The images of the equations on this page were taken from the Wikipedia pages referenced for each equation.
In [1]:
from IPython.display import Image
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]:
${\large f(x,\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}}$
$\mu$ is the mean of the distribution. The $\sigma$ is its standard deviation.
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]:
${\large i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \left[{\Large\frac{-\hbar^2}{2\mu}}\nabla^2 + V(\mathbf{r},t)\right]\Psi(\mathbf{r},t)}$
$\mu$ is the particle's "reduced mass". $V$ is the potential energy. $\nabla^2$ is the Laplacian. $\Psi$ is the position-space 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.
In [4]:
Image(filename='delsquared.png')
Out[4]:
${\large \Delta f = {\Large \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial f}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2f}{\partial \varphi^2}}}$
$\varphi$ represents the azimuthal angle and $\theta$ the zenith angle.