Imagining an Atom

You can picture an atom as empty space filled with an invisible cloud of probability which, when disturbed, instantaneously becomes a point-like electron, or you can imagine an atom as a moving, stormy, kinetic thing which really exists in the same way that a galaxy of stars or the winds of a hurricane exist.

The picture above was constructed by using light pulses to kick hydrogen’s electron out of the atom and onto a detector. After doing this over and over, you can reconstruct a picture of hydrogen. Of course, as in the case of a reconstruction of an image of a black hole there are open questions about how picking and choosing data influences the image. There are also reconstructions from diffracted x-rays, but they are not really ‘pictures’ in a conventional sense.

These reconstructions can be compared to mathematical models of an atom showing regions in which an electron is likely to be found. Two popular models are the Bohr model and Schroedinger’s model. In each model, there is a standing wave between the electron and the nucleus and in Bohr’s model or in a pilot wave interpretation of Schroedinger’s model, the wake of the electron is literally hitting the nucleus and reflected back to the electron. The two models agree in most of their predictions, but each run into complications and make different approximations when dealing with the details of spin.

I can tell you that the angular momentum (mass x velocity x radius) of the orbitals is quantized in amounts defined by an integer times Planck’s constant over 2pi and that Planck’s constant times a unit of mass /time is the surface area of a bubble, but, somehow, I cannot tell you exactly what an atom looks like.

Using a definition of mass from fluid dynamics, you can picture an electron propagating around a nucleus like a boat with a wake, as shown in this picture.

The electron moving around the nucleus will form a closed loop with the wake and when the electron speeds up by a small amount, it will be pushed back into a valley of the wake. If the electron gets a big kick which causes it to jump into another valley, this will produce a quantized absorption or emission of energy from the system. In a sense, the electron rides on its own wake — like a surfer.

This is very similar to the description of processes in a free electron laser or in a synchrotron, yet students everywhere are instructed to ignore the Bohr, planetary model of an atom because the physics of synchrotrons tells us that such a model predicts that the electrons would radiate away all of their energy as synchrotron light and then crash into the nucleus. Is this true?

The standard answer to this question is to roll the eyes and say, “you have to solve the nonlinear Schrödinger equation, of course.” Or just just grunt, Electron degeneracy pressure or Pauli exclusion principle, as though that is a good, satisfying answer.

You probably know that an electron beam which is accelerated (bent, turned) by a magnetic field will emit synchrotron radiation if it is relativistic and cyclotron radiation if it is not relativistic, so why wouldn’t an electron which is traveling around a nucleus do the same thing?

To answer this question, we first need to know how fast electrons are traveling around a nucleus and this turns out to be around 1/137 of the speed of light. This is why non-relativistic estimates work so well for describing hydrogen.

Using the cyclotron as our model, the formula for average energy radiated per unit time is given in terms of the Thomson cross section multiplied by the magnetic field squared and the velocity squared and then all of this is divided by the speed of light times the vacuum permeability.

So, to calculate what this might be in hydrogen, we have to make an estimate for the magnetic field of the nucleus. Let’s say that the magnetic moment for our nuclear spin is 3e-8 eV/T and for our electron, it is 2e-8 eV/T.

An electron or nucleus placed in a magnetic field will wobble (precess) around at the Larmor frequency. We have an electron wobbling in the magnetic field of a proton and a proton wobbling in the magnetic field of an electron. What sort of a dance might they execute? The electron will orbit the proton, but the proton will also be orbiting around a central point.

The energy of the electron in the ground state (lowest possible orbit) is -13.6 eV. So I will use that to estimate a magnetic field of 13.6/(2*2e-8) = 3.5e8 T. This is the field necessary to keep the orbit stable.

Plugging this all in to the cyclotron equation, we get that the energy change per second due to cyclotron radiation is

66e-29[m²]*15984[m/s]*12.25e16[T]²/1.257e-6[Tm/A] = 1 [TAm²]or[VAs]or[J]

That can’t be right. Even if I messed up the calculation, the only acceptable number is zero — otherwise, atoms can’t exist.. unless the same amount of energy is being radiated by the proton at the same time that the electron is radiating. If they are constantly trading energy in the form of photons, that could work.

One can imagine that the electron and proton are each holding the ends of a rope which they are wiggling up and down to emit photons. The proton and electron wiggles are completely out of synch, such that the rope at any given moment is perfectly flat even though they are each shaking it up and down. It has zero kinetic energy between them. If that (stretchy) rope has to get shorter or longer when the electron moves from one orbital to another, it will release or absorb energy (light) in an amount given by Planck’s constant times the frequency with which the rope had been shaken. Such a transition produces, for example, the hydrogen spectral series or, for hyperfine structure, the hydrogen line. The further out the rope is stretched, the higher the frequency and the higher the energy the emitted photons will have when the electron transitions to a lower orbital. By stretching the rope, the emitted or absorbed light wavelength gets shorter because the tension is greater (think of a guitar string).

..and that is why electrons will emit radiation when they make a transition from a higher orbital to a lower orbital, but they don’t emit radiation while they are in a stable orbital.

