Most of us know that the electron is a negatively charged particle that orbits the nucleus in an atom of matter. We also know it as the particle that moves inside conducting metals, creating an electric current, as well as the particle behind electrostatic sparks and lightning. Chemists among us know that the electron is what makes all chemistry possible. Electrons are responsible for the attraction between atoms, creating bonds that range in strength from weak dipole-dipole interactions to practically unbreakable covalent bonds. The electrostatically repulsive cloud of orbiting electrons is what gives an atom its physical size, and it's what prevents us from walking right through walls.
When we visualize what is going on with the electrons in these interactions, it is not difficult to imagine them as tiny physical dust-like particles. Electrons have mass, charge, angular momentum, an intrinsic magnetic moment and helicity, but they have no known substructure. No two electrons can occupy the same space at the same time. They are part of every atom but they can exist separately on their own as well. You can shoot a beam of electrons at a target for example.
At the turn of the last century, this was the emerging, and satisfying, picture of what an electron is. And I dare say this is the picture almost all of us, including many top particle physicists, still carry around in the mind. This picture of the electron recalls a time when physics made sense, before quantum mechanics reared its nonsensical head. We now call this description the classical description of the electron.
Two Models of the Atom
The picture of the electron changed when quantum mechanics came along just a few years later, in the early 1900's. At around this time, physicists were drawing from the Bohr model of the atom, in which electrons orbit the positively charged nucleus within various distinct energy shells. The Bohr Model is shown right.
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| Brighteorange; Wikipedia |
This is a simplified Bohr model of a hydrogen atom, the simplest atom there is, which is composed of just one electron orbiting one proton. Three orange rings, 1, 2 and 3, represent three possible energy levels (called energy shells) where the electron can be located. Closest to the nucleus, n = 1 is the lowest energy shell or ground state of the electron. This is the average state of all the atoms in the chair you are sitting on, for example. Atoms can absorb energy and become excited. This can happen when an atom is struck by a high-energy particle such an electron or a photon, or when it is in a very high-energy (high-temperature) environment. The electron can move up to and orbit within one of several possible higher energy shells (the image above is greatly simplified; even for hydrogen, there are many more than three possible energy states). The atom can return to ground state by releasing orbital energy. The electron can transition from n = 3 to n = 2, for example. It releases energy through the emission of a single photon of electromagnetic radiation. In this case, the 3 → 2 transition photon is in the visible light spectrum. It is red light.
Previous and ongoing work by several others in atomic physics led physicist Paul Dirac to surmise a new model of the electron - a model that was consistent with the "new physics." At the time, many great physicists were working on the same question - what is the atom? Dirac had an impressive list of contemporaries: Ernest Rutherford, Louis de Broglie, Max Born, Werner Heisenberg, Erwin Schrodinger, Albert Einstein and Max Planck among many others. Dirac's goal was fairly straightforward. He wanted a model that took into account the newly discovered fact that electrons possess intrinsic angular momentum, and he wanted the model to work within Einstein's new theory of special relativity.
This model is actually an equation called the Dirac equation. It incorporates Einstein's special relativity as well as the formulations of quantum mechanics.
Dirac's quantum mechanical theory of the atom has been around since the 1930's. As we will see, this means that the theory of the electron as a tiny physical particle is long gone. For many problems in physics, we can still think of the electron this way and we can think of the atom as the Bohr model of the atom, but it is no longer the most accurate conceptualization. The physical electron's long and continuing farewell is in most part thanks to our difficulty with understanding, and accepting, quantum mechanics.
The electron as a physical particle is a good way to begin to conceptualize the electron in junior and senior high school. We must start somewhere, and this concept fits perfectly into the framework of classical mechanics, which is also explored at this level. But once we have mastered that groundwork and we are curious about what exactly matter and energy are, we are ready to get into the gnarly and brilliant head of Paul Dirac. We must seek to understand not only what this model is, but also what it means for contemporary physics. The tenacious belief that no one can possibly understand quantum mechanics is holding us back from vast new undiscovered territory in theoretical physics.
What the Quantum Mechanical Electron "Looks" Like
Below we can see our ground state hydrogen atom as a quantum mechanical model based on the Dirac equation.
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| RJHall;Wikpedia |
This probability distribution tells us something profound about the atom. The electron is never 100% here, there or anywhere. And this is where many students throw up their hands and jump out the nearest window, perhaps fairly enough. This is also where our exploration starts to get very interesting.
The Dirac Equation
The Dirac equation is an enormous breakthrough in physics, and for this reason it deserves some effort in understanding and appreciating it. It is not well known outside of graduate-level physics, and that is a shame. Much of the problem is that it is complex. A symbolic equation that at first glance seems straightforward breaks down into a myriad of complex equations which can be overwhelming for those of us who are not mathematicians. However, these equations not only tie it into several prior disparate theories but also introduce new mathematical objects with new descriptive powers. Here, I aim only to introduce its essence to you, but if you are mathematically inclined, try mathpages.com. It goes through the process of taking a classical equation for the motion of an object and turning it into a relativistic quantum mechanical description based on the Schrodinger equation. Even reading the descriptions of the mathematical process there will offer you an idea of the power of this equation.
This will hopefully be a fascinating but discomfiting journey. Dirac's equation leaves many physicists with a sense of unease. Bringing its solutions into the physical world means that several disturbing paradoxes must come along with.
