Ryan Reece's Sandbox

Here you will find notes from my experiences computing in ATLAS. You can find some more notes on ROOT, python, pyROOT, and makefiles at my website: http://www.hep.upenn.edu/~rreece/computing.html.


Athena Notes


AthenaFramework

void main()
{
    somecode();
}

EventView


Latex


Here I test out using LaTeX in a TWiki.

 We next turn our attention studying the solutions of the Klein-Gordon equation. Consider the following. \begin{equation}     \label{eq:fk}     f_{k}(x) \equiv \frac{1}{\sqrt{(2 \pi)^3 \: 2 \omega_k}} \: e^{- i \: k \cdot x} \end{equation} where  \begin{equation}     k^\mu \equiv \left(\omega_k, \vec{k}\right)^\mu \end{equation} and \begin{equation}     \label{eq:omega-k}     \omega_k \equiv + \: \sqrt{\left.\vec{k}\right.^2 + m^2} \end{equation} Plugging in the \textbf{plane-waves} $f_{k}(x)$ in for $\phi(x)$ shows that they are solutions. The functions $f_{k}(x)$ form a complete basis for a complex function space. The appropriate inner product in this space involves the following operation. \begin{equation}     \label{eq:arrow-derivative} a \stackrel{\leftrightarrow}{\partial_0} b \equiv a \: \partial_0 b - \left(\partial_0 a\right) b \end{equation} The completeness relation is derived by pluggin equations (\ref{eq:fk}) and (\ref{eq:arrow-derivative}) into the following. \begin{eqnarray*}     \int  d^3x \: f^*_{k'}(x) \: i \stackrel{\leftrightarrow}{\partial_0} f_k(x)     &=& \frac{i}{2 \: \sqrt{\omega_k \: \omega_{k'}}} \: \frac{1}{(2\pi)^3}         \int d^3x \left[ e^{i \: k' \cdot x} \: \partial_0 e^{- i \: k \cdot x}         - \left(\partial_0 e^{i \: k' \cdot x}\right) e^{- i \: k \cdot x}\right]\\     &=& \frac{i}{2 \: \sqrt{\omega_k \: \omega_{k'}}} \: \frac{1}{(2 \pi)^3}         \int d^3x \: (-i) \: \left(\omega_k + \omega_{k'}\right) e^{i (k' - k) \cdot x}\\     &=& \frac{\omega_k + \omega_{k'}}{2 \: \sqrt{\omega_k \: \omega_{k'}}} \:         e^{i(\omega_{k'} - \omega_k) t} \:         \underbrace{\frac{1}{(2 \pi)^3} \int d^3x \: e^{- i \left(\vec{k'} - \vec{k}\right) \cdot \vec{x}}}_{             \delta^3 \left(\vec{k} - \vec{k'}\right)}\\     &=& \frac{2 \: \omega_k}{2 \: \omega_k} \: e^{0} \: \delta^3\big(\vec{k} - \vec{k'}\big) \end{eqnarray*} \begin{equation} \therefore \qquad \int d^3x \: f^*_{k'}(x) \: i \stackrel{\leftrightarrow}{\partial_0} f_k(x)     = \delta^3\big(\vec{k} - \vec{k'}\big) \end{equation} Similary, the orthoginality relation is derived by \begin{eqnarray*}     \int  d^3x \: f_{k'}(x) \: i \stackrel{\leftrightarrow}{\partial_0} f_k(x)     &=& \frac{i}{2 \: \sqrt{\omega_k \: \omega_{k'}}} \: \frac{1}{(2\pi)^3}         \int d^3x \left[ e^{- i \: k' \cdot x} \: \partial_0 e^{- i \: k \cdot x}         - \left(\partial_0 e^{- i \: k' \cdot x}\right) e^{- i \: k \cdot x}\right]\\     &=& \frac{i}{2 \: \sqrt{\omega_k \: \omega_{k'}}} \: \frac{1}{(2 \pi)^3}         \int d^3x \: (-i) \: \left(\omega_k - \omega_{k'}\right) e^{- i (k' + k) \cdot x}\\     &=& \frac{\omega_k - \omega_{k'}}{2 \: \sqrt{\omega_k \: \omega_{k'}}} \:         e^{- i (\omega_{k'} + \omega_k) t} \:         \underbrace{\frac{1}{(2 \pi)^3} \int d^3x \: e^{i \left(\vec{k'} + \vec{k}\right) \cdot \vec{x}}}_{             \delta^3 \left(\vec{k} + \vec{k'}\right)}\\     &=& \cancelto{0}{\frac{\omega_k - \omega_k}{2 \: \omega_k}} \: e^{-i \:2\:\omega_k} \: \delta^3\big(\vec{k} + \vec{k'}\big) \end{eqnarray*} \begin{equation} \therefore \qquad \int d^3x \: f_{k'}(x) \: i \stackrel{\leftrightarrow}{\partial_0} f_k(x) = 0 \end{equation}


Major updates:
-- RyanReece - 21 Dec 2007
Latex rendering error!! dvi file was not created.
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