Ethan's Notes

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    #Math

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    $$
    $$
    \text{Injective (one-to-one):}\forall x_{1},x_{2}\in X, f(x_{1}) = f(x_{2}) \to x_{1} = x_{2}
    $$
    $\text{Injective (one-to-one): }\forall x_{1},x_{2}\in X, f(x_{1}) = f(x_{2}) \to x_{1} = x_{2}$
    $\omega=2\pi f=\frac{2\pi}{T}$
    $k=2\pi \xi=\frac{2\pi}{\lambda}$
    $v_{\text{phase}}=\frac{\omega}{k}=f\lambda=\frac{\lambda}{\tau}=\frac{2\pi f}{k}=\frac{\omega \lambda}{2\pi}$
    $v_{\text{group}}=\frac{ \partial E }{ \partial p }=\frac{ \partial \omega }{ \partial k }$
    $v_{\text{phase}}v_{\text{group}}=c^2$
    $v=\frac{\omega}{|k|\cos\theta}=\frac{\lambda}{T\cos\theta}$
    $E=\hbar\omega=\frac{h}{T}=hf$
    $p=\hbar k=\frac{h}{\lambda}=h\xi$
    $n\lambda=d\sin\theta$
    $\lambda=\frac{\lambda_{C}}{\sqrt{ \frac{\kappa}{mc}+\left( \frac{K}{mc^2} \right)^2 }},,\lambda_{c}=\frac{h}{mc}=\frac{2\pi}{k_{C}}$
    $m_{e}c^2\approx 511\text{ keV}$
    $\frac{1}{\lambda_{t^2}}-\frac{1}{\lambda^2}=\frac{1}{\lambda_{C}^2}$
    $R=\frac{\lambda}{2\text{NA}}$
    $\frac{e^2}{4\pi \varepsilon}=1.44\text{ eV}\cdot \text{nm}$
    $\hbar c=197\text{ eV }\cdot \text{nm}$
    $\alpha=\frac{e^2}{4\pi\varepsilon \hbar c}\approx \frac{1}{137}$
    $E\lambda \approx 400\pi \text{ eV} \cdot \text{nm}$
    $\Delta t\Delta\omega \geq \frac{1}{2},,\Delta x\Delta k \geq \frac{1}{2}$
    $\Delta t\Delta E\geq \frac{\hbar}{2},,\Delta x\Delta p\geq \frac{\hbar}{2}$
    $f(x)=\frac{1}{\sqrt{ 2\pi }}\int_{-\infty}^{\infty} a(k)e^{ikx} , dk$
    $a(k)=\frac{1}{\sqrt{ 2\pi }}\int_{-\infty}^{\infty} f(x)e^{-ikx} , dx$
    $\text{To time: }x\to t,, k\to\omega,, i\to-i$
    $f(x)=\frac{1}{2\pi}\iint f(x')e^{}$
    $\text{Commutative: }A\cup B=B\cup A,,A\cap B=B\cap A$
    $\text{Associative: }(A\cup B)\cup C=A\cup(B\cup C),,(A\cap B)\cap C=A\cap(B\cap C)$
    $\text{Distributive: }A\cap(B\cup C)=(A\cap B)\cup(A\cap C),,A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$
    $\text{De Morgan's Laws: }(A\cup B)'=A'\cap B',,(A\cap B)'UBKCK=A'\cup B'$
    $\text{Relation: }aRb=(a,b)\in R$
    $\displaystyle\Delta t\Delta \omega$
    $\Delta t\Delta \omega$
    $E=K$
    $\frac{1}{2}m_{e}v^{2}=\frac{hc}{\lambda}-\phi$
    $\frac{1}{2}m_{e}(0.1c)^{2}>\frac{hc}{\lambda}-\phi$$

    $$
    \sum_{i=0}^{\infty} f(x)
    $$
    $$
    \begin{align}
    1000 \\\underline{\times ,,,,11} \\1000 \\\underline{1000,,, }\\11000
    \end{align}
    $$
    $$
    \begin{Vmatrix}
    2 & 2 & 2 \\3 &
    \end{Vmatrix}
    $$
    $\displaystyle \sum 2$
    $$ \mathbf{A}^{-1}=\mathbf{A}^{\top}\rightarrow \mathbf{A}\text{ is orthonormal}$$
    $$ $$
    $\displaystyle \frac{kg,m^{2}}{s^{2}}$
    $\displaystyle \frac{1}{\pi\sqrt{ 2a }}\int_{-\infty}^{\infty} \frac{\sin(ka)}{k}e^{i\left( kx-\frac{hk^{2}t}{2m} \right)} , \mathrm{d}k$
    KK 5.12: Ascent of Sap in Trees
    Find the maxt height that water will rise in a tree
    $\displaystyle \tau \ln\left( \frac{n(h)}{n_{Q}}+Mgh=\tau \ln\left( \frac{n(0)}{n_{Q}} \right) \right)$
    $\displaystyle Mgh=\tau\left( \ln\left( \frac{n(0)}{n_{Q}} \right)-\ln\left( \frac{rn(0)}{n_{Q}} \right) \right)$
    $\displaystyle Mgh=\tau \ln\left( \frac{1}{r} \right)\Rightarrow h=\frac{\tau}{Mg}\ln\left( \frac{1}{r} \right)$
    $\displaystyle R_{nl}(r)=\sqrt{ \left( \frac{2}{na_{0}} \right)^{3} \frac{(n-l-1)!}{2n(n+l)!} }e^{-r /na_{0}}\left( \frac{2r}{na_{0}} \right)^{l}\left[ L_{n-l-1}^{2l+1}\left( \frac{2r}{na_{0}} \right) \right]$