The Dirac equation is a relativistic wave equation. This means it can describe particles such as electrons traveling close to light speed based on Einstein's theory of special relativity. It describes not only electron behaviour, but the behaviours of all fermion particles as well. Fermions are subatomic particles with spin 1/2. This means they have parity symmetry according to the Standard Model - a particle will possess one of two possible intrinsic spins - up or down. This is significant to us because it means that the Dirac equation describes both the electrons AND quarks - the two kinds of fundamental fermionic particles that make up an atom. Therefore it describes all matter at its most fundamental level. What makes it especially powerful to physicists is that it describes atomic behaviour in the context of BOTH quantum mechanics and special relativity. This allows physicists to describe in detail what is going on inside the electron cloud of the atom, where in energy shells further out, electrons maintain velocities approaching the speed of light. Large atoms with lots of electrons contain at least some relativistic electrons. These relativistic electrons significantly affect the physical and chemical properties of these atoms.
This theory also describes interacting magnetic fields inside the atom as a result of its multiple tiny moving magnetic dipoles (each electron is a tiny magnet or magnetic dipole). It describes the intrinsic quantum spin of the electron and it predicts the existence of antimatter.
The Mathematics of the Dirac Equation
To begin to appreciate the Dirac Equation, we must explore it mathematically. Below is Dirac's original equation in its simplest symbolic form.
It is fairly simple in this form. It describes the electron as a wave function, ψ. A wave function describes, in turn, the quantum state of either a particle or a system of particles, and it contains all the information about that system. This equation therefore can completely describe an electron with a rest mass m within a space-time coordinate x,t. The three p values are components of the momentum of the electron. "c" is the speed of light (incorporating special relativity) and ħ is Planck's constant divided by 2 π (incorporating quantum mechanics).
When Dirac set out, he simply wanted to describe the behaviour of an electron moving close to light speed, and he succeeded. However, the implications of this formula go so much deeper, and, as this article will suggest, they are not yet fully realized.
You can reduce Dirac's equation to a version of Schrodinger's equation that works inside Einstein's special relativity. The most common version of Schrodinger's equation, for a single particle moving in an electric field, is shown below.
This equation describes the wave function of a particle as a solution to the equation, again symbolized by ψ. It is derived from classical wave mechanics and based on de Broglie's hypothesis that any particle can be described as a wave, but Schrodinger's equation is unique because it goes much further to describe at once all the information that can be known about a system.
To this already powerful equation, Dirac added new components such as matrices (ak and B) as well as reformulating the wave function (ψ) into four complex number components, which transform under Lorentz transformations. This gives you space-time transformation in special relativity - time dilation and length dilation and so on. Measuring ψ at any point in space-time gives you something called a bispinor. The bispinor can be thought of as a superposition of four particles - a spin up electron, a spin down electron, a spin up positron and a spin down positron. In this bispinor modern physics caught its first theoretical glimpse of the antiparticles of antimatter.
Dirac's equation can be unraveled into four coupled partial differential equations, one for each of four values that make up the wave function (the complete description) of an electron. This four-component matrix describing the wave function was a giant breakthrough, and it was an entirely new mathematical object in physics. The introduction of this geometric algebra, today called Dirac algebra, which describes a wave function in four-dimensional space-time as a matrix, was an enormous breakthrough for quantum theory in general. It can describe the relativistic mass of a particle as well as how its wave function transforms as it approaches the speed of light.
The Dirac equation, beautiful as it is, also comes with prickly thorns. One thorny example is the Dirac Hole. When used to describe electron dynamics, for every quantum state possessing a positive energy state, there is a corresponding state with negative energy. An electron described by the Dirac equation will therefore have some components of its wave function in a negative energy state. As a consequence of this, there is nothing mathematically inconsistent about a free electron in a (positive energy) electromagnetic field spontaneously decaying into successively lower and lower, and eventually negative, energy states. All the while it would continuously emit all the excess energy it had as photons as it went into lower and lower energy states - a process that theoretically could continue forever without end. Today we can dismiss this crazy scenario because we know that electrons don't have negative energy, but Dirac must have felt this thorn acutely.
To address the problem, he introduced a new theory called the Hole theory. Here I can only imagine sweat on his brow and an almost panicked look on his face. Simply put, space-time is imagined as a many-body quantum state in which all the negative electron energy states are already filled. To introduce an electron, we must therefore put it into an unoccupied positive energy state. That keeps it nicely positive. This still allows an electron to lose energy by emitting photons but only until it reaches zero energy and no further. Dirac also theorized that a hole could exist where all the negative energy states are occupied except for one. That "hole in the sea" would respond to electric fields as if it were a positive particle. It would act like a positron in other words. This is the serendipitous discovery of antimatter. The discovery of actual antimatter (positrons) in a cloud chamber by Carl Anderson just a few years later strengthened Dirac's Hole theory, although Dirac himself incorrectly thought the positive particle was a proton. This theory, while serving up the discovery of antimatter, introduces a troubling new scenario - space-time as a vacuum containing infinite energy.
Quantum field theory reformulated the Dirac equation in a way that the positron is treated as a "real" particle rather than the absence of one. It also reimagines the vacuum of space-time as a sea where no particles exist rather than a sea of infinite particles. However, it does not address the problem of having a vacuum of infinite energy. Vacuum energy is now treated to a mathematical process called renormalization, a process in which you add a constant, for example, to an equation to get a solution that approaches either experimental results or at least something logical. It is often a warning signal that something in the equation is amiss or the theory itself is incomplete. In this case, renormalization is based on the assumption that we can only measure energy in a relative sense anyway, so an absolute value for the universe's vacuum energy is neither required nor possible. There are many additional thorny issues around vacuum energy, considering that it should exert gravitational force, which would affect the inflation and expansion of the universe, and it should have a nontrivial effect on the cosmological constant.
