
3D Electron Orbitals of Hydrogen - c0nrad
https://blog.c0nrad.io/hydrogen/
======
dnautics
a bit of lay-hn-reader explanation from a chemistry/math major-now-dev (my
physics might be a bit wrong, apologies in advance).

These diagrams show the probability density of the electrons around a hydrogen
nucleus, which is the simplest (and a pretty good in general) model for how
electrons live around atom nuclei. The more dots, the denser the probability,
aka: how likely or not one might find an electron in this particular position.

In the upper right corner, there's the psi(n, l, m) selector which lets you
pick the geometry.

n is the "principal quantum number" which corresponds to "the gross energy
level/frequency" of electron. The way to think about this (I think) is this:
If you are plucking a string on a guitar, or play a wind instrument, the more
nodes that it has, the higher the energy of the vibration. Similarly for
electrons around atoms. As you pick diagrams with a higher n, you'll see more
nodes (internal regions with zero density) in the distribution. These are also
higher energy states. Generally, if you look carefully you should be able to
find (n - 1) surfaces, though for the (n, 0, 0) diagrams some of these node
surfaces are tiny spheres close to the nucleus, so you might not see them.

l is the angular quantum number. This number determines how many of those
nodal surfaces are "not spherical". So in a (n, 1, X) diagram, you should
eventually see a plane cutting through if you play around with the
orientation; In an (n, 2, x) you should see two intersecting planes cutting
through, or in some cases a cone (more on that later).

m is the magnetic quantum number, and presumes that the atom is sitting in a
nonzero magnetic field, and selects for different energies that relative
orientations in that magnetic field have. This splits the different
possibilities based on direction relative to magnetic field, and not curve
qualities (number of nodes; shape of nodes).

There's another quantum number, which is the "spin quantum number" that has to
do with the Pauli Exclusion principle, that two electrons can share an orbit
simultaneously. This doesn't really change the shape of the orbital, so I
presume that's why it's not there.

(1, 0, 0) is possible, but probably not shown because it's boring.

As for why you could have a "plane" or a "cone"; the display coordinate
systems are somewhat arbitrary, and as with most quantum mechanics, "reality"
is actually a weighted linear sum (superposition) of all of these
possibilities; so a "plane" and a "cone" are roughly equivalently "surfaces",
but the cone is a linear combination of a bunch of planes rotated around a
line but is selected because it's a convenient and easy basis component with
the other "planes" to generate coverage of the vector space of all
possibilities. To really butcher the explanation: It turns out that you have
to play that "rotate trick" because the space of "all possible probability
distributions" has a fixed dimension, and you run out of ways to chop up three
dimensional spaces with planes, so you have to mash them together to get
correct coverage of the space of distributions.

How this corresponds to the periodic table. The S block (left side) elements
are mostly filling their (row, 0, 0) orbitals, then P block (right side)
elements are filling their (row - 1, 1, _) orbitals. The transition metals are
filling their (row - 2, 2, _) orbitals, and the inner transition metals are
filling their (row - 3, 3, _) orbitals . Although it seems elegant, reasoning
for the "row - X" and not "row" is a bit complicated, empirical and not
theoretical, and if you'd like to understand why, look up "aufbau principle".

~~~
op03
I got lost here -> the more nodes that it has the higher the energy of the
vibration. What is nodes?

~~~
hrjfigkt
Nodes - stationary nodes, think standing wave, points who remain at zero while
the plucked string is vibrating.

[https://en.m.wikipedia.org/wiki/Node_(physics)](https://en.m.wikipedia.org/wiki/Node_\(physics\))

~~~
op03
ah thanks!

------
steerablesafe
If this is supposed to be the probability density then it looks way off. I
suspect that there is a mistake in calculating the absolute value of the wave
function.

Edit: In this base the absolute value of the wave function is supposed to be
rotational symmetric around the z axis.

~~~
raverbashing
Why? The PDE should be Psi^2 no? (which is what it's being plotted)

> In this base the absolute value of the wave function is supposed to be
> rotational symmetric around the z axis.

For the D orbital?

~~~
steerablesafe
The phi-dependence of the wave-functions are exp(i m phi), it has magnitude 1.
|Psi|^2 = conj(Psi)*Psi, the phi-dependency cancels out.

------
puzzledobserver
What assumptions does this visualization make? That it is a lone hydrogen atom
suspended in a homogenous magnetic field?

Wouldn't the probability of finding an electron be spherically symmetric,
regardless of the orbital number, if no external fields were present?

------
ur-whale
Very nice, but: what am I looking at? A probability density? And what do the
control represent? Some sort of energy level? An isosurface of the modulus of
the wave function? If so what does the number mean from an intuitive
perspective?

