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-u)^2}{2\sigma^2}} \end{equation}

In this equation, $\mu$ is the mean or expectation of the distribution, $\sigma$ is the standard deviation and $\sigma^2$ is the variance.

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{\partial}{\partial t}\Psi(\pmb{r},t) = \biggl [\frac{-\hbar^2}{2\mu}\nabla^2 + V(\pmb{r},t)\biggr ]\Psi(\pmb{r},t) \end{equation}

In this equation: $\hbar$ is the Planck constant divided by $2\pi$, $\frac{\partial}{\partial t}$ is the partial derivative with respect to time, $\Psi(\pmb{r},t)$ is the wavefunction of the quantum system, $\pmb{r}$ is the position vector, $t$ is the time, $V(\pmb{r},t)$ is the potential energy as a function of position and time, and $\mu$ is the reduced mass of the particle.

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}\biggl (r^2\frac{\partial f}{\partial r} \biggr) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\biggl (\sin\theta\frac{\partial f}{\partial \theta}\biggr) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \varphi^2} \end{equation}

In this equation, $f$ is the scalar field, $r$ is the radius, $\theta$ is the polar angle, and $\varphi$ is the azimuthal angle.