Another way to look at negative energy solutions is to think of them as positive energy moving backward through time. Most physicists, if they believe this is possible, limit it to virtual particles and thus remove the possibility of backward time travel from macroscopic systems and eliminate the horror of what backward time travel would do to the laws of thermodynamics. Richard Feynman famously incorporated backward and forward time arrows in his (extremely useful for visualizing) Feynman diagrams of particle dynamics.
Some experts consider Dirac's Hole theory less than elegant. But the Hole concept comes back to us when we explore solid-state physics. A sea of conducting electrons (not negative energy electrons in this case) in an electrical conductor contains unfilled spots or holes. An unfilled spot in this sea acts like a positively charged electron, although it is called a hole rather than a positron. In this case, Dirac's underlying arguments for his Hole theory work perfectly well.
In Dirac's equation, the electron is treated as a point mass with no physical dimension. It has no girth or diameter in other words. I, and others, find this puzzling because it seems that this state, where mass is confined to a point-like space, also describes a black hole, albeit a tiny microscopic one. There is an interesting exchange on physics.stackexchange.com where electrons are compared to completely evaporated black holes. What is even more fascinating to me is that this "black hole" obeys the laws of quantum mechanics, whereas Stephen Hawking's black holes are strictly objects of general relativity. These two theories, at least so far, refuse to commute with each other.
When we think about this further, we discover that the electron wave function and the black hole of general relativity (GR) are two very different species. In Dirac's equation, the electron is treated as BOTH a point AND a wave function. A GR black hole has no corresponding wave function description. The electron's wave function collapses, for example, during a collision, at which point it is a point but then it immediately reforms into a wave function once again. Once again as a wave function, we can no longer say that its mass is concentrated on one point because there is no absolute location of this one point. It is a spherical smear. A black hole, in contrast, has no smear. It is treated as a single point (a singularity) at the end of an infinite gravity well, inside of which the laws of physics break down and give us nothing but infinities for solutions. This comparison helps to bring into stark relief the clash between general relativity and quantum mechanics. Dirac's equation does not take into account general relativity. If it did, we would finally have a complete theory of particle physics.
Where Is the Real Electron?
If we return to the two contemporary models of the electron - the Bohr model and the quantum model - they seem mutually exclusive at first glance. The quantum model tells us that the location and energy (also described as momentum or velocity) of the electron cannot simultaneously be measured with prefect accuracy for an individual electron. One must give way to the other so that either location is known or energy is known but not both. The Bohr model tells us not only that the energy of the electron is precise, but it can only adopt a specific set of precise quantized energies as well. It cannot exhibit energy in between one of the energy shells.
Each model is powerful and accurate in terms of describing and predicting specific behaviours of the electron, and both are used in contemporary physics. So how can they both be accurate descriptions of the electron?
If we look more deeply into the Bohr model, we find that Niels Bohr devised his atomic model as a solution to a glaring problem with the prevailing atomic model at the time, the Rutherford planetary model. In this model, electrons orbit the nucleus in stable planet-like orbits. A Rutherford model of a lithium atom, with three each protons, neutrons and electrons, is shown below.
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| Fastfission;Wikipedia |
Unlike planets in gravitationally stable orbits around the Sun, however, electrons are charged particles. The electron is in a state of acceleration (just as a planet is) and according the Larmor formula, any accelerating charged particle radiates electromagnetic energy. The electron's orbit should decay inward as it loses energy and the radiation should increase in frequency as it does so. This would not only result in an unstable atom but its electromagnetic radiation would be a smear across wavelengths rather than the discrete frequencies that had been experimentally observed by that time.
The Bohr model offers stability to the orbiting electron by describing its orbit as a standing wave. A standing wave is a stationary wave that stays in a constant position. Each higher energy orbit (or shell) in an atom is also a standing wave. He found that the energy spacing between these standing wave energy shells was a fixed integer multiple of the lowest energy shell. The energy of an electron shell (each orange ring in the earlier Bohr diagram) is the AVERAGE energy of the standing wave it represents. In the image below, the standing wave on the left fits the averaged energy shell of the electron. The wave on the right is not a standing wave (does not connect) and it does not represent an allowable energy level for the electron.
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| CK-12 Foundation;Wikipedia |
While we can fairly easily picture an electron as a tiny physical particle orbiting around a proton nucleus and we can imagine it jumping up and down the higher and lower energy shells of the Bohr model, our imagination fails us when we deal with the more sophisticated and technologically useful quantum model. This wave function model tells us that the electron is neither here nor there. It cannot be reduced to any kind of point particle unless its wave function collapses, such as when it collides with another particle. Even then we have only a ghost image of where and when the collision occurred. According to Dirac, at any point in space, the electron neither exists nor doesn't exist. It can only be described as a mathematical function. The same is true for the quarks that make up the atom's nucleus, as they too are fermions, which behave according to the Dirac equation.
Knowing this, it seems miraculous that almost all electron behaviours can still be reliably measured and observed, that we have any working physical description of them at all. Most observations are of large numbers of electrons behaving at ordinary energies where their inner quantum nature is hidden from view.
