#spherical harmonics

Okay, so for a transition between energy eigenstates, there needs to be an exchange of a photon with the correct energy. I'm assuming you know that.

Since photons are waves of the electromagnetic field, they impose an electric moment on charged particles, which in a vast majority of cases can be modeled as a simple dipole moment.

Now here's where the quantum mechanics starts: The dipole moment is expressed as a linear operator, which when applied to the wave function of a particular state gives you back the eigenvalue for the dipole moment of that state. However, since we want to describe the transition between states, and the operator only applies to the ket of the wave function, the bra and the ket which it is nestled in between are of the different states, aka the starting and the final electron state.

The operator applies to the starting state ket, which can then be completed on the left with the final state bra, and then integrated over to obtain the transition dipole moment integral.

This integral will tell you the expectation value for the transition. For the selection rules, you don't actually have to precisely calculate this integral, you just have to find out whether or not it is zero, because if it is, that means you have an impossible transition on your hands.

Depending on your representation, the dipole operator as well as the wave functions will look different, as will the space you integrate over.

So first, find which representation (spatial, spherical, momentum space, etc.) you are working with, how the dipole operator looks for that representation, and then pick the two states you want to see if a dipole transition exists between them.

The tricky part is usually to get the wave function representation right, and then to leverage the symmetries of that function to determine if the value of the integral is zero or not. The representation that i find most common for tasks like this is this one, which separates the wave function into a radial and two angular parts. It is also already conveniently expressed in terms of 3 quantum numbers, that being the main quantum number n, the orbital angular momentum number l, and the magnetic quantum number m.

I am afraid you'll have to learn the quirks and symmetries of the generalized Laguerre polynomials as well as the spherical harmonics, in order to make statements about the transition dipole moment integral. However, once you get a feel for their symmetries and remember in what special cases integrals vanish (like integrating an odd function over a symmetric interval, etc.) you will be able to derive the selection rules.

My abstract artwork based on spherical harmonic functions is now available on INPRNT (who seem to have another sale right now)!

Frozen Harmonics

You encounter spherical harmonics when calculating the quantum mechanical wave function for the hydrogen atom. These functions are the building blocks of the lobe-shaped orbitals known from chemistry.

Just as any periodic function – as spiky as it may be – can be built up from sine and cosine waves of different frequencies, any function on a sphere can be built from a basis of spherical harmonic functions.

Here, I’ve tried to use just one these basic functions. Each set of longitude and latitude is assigned a value – like the distance from the origin at that combination of angles.

I visualize surfaces via delicate contour lines: I draw lines on all these lobes, keeping either the longitude or the latitude constant. This is a mesh of lines in three-dimensional space, a virtual sculpture.

Frozen Harmonics Floating in Deep Space

It took me a while to find a strategy for "harvesting" the so-called spherical harmonic functions (that pop up in different sub-fields of physics).

I've tried to use just one these basic functions, visualize it via delicate contour lines, and rotate and pan to find an interesting view. But I feel they are just too regular.

Epiphany: I can combine several of them - they form a basis that lets you build up any function on a sphere (a function of two angles, like a varying "radius" for different points of unit sphere). Epiphany 2: But then, I could actually also pick a more interesting function on the sphere! :-) This is an example, made up from different sine and cosine functions of the two angles, different frequencies, different powers.

Art from Orbitals - work in progress / test

Autumn, exploding spherically harmonically

I created an autumn color palette to draw contours on a spherical harmonic function.

The surface is defined by a function based on sine and cosine functions of two angles (like latitude and longitude).

My palette is defined by three colors - light blue, orange, and black. I am interpolating between these colors and distribute them evenly as I walk about the surface, from contour to contour.

It's a single surface covered with contour lines Usually, I am nesting several of these structures of different size, and I use less contours on each of them. Now I am just using a single surface and many contour lines.

Orbital Autumn

code art by elkement, 2024.

An invasion of spherical harmonic functions, more specifically basis function Y_40.

I am drawing contour lines on the surface of this shape, viewing it from different directions. Created with javascript code based on threejs (no AI). In the end I create a collage from several of these images.

These are just plain lines, I am not rendering anything related to light and shadow. Yet, colored lines create an illusion of light, reflection, and shadows.

The colors are chosen for artistic reasons only: I am re-using my personal autumn color palette introduced in the last post: Three colors - orange, light blue, and black - and interpolations between them.

Prints are available on INPRNT:

Balloon Dog Supernova

Playing with the surfaces that represent spherical harmonic functions! Using a less crazy combination of sine / cosine components than last time.

"Spherical" and "Harmonic" would imply so many poetic artwork names. But no, I see a "Balloon Dog Supernova" :-)

These functions are described by the distance of the balloon dog surface from the origin, as this "radius" varies with two angles. I am stacking lots of balloon dogs of gradually changing color. But I do not show a "surface" as something continuous, like the shiny surface of a balloon. Instead I am drawing contour lines on the balloon dog.