[plus]

What's being plotted is the amplitude of a "hydrogenic" orbitals Ψ_{n,l,m}(x, y, z)^2. Hydrogenic orbitals are the eigenfunctions (wavefunctions) of the 1-electron 1-nucleus Hamiltonian (many-electron wavefunctions are visually similar to this but due to electron-electron correlation become waaaaay more mathematically complex).

This is a nice system because the result is analytic (ignoring relativistic effects and assuming a point-like nucleus). Specifically, Ψ_{n,l,m}(r, θ, φ) = L_n(r) * Y_l^m(θ, φ), where L_n(r) is the n-th order Laguerre polynomial and Y_l^m(θ, φ) is the m-th spherical harmonic with angular momentum l.

From a chemical perspective, n indicates which electron "shell" you are in, l indicates which type of orbital (l=0 is an s-orbital, l=1 is a p-orbital, l=2 is a d-orbital, etc.), and m indicates which of the different orbitals within that shell having that angular momentum (e.g. p_x vs p_y vs p_z).

[ur-whale]

I think you misunderstood my question:

1. The fact that the function is easy to compute because there is an analytical solution to the ODE when the atom is simple enough tells precious little about what the picture actually represents.

2. The fact that the function you talk about has 6 parameters and this is a 3D visualization (3 degrees of freedom) is confusing.

3. The chemistry lesson about orbitals is also an interesting fact but still not properly correlated to the interactive depiction. Notoriously missing: where are m,n,l actually depicted in the story? Am I looking at one specific choice for those? What are the menu entries?

I think there is something that would truly help: if one would take a volume integral over a infinitesimal cube of the 3D interactive representation, what physical units would the result be in?

[antepodius]

The 3-d space shown is just physical space.

n,l,m are the triplet of numbers you can select in the top right. They're called quantum numbers, and they describe the state of this particular system, in particular the state of the electron.

Basically: n is how much energy the electron has (the higher, the further from the nucleus). l and m further describe which orbital the electron is in; l is related to the electron's angular momentum, m is related to its angle to the x-y plane in this visualisation. Only certain combinations of these numbers are allowed by physics (angular momentum, l, has to be smaller than energy, n, for example). n,l,m together describe the state of the electron inside the hydrogen atom.

So what are the dots? Basically, they're meant to represent a cloud. Where the cloud is denser, the electron has more measure; the electron is more there than in other places. Practically speaking, if you made a measurement to see where the electron was, your results would probabilistically correlate with the density of the cloud. The process whereby the electron goes from being a probabilistic cloud to a point particle interacting with your test particle back to a probabilistic cloud is called 'wave function collapse' in the Copenhagen interpretation, or, more generally, 'magic'.

(Or it's just how the universal wavefunction's branches look from the inside.)

A volume integral would be unitless, by definition: the value of the square-absolute-value of the wavefunction at any point (what's represented by this graph) is the probability of finding the electron at that point per cubic metre. A volume integral from negative to positive infinity in x, y, and z gives 1 (no units).

[plus]

> n is how much energy the electron has (the higher, the further from the nucleus)

I disagree with this description. It is true that higher values of n correspond to wave functions with higher energy, but higher values of l also correspond to wave functions with higher energy. n indicates the number of radial nodes.

> A volume integral would be unitless

The volume integral has units of "number of electrons". Calling this unitless is unnecessarily misleading I feel, even if it is technically correct, since in physics we tend not to give units to quantities like this.

[antepodius]

You're right.

[plus]

It is not a 6-parameter function, it is a family of 3d-functions each characterized by 3 parameters. Those parameters can be modified using the dropdown in the top-right corner of the page.

The Hamiltonian is an operator that describes the energy of a system. Eigenfunctions of the Hamiltonian are quantum states, referred to as wave functions. The squared amplitude of a wave function is a probability distribution function. When discussing the wave functions of electrons, the probability amplitude is sometimes referred to as the electron density. You are looking at a sampling from the electron density of the wave functions of the 1-electron 1-nucleus Hamiltonian operator. There are different wave functions (different entries in the dropdown box at the top-right corner of the screen) because the Hamiltonian operator has more than one eigenfunction. Each eigenfunction is characterized by the 3 "quantum numbers": n, l, and m. "n" indicates the number of radial nodes -- areas of a given distance from the nucleus where the electron density is 0. "l" indicates the number of angular nodes -- areas arranged in a certain angular pattern around the nucleus where the electron density is 0.

> I think there is something that would truly help: if one would take a volume integral over a infinitesimal cube of the 3D interactive representation, what physical units would the result be in?

Number of electrons (possibly fractional, if you aren't sampling the whole space). For this particular Hamiltonian, the integral over all space should be numerically 1 for any given eigenfunction, since we are looking at the 1-electron Hamiltonian.

[ur-whale]

> It is not a 6-parameter function, it is a family of 3d-functions each characterized by 3 parameters. Those parameters can be modified using the dropdown in the top-right corner of the page.

Sorry to be pedantic, but these two things are the exact same thing.

[plus]

You were asking how it is to be interpreted, and it is as I said: a family of 3D functions each characterized by 3 parameters. What you are saying is technically correct, but misses the point of what I was trying to convey.

Edit: also, three of the parameter (x, y, z or r, θ, φ depending on whether you are using a Cartesian or spherical coordinate system) are continuous, real, and unbounded (well, unbounded in a Cartesian sense anyway). In contrast, n, l, and m are discrete integer-valued quantum coefficients that obey the relations n > 0, 0 <= l < n, -l <= m <= l.

[o_v_o]

2. Basically it has six parameters because the wave function changes based on the first three parameters (n,l,m) and can then be solved w.r.t. x, y, and z, though it'll be easier to do in spherical coordinates.

3. This hydrogen atom has a nucleus and one electron. Think of n as the energy level of that electron - electrons have discrete energy levels, so as n increases the electron occupies the next discrete energy available to it.

l is another quantized value which corresponds to what we call the orbital angular momentum of the electron, which partially determines the shape of the orbital. This is a big part of the visualization you see - as you change the value of l, we see different shapes, and if you increase the number of particles in the visualization, you get changes in those shapes. These different shells have names - s, p, d, etc - that correspond to the integer value of l - 0, 1, 2, etc.

Importantly, what's being graphed in the visualization is a solution to the specified wave function. It's a 3D probability map, effectively. Where there is a higher chance of the electron being located, the particles are more concentrated, whereas lower chance regions have lower populations of particles.

m is called the magnetic quantum number and can have integer values from -l to +l, and further specifies the particular state of the electron in its "shell" - s, p, d, etc again. If the wave function has n=2 and l=2, then it's in the d shell, and can have values of m from -2 to +2. The actual value of m determines the final "shape" of the orbital, again depicted as a probability map - every dot you see plotted can be a location of the electron, so plotting a lot of them based on the probability distribution gives you a visualization of the regions available to that electron.

So the menu entries are just values of n,l,m that aren't separated by commas.

I hope that clarifies some things!

[_Microft]

> 2. The fact that the function you talk about has 6 parameters and this is a 3D visualization (3 degrees of freedom) is confusing.

Why is that a problem?

It's no different than defining a family of linear functions f_a,b(x) = a x + b and then letting the user pick values for the parameters a and b before plotting it onto the xy - plane.