The images of the equations on this page were taken from the Wikipedia pages referenced for each equation.
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from IPython.display import Image
from IPython.display import display
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.
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Image(filename='normaldist.png')
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The probability density of the normal distribution is $\begin{equation*} f(x,\mu,\sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^-\frac{(x-\mu)^2}{2\sigma^2} \end{equation*} $ where $\mu$ is the expectation of the distribution and $\sigma$ is its standard deviation with $\sigma^2$ as 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.
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Image(filename='tdseqn.png')
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This is the time-dependent Schrodinger equation for non-relativistic particles, $\begin{equation*} i\hbar \frac{\partial}{\partial t} \Psi (\mathbf{r},t) = \left[ \frac{-\hbar^2}{2 \mu} \nabla^2 + \mathbf{V}(\mathbf{r},t)\right] \Psi (\mathbf{r},t) \end{equation*}$ where $\Psi$ is the wave function of the system, $\mu$ is the particle's reduced mass $\mathbf{V}$ 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.
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Image(filename='delsquared.png')
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This is the equation for the Laplacian squared acting on a scalar field; $\begin{equation*} \Delta f = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 \frac{\partial f}{\partial r}) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial\theta} (\sin\theta \frac{\partial f}{\partial\theta}) + \frac{1}{r^2 \sin^2\theta}\frac{\partial^2 f}{\partial\varphi^2} \end{equation*}$ where $\theta$ is the zenith angle and $\varphi$ is the azimuthal angle.
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