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*}$$ \mu = \text{Mean of the distribution} $$$$ \sigma = \text{Standard deviation} $$$$ x = \text{Independent variable} $$

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{\partial}{\partial t} \Psi(\boldsymbol{r},t) = \biggl[\frac{-\hbar^2}{2\mu}\nabla^2 + V(\boldsymbol{r},t) \ \biggr]\Psi(\boldsymbol{r},t) \end{equation*}$$ \boldsymbol{r} = \text{3D position vector}$$$$ t = \text{Time}$$

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*}$$ r = \text{Distance from origin}$$$$ \theta = \text{Polar angle}$$$$ \varphi = \text{Azimuthal Angle}$$$$ f = \text{Scalar field}$$