D)}\\
2 \langle T \rangle = n \langle V \rangle \quad (V\propto r^n) \\
\langle T\rangle = -E_n, \quad \langle V \rangle = 2 E_n \quad \text{(hydrogen)}\\
\langle T\rangle = \langle V \rangle = E_n/2 \quad \text{(harmonic oscillator)}\\
\end{align*}
\section{Math}
\subsection{Legendre Polynomials}
Domain: $(-1,1)$ \\
Even, Odd, Even, Odd ...
\begin{align*}
P_0(x) &= 1 \\
P_1(x) &= x \\
P_2(x) &= \frac12 (3x^2-1) \\
P_3(x) &= \frac12 (5x^3 - 3x)
\end{align*}
\subsection{Hankel Functions}
Solution to Radial Shrodinger Equation:
\begin{align*}
-\frac{\hbar^2}{2m} \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 R_{El}) + [V(r) + \frac{\hbar^2 l (l+1)}{2m r^2}]R_{El} = E R_{El} \\
V = 0 \to R_{El} = j_l(kr) \\
V \neq 0 \to R_{El} = j_l(kr+\delta_l) \\
r \to \infty \Rightarrow R_{El} = \frac{\sin(kr-l\pi/2+\delta_l(E))}{kr}
\end{align*}
When $kr >> 1$
\begin{align*}
j_l(kr) &\to \frac{\sin{kr-l\pi/2}}{kr} \\
\eta_l(kr) &\to \frac{-\cos{kr-l\pi/2}}{kr} \\
h_l(kr) &\to \frac{e^{i(kr-l\pi/2)}}{ikr} \\
h_l^*(kr) &\to \frac{e^{-i(kr-l\pi/2)}}{-ikr} \\
j_l(kr) &= \frac{1}{2} (h_l(kr)+h^*(kr))
\end{align*}
\subsection{Hermite Polynomials}
Domain: $(-\infty,\infty)$ \\
Even, Odd, Even, Odd ...
\begin{align*}
H_0(x) &= 1 \\
H_1(x) &= 2x \\
H_2(x) &= 4x^2-2 \\
H_3(x) &= 8x^3-12x
\end{align*}
\subsection{Spherical Harmonics}
\begin{align*}
|l,m \rangle &= Y_l^m(\theta, \phi) \\
Y_0^0(\theta,\phi) &= \frac12 \frac1{\sqrt{\pi}} \\
Y_1^0(\theta,\phi) &= \frac12 \sqrt{\frac3\pi} \cos{\theta} \\
Y_1^{-1}(\theta,\phi) &= \frac12 \sqrt{\frac{3}{2\pi}} \sin{\theta} e^{-i \phi} \\
Y_1^{-1}(\theta,\phi) &= -\frac12 \sqrt{\frac{3}{2\pi}} \sin{\theta} e^{i \phi}
\end{align*}
\subsection{Green's Function}
For a Linear Operator $\hat{D}_x$
\begin{align*}
\text{Homogeneous solution: }\hat{D}_x \psi_0(x) = 0 \\
\text{Hard Problem: }\hat{D}_x \psi(x) = f(x) \\
\text{Simple Problem: }\hat{D}_x G(x,x') = \delta(x-x') \\
\psi(x) = \psi_0(x) + \int_{\text{f Domain}} G(x,x') f(x') dx'
\end{align*}
\subsection{Some Integrals}
\begin{align*}
\Gamma(n+1) &= n! \\
\Gamma(z+1) &= z \Gamma(z) \\
\int_0^\infty x^n e^{-ax} dx &= \frac{n!}{a^{n+1}} \\
\int_0^\infty e^{-ax^b} dx &= a^{-1/b} \Gamma(1/b+1) \\
\int_0^\infty e^{-ax} \sin{bx} dx &= \frac{b}{a^2+b^2} \\
\int_0^\infty e^{-ax} \cos{bx} dx &= \frac{a}{a^2+b^2} \\
\int_{-\infty}^\infty e^{-ax^2+bx} dx &= \sqrt{\frac{\pi}{a}} e^{\frac{b^2}{4a}} \\
\int_0^\infty e^{-ax^2}x^n dx &= I_n(a)\\
I_0=\frac12 \sqrt{\frac{\pi}{a}}, I_1&=\frac{1}{2a}, I_2=\frac1{4a} \sqrt{\frac{\pi}{a}}, I_3=\frac{1}{2a^2}
\end{align*}
# A "Big" Test of Words
**Source: `clipr::write_clip(stringi::stri_rand_lipsum(10))`**
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scelerisque ante sem cras. Donec, ligula diam non finibus mauris duis platea
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scelerisque ut erat facilisis. Metus vestibulum bibendum in non. Ac, leo viverra
orci risus sapien vitae. Luctus vitae cum, dictum ligula sapien. Vitae sem massa
faucibus netus morbi habitasse mi vehicula. Sollicitudin aliquam non egestas
litora cursus non diam. Lacus vehicula magna sit non lobortis non auctor
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tincidunt amet. Semper, porta, aenean orci ut, ullamcorper tincidunt. Etiam