Young's Double Slit Experiment Peers Into a Secret Quantum World
There are experiments elegantly designed to offer us a glimpse into the hidden world of particle behaviour. Young's famous double slit experiment is an excellent example. Below, Dr. Quantum offers an easy-to-follow 5-minute introduction to this experiment.
A beam of light or electrons is shot through two parallel slits in a plate. Either photons or electrons go through the two slits and hit a detector screen behind the plate. Both electrons and photons build up interference patterns, which beautifully demonstrate their wavelike nature. However, when the beam of particles is reduced so that just one particle is shot at a time, something purely astounding is observed - the interference pattern is still created! The results of a double slit experiment are shown below right where an interference pattern is built up by electrons shot one at a time.
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| user:Belsazar;Wikipedia |
This demonstrates not only the dual wave-particle nature of particles but it also indicates that individual particles behave in a clearly probabilistic (building up the interference pattern) manner, which only quantum mechanics can account for. This experiment has been performed using "particles" as large as molecules called buckyballs.
The Young's double slit experiment, discussed in high school physics, may have a simple setup but it leaves us with very fascinating questions. It demonstrates to students how particles of light and matter exhibit both a particle nature and a wave nature depending on how they are measured, and this is where most discussions end on the matter. Slowing the beam down to individual particles shot one at a time reveals their true and bizarre quantum mechanical nature to us.
In describing this experiment, Wikipedia includes a quote from Richard Feynman, [this result is] "a phenomenon which is impossible [...] to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]." (Feynman, Richard P.; Robert B. Leighton; Matthew Sands (1965). The Feynman Lectures on Physics, Vol. 3. US: Addison-Wesley. pp. 1.1-1.8. ISBN 0201021188.) Wikipedia goes on to say that Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment. In 2002, Physics World readers voted the single electron version of the experiment the "most beautiful experiment" and I whole-heartedly agree with them.
Many variations on this experiment have been done since Young first performed it in the early 1800's. One of my favourite variations is called the delayed quantum eraser. This experiment is in itself a paradox UNLESS it is interpreted in a certain way.
First, two statements are made. If an electron manifests itself as if it came through just one slit, it must have entered the setup as a particle. If, on the other hand, the electron manifests itself as if it came through as two indistinguishable paths through both slits then it entered as a wave. Now the experimental apparatus is reversed mid-flight. And . . . the electron reverses its decision (AFTER going through one or two slits!) to be a particle or wave.
How can a measurement made in the present reverse a "decision" made by the electron in the past? This is a time paradox that can only be resolved if we consider the electron inflight to be in a superimposed state in which it is neither particle nor wave but has the potential to be either one. This is the interpretation made by John Wheeler in a series of thought experiments in the late 1970s before the actual experiment was performed. This setup, as well as recent far more complex experiments designed to explore (faster than light speed and backward through time) particle decision-making, concludes that the particle, either photon or electron, exists in a state that is neither particle nor wave, with the potential for both at the same time. Only when a particle is detected, does its wave function collapse and it shows either clear particle or wave behaviour. Detection also means that it is absorbed or deflected and therefore no longer in existence in its original state. These experiments also show that the particle travels both routes (through both slits) as well as just one route (it is detected as a particle when it can only travel through just one slit, creating no interference pattern). Einstein thought this was a clear indication that quantum theory was missing something or it was out-and-out wrong. Richard Feynman, however, saw this two-route scenario as an opportunity to rethink quantum behaviour (as we will see).
This still leaves us with a fundamental question. When two slits are open, how do the particles know where to go in order to work together to build up the precise interference pattern of a wave? At least it seams that this is what they are doing in the single particle experiment version, and it seems to look like a violation of causality, the tried-and-true experience of all cause and effect relationships. An experiment set up by Kim et al in 2000 (Kim, Yoon-Ho; R. Yu, S.P. Kulik, Y.H. Shih and Marlan Scully (2000). "A Delayed Choice Quantum Eraser". Physical Review Letters 84: 1-5. arXiv:quant-ph/9903047. Bibcode:2000PhRvL..84....1K. doi:10.1103/PhysRevLett.84.1) was designed specifically to test this retrocausality possibility. It is a complex setup that is best described on Wikipedia. Not everyone's interpretation will agree with those stated on the Wikipedia page, but the results of this experiment say two important things. First, there is no point at which a decision is made by the particle in-flight whether to be a particle or a wave. Second, only when the final signal at the detector plate is observed, regardless of how the particles are manipulated along the way, do the wave functions of the particles finally collapse into a total non-interference pattern (at Do). These manipulations include long delays in detecting one of two entangled particles. The researcher's interpretation is that this final signal comes to the observer as an ordinary beam of light. It is the only "real" component of the particle's behaviour, which travels no faster than light speed and, therefore, does not violate causality. As Wiki states, there are numbers of physicists that do not agree with that interpretation and do believe that causality is violated.
I would offer a third conclusion from this experiment, perhaps better stated as a question: Is an electron, then, the entire interference pattern? Does this experiment indicate that individual electrons are not individual at all, so when we observe individual electrons going through the slits, we are seeing what looks to us like separate pieces of what, in reality, is the whole picture? I think this implies the non-reality of time.
While we may agree that causality is not violated in this experiment, it does appear that particles are somehow acting outside of time, as we know it. I mentally find myself going back to the original single particle version. The Kim at al version of the experiment does not answer the question mentioned earlier of how individual particles shot one at a time, and with as long a delay as you want in between, STILL build up an interference pattern. Other variations of this experiment (they are on the Wikipedia page) have also proven that the order in which the particles are shot makes no difference whatsoever. The particles always randomly build up the pattern. There appears to be some kind of "space" in which these particles live all together as one and which is not observable by us. Richard Feynman has done more to conceptualize this space for us than any other physicist.
