LaTeX Exercise 1

The images of the equations on this page were taken from the Wikipedia pages referenced for each equation.

Imports


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from IPython.display import Image

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{2pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}} \end{equation*}

In this equation $\mu$ is the mean $\sigma$ is the standard deviation and x is the position

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)=[-\frac{\hbar^{2}}{2\mu}\nabla^{2}+V(r,t)]\Psi(r,t) \end{equation*}

i is the imaginary number $(-1)^{1/2}$, $\hbar$ is a constant (plancks) $\frac{\partial}{\partial t}$ is the partial derivative with respect to time, $\mu$ is the reduced mass, $\nabla^{2}$ is the lapalacian operator,Vis the potential energy and $\Psi$ is the wave function

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

r is the radial position, $\theta$ is the angle


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