mauris suspendisse gravida arcu dui vitae. Platea, sodales velit, habitant
tincidunt nulla eu praesent. Eros sociosqu maecenas quis dolor egestas consequat
ligula consectetur netus amet. Arcu, laoreet finibus facilisi primis vulputate
erat in nec. Leo aenean cubilia parturient amet ornare sed sed. Proin, volutpat,
nisi ut metus ut in. Facilisis faucibus nunc, justo lorem dictumst penatibus.
Conubia eu posuere vivamus et class, primis ac nec, hendrerit. Mollis sed
primis. Vitae mus conubia pellentesque. Nunc in accumsan dignissim. Tristique
himenaeos lorem adipiscing fermentum ornare. Mus dapibus commodo. Convallis
massa inceptos ornare. Fames cum ante finibus netus aliquam egestas.
Sollicitudin vel sodales et ut cras. Vitae iaculis facilisis euismod ligula leo
et sit rutrum porttitor sodales in dui, sed tellus. Sed in est egestas eget
auctor nec turpis. Condimentum maecenas, dolor fusce dictumst. Tempor, justo,
sed magnis interdum interdum id potenti mus penatibus. Vitae magna rutrum netus.
Platea non diam fames metus nisl convallis suspendisse. Ac quisque quis id
cubilia, porta, et nam quam fermentum elit. Et malesuada taciti sagittis lacus
lacus erat magnis. Nulla eros nec litora vestibulum nam et rutrum mollis maximus
accumsan. Nibh ut ridiculus. Donec praesent, est sed semper consequat leo
egestas interdum, vel penatibus. Arcu nec leo orci aenean sapien. Mi, nisi massa
donec vulputate gravida. Dui conubia ipsum. Aliquam feugiat vestibulum nostra
pellentesque donec. Dapibus efficitur integer in, condimentum justo quam sociis,
eu. Aenean sollicitudin nulla sed ante pellentesque est class. Sit, nec egestas
urna quis aptent pretium nibh feugiat imperdiet eu sodales. Pellentesque varius
et consequat a aliquet, ut duis maximus nec. Ac ac pharetra himenaeos felis, ut
morbi amet diam non. Class, venenatis venenatis mollis sed. Ipsum ut elementum
sem luctus aliquet augue! Aenean at, ligula tempus urna litora. Volutpat
consequat eu auctor eu. Aliquam accumsan tincidunt amet ac in felis dignissim,
aliquam, tincidunt congue. Viverra et amet nulla pellentesque et. Ipsum sed
aliquam non. Lacinia, cras tortor vel ante ante ac curae laoreet, ipsum sodales.
Ligula metus ac lacinia sagittis in fusce fames montes in habitasse, quam. Nunc,
id! Cubilia varius eget, gravida quis phasellus pretium ex dictumst. Sit,
mauris, in lacinia cubilia vehicula non maximus. Velit sed, aliquam fringilla
nibh duis dolor ac parturient lorem finibus. Non amet magna pretium, montes
tortor, tincidunt, ante interdum vitae vitae. Velit vitae dignissim nulla eu
nunc ullamcorper. Penatibus convallis nam eu eu augue. Pellentesque rutrum justo
finibus enim pulvinar libero pharetra convallis curabitur. Non velit
pellentesque amet est. Felis curae natoque, conubia curabitur dictum eros, nisl
nibh, sed nisi efficitur dapibus. Eu et auctor nostra et. Fermentum libero dolor
nibh vitae metus in elit hac vulputate! Tempus id tempor netus vestibulum
lectus et, donec! Habitasse arcu pulvinar viverra eleifend gravida dui sit
volutpat in inceptos. Quis nibh elit purus ac ultrices. Lacinia, cursus
parturient ut purus et, et ullamcorper quis viverra et nulla in. Ac semper
risus. Est vel dapibus vitae aliquet felis ullamcorper ut augue cubilia blandit
ipsum. Vitae porttitor eros dignissim neque habitant habitasse laoreet erat.