Another nice model of an atom can be found in Chladni plates and the mathematical model of this system developed by Sophie Germain is markedly similar to the model which made Erwin Schroedinger famous.

https://www.youtube.com/watch?v=tFAcYruShow

The Chladni plate was also studied by Leonardo da Vinci.

Or, if you prefer more ancient models, try that of Cleopatra the Alchemist

Now that we have some mental images of atoms, maybe we can make some pictures of electrons to go along with them. We know that electrons have charge and that they spin and act like magnetic dipoles. But how is an electron always spinning without losing any energy?

One answer is that the ether, zero-point energy, electron degeneracy pressure, vacuum energy, virtual particles, or whatever you call the fluctuating energy density of space is constantly driving a pressure gradient around the electron. When a pressure gradient encounters a point-like disruption like a stone in a stream, the pressurized space will swirl around the stone. When the pressure gradient tends to come from one direction, you get a clockwise swirl which makes an electron or a “spin up” or negatively charged particle and when it tends to come from the other direction, you get a counterclockwise swirl which makes a positron — which is the same thing as a “spin down” particle.

As in Feynman’s explanation of how magnetic fields are created by moving charges due to length contraction, electrons act like (are) each other’s anti particles when they are moving in opposite directions relative to each other — as they are known to do when they share a ‘shell’ while orbiting a nucleus.

In order to understand why the behavior of spin is quantized, it is helpful to first understand the concept of a drag crisis where when the velocity to viscosity ratio increases above a certain threshold, the drag coefficient sharply drops and this allows a vortex to persist. While I have been through all of the quantum field theory jargon, I prefer a fluid dynamics way of thinking to the “spin isn’t really spinning, it is an abstract concept with no physical analogue” ways of quantum field theory.

One can entirely avoid quantum field theory by using the model from condensed matter physics in which the electron splits into the spinon, the orbiton and the holon via Spin–charge separation. This picture caused me to start thinking of the electron as a solid, spinning object (spinon) surrounded by a swirling fluid (holon) and separated from the nucleus by some space (orbiton) and this is very similar to a planetary model.

Something pulls us down to earth because there is an exchange of gravitational waves between collections of atoms. When averaged out, the standing wave looks like your standard, general relativity, curved spacetime picture. The waves are exchanged because space itself is oscillating.

Something in the earth pushes up on us because there is an exchange of electromagnetic waves between an electron and the nucleus. The waves are exchanged because space itself is oscillating.

It seems to me that particles have angular momentum and charge because there is a rigid, spinning bubble-like thing and a current of fluid which swirls around it. The rotation of the rigid thing corresponds to our standard notion of angular momentum. The swirling current of fluid corresponds to our notion of charge and flowing charge creating a magnetic field. Opposite/like charges are swirling in opposite/like directions and the currents associated with those swirls attract/repel one another. When the swirl is fixed in one location, the swirl only has a magnetic moment, but when the location of the swirl begins to move, it acquires an electric dipole moment.

You might have noticed that I was careful not to use the word spin in the above text. The word spin has been coopted in an unfortunate way so that it is more closely associated with charge than it is with angular momentum. This is the reason why the difference between positrons and electrons is the same thing as the difference betwen spin up and spin down particles.

If this is hard to understand, it helps to remember that angular momentum is a property of the flow of mass while the magnetic moment is determined by the flow of charge around the particle center. They interact, but they are not the same. They have a constant of proportionality (Gyromagnetic ratio) which is not really constant (Larmor precession) as when particle spin precesses due to its magnetic moment and gravitational moment in the same way that the axis of a spinning planet precesses due to its magnetic moment and gravitational moment.

We know that spin flip is quantized. As in, it happens instantaneously. But why is it quantized?

I think this becomes clear with the proper interpretation of the Stern–Gerlach experiment.

The experiment showed that since the charged particles were only ever deflected down or up and nothing in between, the charged particle spin is always quantized.

In the rest frame of a particle bunch, the individual charged particles are travelling either forwards or backwards due to their shot-noise. Since an electron travelling backwards looks like a positron to the electron travelling forwards, the reason that the particles are deflected either up or down and nothing in between is because they are either electrons or positrons and it is not possible for some intermediate sort of particle to exist.

The reason that electrons/positrons/spin cannot exist in an intermediate state is because they cannot be going in two directions at the same time. If a particle is moving away from you, it is a positron and if it is moving towards you, it is an electron. There is nothing unique about these two particles other than the direction of their travel relative to you.

Wait, you might say: if the bunch of particles is made up of electrons and positrons in the rest frame of the bunch, then why don’t they annihilate? The reason is that a high energy beam of particles is largely frozen in its rest frame, so there will be no collisions at high energy. While a low-energy beam of particles tends to transversely diverge too quickly for the forwards and backwards moving particles to interact in a destructive fashion. It is only in the high-energy environment of a collider that electrons and positrons annihilate.