From Young's Experiment to Path Integral Formulation
Richard Feynman built on the results of Young's experiment to come up with the path integral formulation (PIF) of quantum mechanics. The classical behaviour of light (reflection, refraction, etc.) relies on the concept of a single trajectory of photons. PIF, in contrast, reflects not each actual photon's individual path but the path of greatest probability. I attempt to describe it in detail in the article The Behaviour of Light. Scroll down to Wave-Particle Duality in that article and read on from there, although I will describe it here as well. Like the Young experiment, this formulation is valid for all particles, including photons and electrons. It is mathematically derived from Schrodinger's equation.
An electron, for example, doesn't travel in a single straight line from A to B. It can potentially take one of an infinite number of paths, which can be curved or straight and can be any length, to get from A to B (and according to this theory it DOES take every possible path). The figure below illustrates just three of an infinite number of possible random particle paths. These paths are virtual paths rather than actual paths that the particle takes from A to B. That being said, neither does the particle move from A to B in a straight trajectory (the classical interpretation). The probability amplitude of the particle, however, does travel in a straight trajectory. I've drawn that amplitude arrow in blue below left.
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| adapted from Matt McIrvin;Wikipedia |
The path integral not only offers us a way to help us visualize the secret quantum world of the electron but it also offers insight into how it "experiences" space-time. The quantum world is symmetric with regard to space and time, a quality with huge ramifications. Our classical world is symmetric with regard to space but not time. We experience time as a one-way arrow, and all thermodynamics and mechanics are based on that forward arrow. A broken wine glass never rebuilds itself. Neither Schrodinger nor Heisenberg considered the symmetry of time in their formulations of quantum mechanics. The path integral formulation reproduces the Schrodinger equation and the Heisenberg equations of particle motion and not only shows that they are compatible with special relativity, where time is a variable (as in the Dirac equation), but it also shows that time is reversible at the quantum level.
The electron goes much further than "sensing out all possible routes" in the Young experiment. As a virtual particle, it takes all possible routes. As a real particle, it takes no route at all. The "real" route taken, the one we desperately look for in that double slit experiment, is nothing more than a probability amplitude. The only "real" particle is that which we detect, and as stated earlier, we can detect nothing more than the aftermath of a collapsed wave function, while the particle itself has already been absorbed or deflected and if it still exists, it exists in some other state.
The fundamental point about quantum particles: it is not the individual particle - a photon, a quark or an electron - that moves from place to place but the probability of that particle that moves. Put another way, all the particle behaviours we observe experimentally are actually our observations of their probabilities, and not of them as individual particles. The classical electron as an individual physical entity is gone. In its place is an interpretation of a quantum world inhabited by virtual particles which are no more than and no less than infinite potentialities of wave functions. Our senses contract this infinite and undefined reality into the classical world we comprehend. But the underlying reality is incomprehensible to ordinary scientific thought and a new approach is required. If we can even begin to think of the electron as a quantum entity, the fact that two particles separated by great distances can demonstrate quantum entanglement, a form of communication that far exceeds the speed of light, starts to seem just a bit more plausible.
The Electron Is Not a Physical Particle
So far we have explored the electron as an individual wave function and what this wave function means.
At everyday energy and in large numbers, electrons act like particles or waves, or sometimes both as in the case of the Young experiment. It depends only on how they are measured. But as we've seen, it is not the electrons themselves we are observing but their probabilities instead. We can now think of the electron as a probabilistic wave function. We have done what we can to build a concept of a wave function in our minds. However, it is tempting at this point to now think of the wave function as its own separate and distinct entity. Two interacting electrons, then, are simply two wave functions acting on each other across a distance. As I hope we will see, this too is a way of thinking that will require updating.
In addition to the Young double slit experiment, we can explore electrons and other particles, as well as atoms, at energy extremes. Particles near the coldest temperature possible, absolute zero, betray their secret quantum nature, and we can take advantage of that. In fact, it turns out that at such ultra-low energies, the curtain of random atomic kinetic movement pulls back just enough to reveal what might be the true nature of matter.
Do Electrons Stop at Absolute Zero? Is There Such A Thing As A Stationary Electron?
An electron can theoretically reduce to what most physicists would call a zero-energy state, where it would be perfectly stationary, even though some energy would be present as a mass equivalent (and there is irreducible momentum in its intrinsic spin). Experimentally, an electron is never found to be motionless. A Penning trap, which uses strong magnetic and electric fields to trap charged particles, can confine an electron's movement to a small elliptical-like plane, but it cannot suspend it in a completely motionless state. Cooling the electron can further reduce its kinetic energy, but an electron as a wave function can never maintain an absolutely still position. Thanks to its quantum mechanical nature, its position cannot be perfectly localized.
What happens to an electron in an atom when that atom is cooled to absolute zero? Can an electron in an atom ever slow down and stop moving? Until recently, it was thought that absolute zero could not be achieved but it could be approached using a variety of processes. Laser cooling, for example has achieved temperatures of less than a billionth of a kelvin. We expect all molecular motion to stop at absolute zero, as individual atoms achieve the lowest ground state possible. Atoms should stop moving at absolute zero because such a system should possess zero kinetic energy from the random movement of atoms. There should be no state colder than absolute zero.