~~~
nsxwolf
Is this really what hydrogen "looks" like in space, or is this just another
impossible for the layperson to understand abstraction?

I never even know how to ask the question I want to ask, just like it's
impossible to ask if the colors in the photograph of a planet are "real" or
not. I've just given up and decided that all of space is in black and white
except for Earth.

~~~
throwaway_pdp09
An excellent point (the first sentence I mean). I'd appreciate a physicist's
opinion.

~~~
antepodius
It's a data visualisation of a property of a model of hydrogen. The dots are
denser where the absolute value of the 'wavefunction' (which is just a
function that takes in space and time coordinates and returns a complex value)
is higher.

It's not 'real' in the same way a simulation of a tennis ball flying through
the air, rendered with dots isn't, but worse: that's a visualisation of a
model, too, but if you imagine it being a simulation of possible sensory data
it makes more sense- the world is set up so it's possible for a tennis ball to
be seen by a conscious being, but that isn't true for a hydrogen atom.

In a model of a tennis ball, you might have a black screen, and then draw a
dot where the tennis ball is, according to the model.

In this model of the hydrogen atom, you have a black screen, and then you draw
a cloud with denser and less dense regions to represent where the electron
'is', according to the model. The problem is that electrons aren't point
particles; in this model, an electron is a cloud- it's described not by some
vector describing its position, but by the aforementioned wavefunction. It's a
cloud in space, (except every point is complex-valued- they're taking the
magnitude for this rendering) that changes (or doesn't) over time.

There's layers here. To what degree is a simplified model 'real'? To what
degree is a visualisation of a model a picture of a 'real' thing, even if that
model were true and complete?

~~~
throwaway_pdp09
I wasn't clear - sorry. It obviously can't be seen, and I (just about) get
it's a probability cloud. My question is, is the underlying probability cloud
real, in the sense that we can think of it existing and in it's actual
peculiar shape, so if we could probe it we would indeed find something shaped
like that, or is it just an abstraction/model 'that just works'?

It's not even an easy question to ask, come to think of it.

~~~
mncharity
As commented elsewhere, the OP renders look perhaps not quite right. But in
general...

Atoms are little balls. With electron density that's mostly spherically
symmetric. It's very high at the nucleus, and falls off exponentially outward.
Down by several orders of magnitude by the time you reach distances at which
atoms hang out together.

A 2D analogue might be a stereotypic volcano, if height were density. I wish I
knew of a better one. Few familiar objects have this degree of fuzzy.
Diffusing smells, but you can't see those.

The common representations with a solidish surface at some large distance from
the nucleus is... useful when doing chemistry, but badly misrepresents the
physical object.

Electron density manifests clearly and concretely. For example, you can poke
at it with the vibrating tip of an Atomic Force Microscope.

Electron states, orbitals, seem less often encountered that directly. Rather
than at one step remove - seeing density, or some other phenomena, and
explaining it with states.

Though there's a fun STM image I'm just now failing to quickly find. An STM
scans a tip across a sample, measuring and mapping the tunneling current
between them. Usually with a boringly symmetric ball of a tip, so the
interestingness is all sample. In this case however, the sample had a grid of
boring s states, and tip conduction was through a tilted f state. So the
sample repeatedly mapped the tip. The image is thus a grid of little images of
the tip's f state, laid out like rows of little buoys. EDIT: Maybe this is it:
[https://arxiv.org/pdf/cond-mat/0305103.pdf](https://arxiv.org/pdf/cond-
mat/0305103.pdf) fig 6 page 12 (though it doesn't entirely match what I'm
remembering).

Just to be clear, representing density by dots, rather than say by a color
gradient on voxels, is purely a data-visualization rendering choice. Like old
newspapers using halftone images. Which I mention only because there are
misconceptions around the individual dots themselves representing something
about the electron.

------
phonon
[https://daugerresearch.com/orbitals/index.shtml](https://daugerresearch.com/orbitals/index.shtml)
is a much nicer (and I believe more accurate) visualization.

------
mncharity
Some similar work: [https://www.willusher.io/webgl-volume-
raycaster/#Hydrogen%20...](https://www.willusher.io/webgl-volume-
raycaster/#Hydrogen%20Atom) , though caveat, that while labeled "Hydrogen
Atom", the density shown is of just one state of the electron;
[http://phelafel.technion.ac.il/~orcohen/DFTVisualize.html](http://phelafel.technion.ac.il/~orcohen/DFTVisualize.html)
.

There are many more visualizations that show total electron density rather
than individual states. Though arguably far far too few, given how pervasively
unsuccessfully these topics are taught.

