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*}

The above equation is the probability density for x, where $\mu$ represents the mean of the distribution, and $\sigma$ is the standard deviation with $\sigma^{2}$ being 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\bar{h}\frac{\partial}{\partial t} \Psi (\mathbf{r},t) = \left[\frac{-h^{2}}{2\mu}\Delta^{2} + V(\mathbf{r},t)\right]\Psi(\mathbf{r},t) \end{equation*}

In the above equation i is an imaginary unit, $\bar{h}$ is placks constant divided by 2$\pi$, $\Psi$ is the quantum wave function of the system, $\mu$ is the reduced mass, $\Delta^{2}$ is the Laplace operator, and V is the potential energy.

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}{\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{equation*}

In the above equation $\theta$ represents is the zenith angle and $\varphi$ is the azimuthal angle