# LaTeX Exercise 1

## Imports



In [1]:

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.



In [2]:

Image(filename='normaldist.png')




Out[2]:



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.



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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.



In [3]:

Image(filename='tdseqn.png')




Out[3]:



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.



In [4]:

Image(filename='delsquared.png')




Out[4]:



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.



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