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.

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

$\mu$ is the mean of the distribution and $\sigma$ is the standard deviation.

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.

$$\begin{align} i\hbar\frac{\partial}{\partial{t}}\Psi(r,t)=\left[\frac{-\hbar^2}{2\mu}\nabla^2+V(r,t)\right]\Psi(r,t) \end{align}$$

$\hbar$ is Planck's constant over $2\pi$. This is multiplied by the partial derivitave of the wave function $\Psi$ with respect to time. $\mu$ is the particle's mass. $V$ is the potential energy and $\nabla^2$ is the Laplacian

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.

$$\begin{align} \Delta f=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial f}{\partial r}\right) +\frac{1}{r^2 \sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial f}{\partial\theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial\varphi^2} \end{align}$$

$r$ is the distance from the origin, radius, $\theta$ is the angle in the xy plane from the x axis, and $\varphi$ is the angle from the z axis.