An atom, which possesses its own wave function, will behave like any other quantum mechanical particle. Thanks to the either-or nature of quantum mechanics, once an atom's velocity reduces to zero, its location wave function should expand out toward infinity (this would also theoretically happen to an electron's wave function). An atom cooled to absolute zero should theoretically exist everywhere at once across the universe. Matter would not be recognizable as we know it.
This prediction was recently tested. In 2013, German researchers not only achieved absolute zero, but they went below it, a feat that seems logically impossible. They achieved negative kelvin temperatures in gas atoms, with mind-blowing results. Temperature depends on the kinetic movement of atoms (it is the measurement of their average kinetic movement) but it also depends on pressure and potential energy as well. By cooling atoms to a within a few nanokelvin and controlling their behaviour with magnetic fields, the scientists could manipulate these aspects of temperature. By maintaining negative pressure and limiting potential energy by introducing a special atomic lattice, they created something that was previously thought impossible - a negative kelvin environment.
What they found was astonishing. At everyday temperatures, atoms tend to settle in a state of minimal allowable kinetic energy. Some atoms have a little more energy and some have a little less, but on average they settle at the lowest possible potential energy state (this is called a positive Boltzmann distribution), and the temperature reflects their average kinetic energy. At (theoretical) infinite temperature, atoms are expected to exhibit an even spread of possible potential energies because all energy states within the upper and lower bounds are equally possible. At negative kelvin temperatures, they observed that atoms settle at the upper limit of their potential energy, exhibiting a negative Boltzmann distribution, an entirely unexpected result. In a sense, negative kelvin temperatures are not cold at all. They are hotter than infinite temperature. If a negative temperature system comes into contact with a positive temperature system (at any temperature), heat will flow from the negative system to the positive system. Negative temperature systems are also unique in regard to their dynamics. Instead of contracting under the force of gravity (as positive temperature systems do), negative temperature gases are stabilized against contraction, and this offers a possible clue into the nature of dark energy, the accelerating expansion of the universe.
According to this new research, it appears that sub-zero atoms not only do not stop but they speed up to their maximum potential (and kinetic) energy limit. Does this mean that there is no reality to the idea of an atom at full stop at absolute zero, with a corresponding wave function as large as infinity? Yes, for a single atom this would be true because negative kelvin temperature is possible, but it is possible only for systems that possess limited possible energy states, such as a tiny sample of a few gas atoms. Any larger system of atoms or particles cannot achieve negative kelvin temperature.
What does this mean for the wave function of the electron at absolute zero? One model, the model of a Fermi gas, predicts the behaviour of electrons in metals at absolute zero. It is odd to think of a gas state within a metal, but electrons in metals exist as a degenerate electron gas, which means that atomic nuclei share electrons equally. These free electrons behave like a gas of particles. Even in the coldest system approaching absolute zero (0 kelvin or -273.15°C), there is still movement. A gas of non-interacting electrons, for example, when cooled almost to absolute zero, is still filled with electrons whizzing around at great velocities.
Electrons are fermions. They are spin-1/2 particles and, as such, all electrons must occupy separate quantum states. This requirement is what contributes to electrons having very high velocities even at absolute zero. They still possess a maximum energy at absolute zero, called Fermi energy. In ultracold metal atoms, all electrons find the lowest possible energy configuration they can. Fermi energy is the energy of the highest occupied electron energy shell in this configuration (the kinetic energy or temperature of these electrons is estimated to be around 80,000 K). Because electrons are fermions, they cannot all fall down together into the same lowest possible energy shell because only two electrons (one spin up and one spin down) can occupy each orbital within a shell. Electron velocity increases as one moves to higher energy shells in an atom. The temperature reflects the electron velocity of the highest energy occupied shell. At absolute zero, the wave functions of these electrons are not affected beyond maintaining a lowest possible energy state. Fermi energy is why electrons inside atoms never stop. A free electron, however, can approach a near-stationary state, however, as described earlier.
Even an electron in a stationary state in a classical sense "moves" in a quantum mechanical sense. Stationary means that the particle has a constant probability distribution for its position, velocity, spin, etc. in a static environment. This does not mean that the wave function itself is stationary however. It is a standing wave, which means it is always changing its complex phase factor. According to de Broglie's relationship, the energy of this wave, which cannot be reduced, is its oscillation frequency times Planck's constant.
The Answer is No and No
You can always slow down a physical object, like a buckyball, to a full stop (at least in the classical sense!). But a particle with a single wave function, such as an electron or even an atom with its own wave function cannot achieve quiescence. Particles have quantum mechanical properties that are fundamentally by their very nature always in motion. Quantum mechanics means that a particle's position cannot be maintained precisely. Even if it could, the particle's wave function is always in motion. Third, an electron has fundamental spin, and in this sense too it always possesses an intrinsic momentum. An electron at full stop is no longer an electron.
Absolute Zero: A Glimpse Into the Weird Quantum World
The quantum world, in which objects behave as both particles and waves, where matter can be in many places at once, where time runs forward and backward and behaviours are based entirely on probability, phenomena are far too tiny to observe directly. A Bose Einstein condensate (BEC) is a rare case in which quantum mechanical effects (the rules of the very tiny quantum world) emerge into the classical world (where the rules of the large rule).
In a BEC, the wave functions of millions of atoms merge into one "giant" wave function about a millimeter across (large enough to observe). Atoms cooled to within a few billionths of a degree above absolute zero and held together in place by lasers and magnetic traps turn into one single wave function. Interestingly, only atoms in which the sum of their neutrons, protons and electrons add up to an even number can be made into BECs.