If I want to extend this picture to describe how an electron undergoes a state-change to become a muon or how a muon melts off a layer to become an electron I have to define four sorts of substance: one is solid, one is space, one is energy, and one is fluid. Energy appears to push space which will turn into fluid when it encounters a solid.

Translating this into more traditional language, we get “Fire appears to push wind which will turn into water when it encounters earth.” Ha.

Is that what Cleopatra the Alchemist’s text says?

I bet Hypatia of Alexandria knew.

However, I doubt that she knew why spin-1/2 fermions must go through 720 degrees to equal the 360 degrees required by spin-1 bosons.

When I was in school, I found it very confusing to be told that a photon in free space is a massless boson with spin-1 which rotates through 360 degrees and acts like a classical particle with respect to its angular momentum while a photon trapped in a two dimensional film is a massive fermion with spin-1/2 which rotates through 720 degrees and has half of the angular momentum which you would expect from a classical particle.

First of all, what is angular momentum for a massless particle?

It means that there is rotation, but it is not localized enough to qualify as massive. When the rotation gets trapped in a surface, it becomes massive.

Second of all, what do the terms “fermion” and “boson” add to my understanding?

Nothing.

What is spin?

When you multiply the “spin” by Planck’s constant, you get the “intrinsic angular momentum” of a particle, a concept which is only loosely related to angular momentum and more closely related to charge — a quantity describing the circulation of electric current around a particle. A photon in free space has intrinsic angular momentum equal to Planck’s constant — the smallest possible change in angular momentum in free space. A photon trapped in a film has intrinsic angular momentum equal to half of Planck’s constant.

Where did half of the angular momentum go when the photon got trapped in the film?

Conservation of angular momentum requires that the photon in the film goes through 720 degrees of rotation while the photon in free space only goes through 360 degrees. In this case, angular momentum is conserved, but intrinsic angular momentum (spin) is not.

In what sense does the spin-1/2 photon behave non-classically?

It only behaves non-classically if you assume that it didn’t change in shape when it transformed from a spin 1 particle into a spin 1/2 particle, and that is clearly not the case, so I don’t know why anyone ever says “It doesn’t behave classically.” (I suppose that the classical comparisons often model an electron as a solid ball of charge, but that is just a straw man.)

Spinning particles act perfectly classically if you treat spin 1/2 particles as spinning surfaces and spin 1 particles as spinning volumes. There is no need to tell students that spin is a mysterious, non-classical, abstract thing. That messes up their intuition.

A rotating volume usually has 1/2 of the moment of inertia of a rotating surface and this is why an intrinsic angular momentum equal to half of Planck’s constant (spin-1/2 fermion) corresponds to a movement which couples to a surface and an intrinsic angular momentum equal to Planck’s constant (spin-1 boson) corresponds to a movement which couples to a volume. You can motivate this with comparisons of the inertia of solid cylinders and hoops and the more mathematically inclined people can entertain themselves by calculating the results for more complicated shapes.

No one has been able to directly measure the radius of an electron, but that is no surprise since bubbles are rather ethereal things. Once you stop disturbing the turbulent flow which has given rise to their cavitation, they reappear.

What I am trying to get at in this post is a return to basic physical intuition that applies at all length scales and that does not confuse approximations with reality. By using words instead of equations, one maintains sufficient uncertainty to capture reality more accurately. When one aims for too much mathematical precision, one invariably misses the mark when trying to imagine an atom.

When I imagine an atom, I imagine each unit of empty space containing a unit of time divided by unit of mass — a quantity that describes a unit of vibration and rotation. Because every unit of space already has some vibration and rotation in it, when something exists, it must interact with the vibrating and rotating space that surrounds it. This forces the object to move around and obey an uncertainty principle in which Planck’s constant is the minimum momentum or position changed allowed in nature. It is the area of a unit of space given by a sphere with a radius defined by the amount of vibration and rotation of the center. When two of these spheres are vibrating in synch, they will be attracted to one another, as in an atom. When they are out of synch, they repel as in two electrons travelling in opposite directions in an orbital. When lots of these spheres are packed together and moving in synch within a potential energy well described by Laplace or by general relativity, there is a gravitational field. When you try to calculate the local, electric field from a synchronous oscillation responsible for gravity, you have so solve the problem of renormalization and this is something that materials scientists are rather good at.

If I reduce this picture down to the essentials:

There is an attraction when vibrating objects move together and repulsion when they move in opposition.

There is an attraction when objects circulate in opposite directions and repulsion when they circulate in the same direction.

Circulation and vibration define a spherical space with a minimum area given by Planck’s constant and a radius determined by the center’s position and momentum. This is a way to describe the uncertainty principle.

The number of such spheres in a given space determines the density and the amount of uncertainty present. Denser space will be heavier, more uncertain, and have a stronger gravitational field. It will also bend light more than less dense, less uncertain space. The more uncertain space is, the more power it has to change the course of things. This is why uncertainty is critical for an understanding of gravity and why an understanding of gravity is necessary if you want to imagine an atom.

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I first posted some of this material on quora.com.

The image in the header is Underwater Explosion at the Unity asset store.