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.

A normal distribution is defined by the equation: $$f(x,\mu, \sigma)=\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2 \sigma^2}}$$ where $\mu$ is the distribution mean, $\sigma$ is the distribution standard deviation, and $x$ is the 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.

The time-dependent Schrodinger equation for non-relativistic particles is: $$\large\dot{\imath}\hbar\frac{\partial}{\partial t}\Psi(\textbf{r},t)=\biggl[\frac{-\hbar^2}{2\mu}\nabla^2+V(\textbf{r},t)\biggl]\Psi(\textbf{r},t)$$ where $\dot{\imath}$ is the complex number, $\hbar$ is the reduced Planck constant, $\Psi(\textbf{r}, t)$ is the wave function of our particle, $\mu$ is the particle mass, $\nabla$ is the laplacian operator, and $V(\textbf{r}, t)$ is the pontential 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.

The equation for the Laplacian squared ($\Delta = \nabla^2$) acting on a scalar field $f(r,\theta, \varphi)$ is: $$\large\Delta f = \frac{1}{r^2}\frac{\partial}{\partial r}\biggl(r^2\frac{\partial f}{\partial r}\biggl)+\frac{1}{r^2 sin \theta}\frac{\partial}{\partial \theta}\biggl(sin \theta \frac{\partial f}{\partial \theta}\biggl)+\frac{1}{r^2 sin^2 \theta}\frac{\partial^2 f}{\partial \varphi^2}$$ where $f$ is our scalar field, $r$ is our radial distance, $\theta$ is the angle swept out from the x-z plane, and $\varphi$ is the azimuthal angle from the z axis.