LaTeX Exercise 1

The images of the equations on this page were taken from the Wikipedia pages referenced for each equation.

Imports

Typesetting equations

In the following cell, use Markdown and LaTeX to typeset the equation for the probability density of the normal distribution $f(x, \mu, \sigma)$, which can be found here. Following the main equation, write a sentence that defines all of the variable in the equation.

\begin{equation*} f(x,\mu,\sigma)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \end{equation*}

This function defines the normal distribution where $\mu$ represents the mean value, $\sigma$ represents the standard of deviation, and x is a variable

In the following cell, use Markdown and LaTeX to typeset the equation for the time-dependent Schrodinger equation for non-relativistic particles shown here (use the version that includes the Laplacian and potential energy). Following the main equation, write a sentence that defines all of the variable in the equation.

\begin{equation*} i\hbar \frac{\delta}{\delta t}\psi(r,t) = [\frac{-\hbar^2}{2\mu}\bigtriangledown^2 + V(r,t)] \psi(r,t) \end{equation*}

This function represents the time dependent Schrodinger equation for non-relativistic particles. $\mu$ is the reduced mass, V is potential energy $\bigtriangledown^2$ is the laplacian, $\psi$ is the wave function.

In the following cell, use Markdown and LaTeX to typeset the equation for the Laplacian squared ($\Delta=\nabla^2$) acting on a scalar field $f(r,\theta,\phi)$ in spherical polar coordinates found here. Following the main equation, write a sentence that defines all of the variable in the equation.

\begin{equation*} \bigtriangleup f = \frac{1}{r^2}\frac{\delta}{\delta r}\Biggr(r^2 \frac{\delta f}{\delta r} \Biggr) + \frac{1}{r^2 sin \theta} \frac {\delta}{\delta \theta} \Biggr( sin \theta \frac{\delta f}{\delta \theta} \Biggr) + \frac{1}{r^2 sin^2 \theta} \frac {\delta^2 f}{\delta \varphi ^2} \end{equation*}

This is the equation of the Laplacian squared acting on a scalar field in spherical polar coordinates. $ \theta$ represents the andgle, r represents the distance in the polar coordinate system