How to Graph an Electron Orbital?

[Ryan Reed]

I've been really into electrons and their orbitals for a few months now, but I've never understood how they come up with all of these 3d models from these complicated equations. I would love it if someone could explain in detail the equations and the values of the variables and constants within these equations.

 

[DrDu]

It is not that complicated. Once you know the orbital wavefunction $\psi(\vec{r})$, you can plot isosurfaces, i.e. the surfaces defined by $|\psi(\vec{r})|=\mathrm{const.}$. You can do this with almost every plotting program, e.g. gnuplot.
Maybe you can lay hands on this article:
http://pubs.acs.org/doi/abs/10.1021/ed077p785?journalCode=jceda8

 

[Ryan Reed]

You misunderstood my question, I was asking what all the symbols mean and I'd like to be able to plot the graphs myself with a given equation. I would like to understand the math.

 

[fzero]

You are suggesting that someone try to explain what would take many hours of lecture in a typical quantum mechanics course. It would be more productive for you to spend some time with a good quantum text like Griffiths or some freely available course notes (http://farside.ph.utexas.edu/teaching/qmech/Quantum/Quantum.html is a good option). Then, when you have questions about a particular item that is giving you trouble, post again with your questions, but try to make it a little clearer what background you have so that the responders know how best to approach the question in a way that you will understand the answer.

 

[Nugatory]

You misunderstood my question, I was asking what all the symbols mean and I'd like to be able to plot the graphs myself with a given equation. I would like to understand the math.

Although the equations look very complicated, they all come down to something along the lines of $\psi(r,\theta,\phi)=...$ where the stuff on the right-hand side is some function of the three variables $r$, $\phi$, $\theta$, and the quantum numbers $n$, $l$, $m$ that define the orbital. So you'll set the values of $n$, $l$, and $m$ to whatever is right for your orbital (for example, the simplest one is the s0 orbital with all three equal to zero) and you'll have a function of $r$, $\phi$, and $\theta$,

$r$, $\phi$, and $\theta$ are just the ordinary spherical coordinates (google for "spherical coordinates" if you don't know what that means) so it's easy enough to draw a picture that indicates the value of a given function of those variables at each point in space. The only trick is that you don't want to show $\psi(r,\theta,\phi)$ in your picture, you want $\psi^*(r,\theta,\phi)\psi(r,\theta,\phi)$. (If you're not familiar with that $*$ superscript, google for "complex conjugate").

The 3d pictures you've been seeing are drawn by coloring every point where $\psi^*(r,\theta,\phi)\psi(r,\theta,\phi)$ is greater than some threshold value, and leaving that point blank otherwise.

After reading this reply, you might reasonably conclude that there's no substitute for working through a first-year QM textbook. You'd be right.

 

[jtbell]

you might reasonably conclude that there's no substitute for working through a first-year QM textbook. You'd be right.

It doesn't have to be a full-blown QM textbook like Griffiths. There are a number of "introductory modern physics" textbooks that are intended to follow directly after a first-year college/university calculus-based introductory physics course (which deals mainly with classical physics). They cover basic concepts of QM like the Schrödinger equation and the wave function Ψ, and lead up to presenting the results for the hydrogen atom orbitals, introducing spherical coordinates along the way. See for example https://www.amazon.com/dp/1118061144/?tag=pfamazon01-20 or https://www.amazon.com/dp/0534493394/?tag=pfamazon01-20.