In an ordinary gas of sodium atoms, for example, the atoms fly around randomly in all directions. In a BEC (also called a Bose gas or an ideal quantum gas), those atoms move together in step as one, like soldiers marching in formation. A BEC is "a single matter wave propagating in one direction."
This is no ordinary matter. When two BECs are added together they do not mix like a gas does. Instead they interfere with each other like two overlapping waves. Within, atoms disappear and reappear somewhere else inside the BEC based on constructive (appear) and destructive (disappear) interference.
In classical mechanics, a collection of identical atoms in a gas can be, in theory, distinguished from one another and each atom's trajectory, velocity and location can be described at any point in time. In theory, the evolution of this collection of gas atoms can be calculated precisely. In quantum mechanics, however, Heisenberg's uncertainty principle makes its presence obvious. The evolution of a quantum mechanical system of gas atoms cannot be determined precisely. The trajectory of each atom cannot be calculated as a precise value, and as the system evolves, that precision grows even fuzzier. Ultimately a collection of atoms is a chaotic and unpredictable system.
And yet, this collection of atoms, at a cold enough temperature, exhibits a level of organization never observed until fairly recently in physics. Near absolute zero, the system's energy is low enough to significantly dampen atomic jiggling. Atoms become almost still. This system of atoms is called a Bose gas. The collection is so "calm" that its underlying fundamental quantum reality is able to reveal itself, and here we see atoms acting like the wave functions they really are. Under the right conditions these wave functions line up in step with each other and form one huge wave function, which in every quantum mechanical way, is one huge atom. This is a Bose Einstein condensate, or BEC. Atoms (fermionic composite particles), bosonic gases (diatomic molecules that act like bosons) and even photons (bosons) have all been successfully condensed into BECs. Ordinary everyday atomic motion obscures the quantum mechanical reality that is always present underneath.
This reality is, to put it lightly, odd. This reality is composed not of objects, time and distances but of probability distributions. And it implies that all particles are interwoven into each other, at least to some extent, through wave functions that cannot be thought of as distinctly separate entities since each one potentially extends to infinity. Wave functions under the right conditions also overlap with each other (and in the case of a BEC they completely overlap into one wave function). We only observe a particle such as an electron as localized because the probability of the electron's location being far away from its cloud drops off very quickly. We observe an atom as localized for the same reason.
The Quantum Vortex and Intrinsic Quantum Spin
BEC's demonstrate another fascinating quantum phenomenon called quantum vortices. Such vortices may hold clues about how particles such as electrons come into existence. As we move toward thinking of the electron as a probability distribution, a new question comes to us. How do these localized distribution clouds arise in the first place? Why isn't the universe a vast homogenous sea of probability (or is it)?
A vortex in a BEC can be created much like any vortex - by stirring the condensate with lasers or by rotating the confining trap. Both actions are like stirring a cup of coffee with a spoon. They are small but detectable - the core of a vortex can be up to ten micrometers across. Unlike macroscopic continuous vortices (such as tornadoes), BEC vortices are quantum in nature. In BEC's the vortex carries quantized angular momentum. Vortices can also exist in superconductors, where the vortex caries quantized magnetic flux.
At the center of each quantum vortex is a phase singularity around which the phase changes by a multiple of 2π. This 2π winding is exactly the same phenomena as the de Broglie matter wave. If you pick any point in space and call it 0 and then go around in a circle, coming back to 0, the phase has changed either by 2π or zero. The phase singularity at the center of a quantum vortex means that there are zero matter waves there. There is no superfluid in the core but a vacuum or nothingness instead, even though there is no pressure-related reason why it cannot be occupied (the core inside a tornado is filled with air).
What is a quantum vortex mathematically? The Schrodinger equation is a non-linear equation that describes waves in a medium. In one dimension, it describes something called a soliton, a self-reinforcing solitary wave. It is a localized standing wave that does not change shape with time, and it acts like a particle. A 2011 paper by Manfried Faber, for example, describes particles as stable topological solitons. In two or more dimensions it describes a vortex. Both solitons and quantum vortices are observed (as macroscopic excitations) in BEC's.
What Do Quantum Vortices Have to Do With Particles?
Intrinsic Quantum Spin Sets Up Our Story
Do quantum vortices exist in all quantum mechanical matter waves, hidden within all matter - are they behind the intrinsic angular momentum of each particle? Most particles have intrinsic angular momentum, or quantum spin, and this alone is a concept that is very difficult to grasp in a common-sense way. We often come across spin-1/2 particles called fermions (which includes the electron) and spin-1 particles called bosons (such as the photon) in particle physics. The Standard Model of particles also now includes the Higgs boson, a spin-0 boson.
Intrinsic quantum spin is not the kind of spin like a spinning top or a spinning ballerina. It is quantized in particles, which means that it is either 1/2, zero or one, or some multiple of these values, but never in between, whereas a spinning ballerina slows down or speeds up on a continuum, transitioning smoothly between spin rates. The quantum spin is also unique because it is intrinsic. A ballerina exists when she is not spinning. The spin of a particle is part of what makes it what it is. How do we even know that the electron, for example, has spin? You can prove it by working backwards from a macroscopic example. If you take any electrically charged object and spin it in the usual sense, you create a magnetic field around it. If you know the charge of an object and how fast it is spinning you can calculate its magnetic field. Now, if you measure an electron's magnetic field and its charge you can find its minimum spin rate (in the usual classical sense!). This little test at least tells us that the electron must be spinning, but it is not the whole story. There is a big problem.
