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*}

Here, $\mu$ is the mean or expectation of the distribution (and also its median and mode). The parameter $\sigma$ is its standard deviation. $x$ is the value in the normal distribution.

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 \bar{h} \frac{\partial}{\partial t} \Psi(r,t) = \left[(\frac{{-\bar{h}^2}}{2 \mu} \nabla^2 + V(r,t) \right] \psi(r,t) \end{equation*}

$\mu$ is the particle's "reduced mass", $V$ is its potential energy, $\Delta$ is the Laplacian (a differential operator), and $\Psi$ is the wave function (more precisely, in this context, it is called the "position-space 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*} \Delta f = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 sin\theta}\frac{\partial}{\partial \theta} \left(sin\theta\frac{\partial f}{\partial \theta}\right) + \frac{1}{r^2sin^2\theta}\frac{\partial^2f}{\partial\varphi^2} \end{equation*}

In this equation, $\varphi$ represents the azimuthal angle and $\theta$ the zenith angle or co-latitude.