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

The function f itself is the probability density of the normal distribution of the variable x, with mu representing the mean of the distribution and sigma representing 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{equation*} 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{equation*}

i is the imaginary number, sqrt of -1, hbar is a constant, t is time, psi is the wave function, mu is reduced mass, V is potential energy, r is the position vector, and the upside down triangle is the Laplacian operator.

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\mathcal{f} = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial\mathcal{f}}{\partial r}\right) + \frac{1}{r^2 \sin{\theta}}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial\mathcal{f}}{\partial \theta}\right) + \frac{1}{r^2 \sin^2{\theta}}\frac{\partial^2 \mathcal{f}}{\partial \varphi^2} \end{equation*}

r is the radius, theta is the zenith angle, and varphi is the azimutal angle.