If you'd like to explore, GPAW
[https://wiki.fysik.dtu.dk/gpaw/](https://wiki.fysik.dtu.dk/gpaw/) can be
useful. Here's a random example of use:
[https://www.brown.edu/Departments/Engineering/Labs/Peterson/...](https://www.brown.edu/Departments/Engineering/Labs/Peterson/tips/ElectronDensity/index.html)

------
tehsauce
A while ago I made an interactive rendering of hydrogen orbitals that is fully
volumetric. Fun feature, the electron pdf is calculated without using any
look-up tables so the algorithm can theoretically be scaled to any energy
level you like.

[https://shaderpark.netlify.app/sculpture/-Lc8sSICWEgoYmFyBm4...](https://shaderpark.netlify.app/sculpture/-Lc8sSICWEgoYmFyBm46?hideeditor=true&hidepedestal=true)

------
supernova87a
It's a great visualization --

I would suggest, though, that some contour lines (or translucent shells?)
might help make the point more apparent to someone trying to learn about the
shapes (which I suppose is the point).

After all, the point is to grasp something visual about it, and just vaguely
discernible clouds of points don't probably convey that sufficiently, although
they are accurate of course.

------
gus_massa
Why the options don't include also the "100" function (aka 1s orbital)? I
guess the error is in
[https://github.com/c0nrad/hydrogen/blob/209ffe5cf14879a4679b...](https://github.com/c0nrad/hydrogen/blob/209ffe5cf14879a4679bc5368ddb0edb7c685838/hydrogen.cpp#L182)

------
vertbhrtn
Is there a way to make these shapes evolve with time?

~~~
ars
They don't. They exist in all the shapes at the same time, which is what the
plot is showing, a sort of combination of all the shapes, with more dots where
the electron "exists" more often.

~~~
plus
It's possible to simulate the time-evolution of the electron density using the
time-dependent Schrodinger equation. That said, if the initial state is chosen
to be an eigenfunction of the Hamiltonian (read: any of the things being
plotted on this page), there will be no time dependence -- the electron
density will remain static. However, if the initial state is chosen as a
superposition of states (read: any configuration that is _not_ given by the
square amplitude of an eigenfunction of the Hamiltonian), you can simulate the
time evolution.

~~~
ellis-bell
that's not entirely true though. you're right about the superposition of
energy eigenstates, and what i'm about to say is uninteresting from the
perspective of the posted simulation, although it is interesting "in the real
world"

in time dependent perturbation theory you can show that electrons can
transition to different energy states through spontaneous emission. for
example, psi(n=4,l=0)-->psi(n=3, l=1) by emitting a photon.

thus, the "change" in the simulation posted here would be uninteresting since
it would just correspond to clicking a different eigenstate in the top right!

~~~
plus
What I'm saying is true -- if the initial state is chosen to be a
superposition of states, then the time-dependent Schrodinger equation enables
one to simulate the time evolution of the electron density.

> thus, the "change" in the simulation posted here would be uninteresting
> since it would just correspond to clicking a different eigenstate in the top
> right!

That is a drastic oversimplification of reality. Spontaneous emission of a
photon is not instantaneous. It is still possible to simulate the time-
evolution of this process.

~~~
ellis-bell
everything is a simplification. of course it's not spontaneous, but that is
literally the word used by physicists to describe the phenomenon.

what does the wave equation of the electron in a hydrogen atom look like
during "spontaneous" emission of a photon? i don't think anyone has any idea.

i'm talking about something entirely separate from a linear combination of two
energy eigenstates. i'm not saying take

\psi = \sin{\theta} \psi_1 + \cos{\theta} \psi_2

where \psi_1 and \psi_2 are eigenstates of the hydrogen atom hamiltonian and
simulate it. i'm saying there is a phenomenon that i'm pretty sure wouldn't be
adequately modelled by a smooth function.

edit: add explanation of \psi_1 and \psi_2

~~~
plus
The hydrogenic orbitals form a complete and orthonormal basis in which you can
expand any arbitrarily-chosen one-electron wavefunction. That is to say that
_any_ arbitrarily chosen initial state for the one-electron wavefunction (and
thus any arbitrarily chosen electron density) can be expressed as a linear
combination of the hydrogenic orbitals. This, in combination with the time-
dependent Schrodinger equation, enables us to simulate the time evolution of
the hydrogenic system starting from _any_ arbitrarily chosen initial state.

> what does the wave equation of the electron in a hydrogen atom look like
> during "spontaneous" emission of a photon? i don't think anyone has any
> idea.

It sounds like you are not aware that this is an entire field of research
within the theoretical chemistry community. Theoreticians have been studying
spectroscopy for a century.