Etiam diam elit vitae vel dui. Vitae gravida felis ante platea eget vestibulum
tristique, in suscipit ac. Lacus convallis neque fermentum primis taciti duis
non, vitae euismod, est velit porttitor libero ridiculus. Iaculis quis nam
viverra ac aliquet urna montes. Sed congue accumsan, est lacus proin sed
pellentesque facilisis ac. Amet ipsum penatibus ex dolor tincidunt massa. Dictum
nec vestibulum et vel ultricies potenti ligula himenaeos ac eu pellentesque.
Nascetur mi ut tortor orci? Nascetur at cubilia finibus luctus, in maximus dolor
porta. Proin porttitor tincidunt in suscipit et, class ac eu, sed fames. Quisque
habitant, venenatis nec pulvinar ante ad lobortis fames a dictumst maecenas
dictumst nisl sed. Conubia conubia platea lectus sem ut vitae proin nisl. Orci
curae, per dolor lacus mollis. Posuere lacinia ultrices at tempor, aliquam
tellus quam, sed enim. Aliquet a sem semper duis etiam suspendisse, eget enim et
dolor cras. In porta turpis sit imperdiet eros sed vitae est. Sed vehicula
mauris vivamus. Consectetur lectus urna mi penatibus. Ut consectetur erat ac
justo habitasse praesent primis. Erat at tempor nam augue nulla ut ridiculus
rhoncus, dapibus. Urna eros quam ut, nulla finibus mauris interdum mauris justo.
Vestibulum hac adipiscing nunc sagittis ligula, varius enim a. Amet, feugiat
nulla mauris netus aptent diam. Ut malesuada nulla condimentum nam dignissim.
Nec magna volutpat, magna in. Fringilla lacinia, quis nunc sit enim, massa nec
tristique a scelerisque ac pellentesque sed. Penatibus tristique, quis laoreet
laoreet sollicitudin interdum himenaeos donec est commodo. Per maximus tempor
tristique commodo at, curae a ultricies urna. Vel fringilla donec non velit
maximus diam id ac. Aliquam risus integer, massa, non. Magna commodo turpis nunc
tristique mollis tristique. Malesuada purus quis ut taciti vulputate sed et
auctor. Mauris ex id primis feugiat nec. In sed mattis tempus risus, amet nulla
penatibus augue magna lacus laoreet. Fermentum phasellus rutrum, mollis pharetra
nullam vel nascetur leo himenaeos dolor pulvinar. Mollis nostra nullam
phasellus, a suscipit nibh ultricies. Cubilia, lorem nulla orci turpis amet
inceptos ad rutrum proin vitae. Ipsum auctor metus mauris non a id, habitasse,
porttitor tincidunt. Nullam nam nunc ullamcorper morbi nam facilisis fusce, non
habitant cubilia nec euismod. Lectus quis accumsan neque habitasse sed leo.
Lectus sed per venenatis sed accumsan viverra malesuada, diam. Ut dui, porta,
blandit ut vestibulum. Volutpat venenatis vitae, suscipit tempor amet venenatis.
Sed mi dui tristique non ligula, ac ridiculus sit parturient. Ut, nibh, et
taciti, at sed bibendum orci potenti, et. Egestas, condimentum laoreet donec,
pharetra nec. Ut quis faucibus sed. Rutrum quis lorem in ut. Sollicitudin,
inceptos posuere parturient class magnis neque id sed etiam. Pulvinar finibus
ullamcorper, scelerisque imperdiet conubia et vitae. Aliquet tortor mi sed
commodo senectus ante nunc ut mattis ornare cursus tempus.