The concept of intrinsic quantum spin comes from Dirac's equation. His new version of the Schrodinger wave equation incorporated special relativity. This relativistically invariant (this means that the charge of an electron is the same no matter how space-time transforms around it) equation offers an elegant solution to a problem with the magnetic field created by the electron. A rotating charged particle gives rise to a magnetic field. The problem is that the electron, then thought to be a physical particle with a size, has to very small. It has to have an upper radius limit of 10-22 meters, based on experimental results. A sphere this small would have to rotate faster than light speed to account for the measured strength of is magnetic field. According to special relativity, nothing can move faster than light speed. So how is this possible? This is yet another hint that the electron does not "live" in our classic three-dimensional world.
In Dirac's equation, the electron is no longer treated as a classical particle with size. It is a zero point particle that correctly predicts the observed magnetic field strength, or magnetic moment, of the electron. The closest thing to visualizing this is to think of an electron as a point in space spinning at infinite speed, but this visualization is incorrect and we need once again to go to the math behind it to get our best sense of what's going on.
The electron's intrinsic angular momentum is treated mathematically as something called a spinor. A spinor is like a vector or a tensor because it can describe a point that transforms in (usually three-dimensional) space. To understand a spatial vector for example, think of a thumbtack stuck in a flat sheet of cork. Let's tie a string to it at one end and tie a marker to it at the other end. Make the string taut and draw a point (at zero). Now you can draw a circle around the tack. You will find that you have to travel 360 degrees before you get back to point zero. A spinor is different. It takes a 720 degree rotation to get back to point zero. This obviously isn't the normal two-dimensional space of our cork sheet.
Are Particles Built Into Space-Time? Is It Einsteinian?
The 720 degree example lives in something called complex vector space. We can visualize ordinary two-dimensional (the flat cork sheet) and three-dimensional (a cube made up of cork sheets) vector space. We cannot visualize complex vector space. Ordinary vector space is made of real numbers, such as 4 or 6.7. Think of a coordinate system with x, y and z vectors. You can define the location of any point by giving it a location along each of the x, y and z axes. Complex vector space, however, is made of complex numbers, in the form a + bi where a and b are real numbers and i is an imaginary number that satisfies the equation i2 = -1 (you can't visualize it). Why go to this length to describe the electron spinor? This seams like a lot of unnecessary complicatedness but it allows you to solve equations that cannot be solved using real numbers alone, and it has many practical applications in physics as well as other sciences. Where does an imaginary number live? Nowhere we can conceptualize, and that is, in a nutshell, why quantum mechanics does not allow us to enter, at least not with our common sense.
The intrinsic spin of an electron traveling well under the speed of light is described as a spinor in three dimensions. A relativistic electron, one travelling close to light speed, is in contrast described by a Dirac spinor in four dimensions. This is the part of the Dirac equation that uses the Lorentz metric, the metric that intimately links space and time together and allows time dilation in special relativity. It is four-dimensional real vector space, often called Minkowski space.
It's intriguing, at least to me, that the spinor, a beast living in complex space, somehow can live in real-vector (no imaginary component) space-time. The mathematical reason it can is that complex numbers can represent an ordered pair (two points you can map) where sum and scalar product work just fine and can be mapped into real vector space.
We explored complex numbers and complex space in a previous series of articles called The Fractal Universe. There we found out that we can use complex numbers to describe spatial dimensions that "live" in between the whole integer dimensions (one, two and three dimensions, and so on) we are familiar with. These complex numbers create a geometry called fractal geometry. Many examples from nature as well as the Dirac electron, itself at the heart of all nature, suggest that the natural world may be better described in a non-linear, non-differentiable and non-Euclidean framework. Most theory in physics (including Einstein's space-time), in contrast, is linear, differentiable and Euclidean, owing to a centuries-long tradition of using these mathematical structures.
Complex mathematical structures such as intrinsic quantum spin may seem esoteric and hidden to us but they are not. Consider the rotation of our planetary system and the rotation of our galaxy. Both examples of classic spin owe their origin to the summed intrinsic angular momentum of all of the countless particles making them up.
How do quantum vortices correspond to two built-in properties of particles, quantum spin and magnetic moment? Is the particle itself some kind of stable quantum vortex in space-time, with no physical reality beyond its mathematical geometry? If so, what is doing the (quantized!) stirring and what is the medium being stirred? However these questions are answered, they seem to point us toward the possibility that the physical world we experience and measure is not fundamental reality, but instead an artifact of our perception.
Conclusion
What do we make of the electron? We find that all the matter around us, including ourselves, owes its properties and behaviours, in large part to the electrons within each atom building block. The physical, chemical, electrical and magnetic properties of all matter owe themselves to the electrons inside atoms. We expect the atom to be a physical building block and the electron to be a physical sub-component of the physical world we experience every day, and we are jolted to find it is not. The electron and the atom do not operate according to everyday classical mechanics and dynamics but according to quantum rules instead, in a world we cannot visualize or even conceptualize, where time and space lose their ordinary meaning and uncertainty means that rules are meant to be broken. This seemingly chaotic world translates into a reliable classical world at larger scales as a wonderful consequence of all possibilities averaging out. The stability of objects and the predictability of motion and forces come not from what individual subatomic particles themselves are doing but instead from a cloud of averaged activities. We cannot even know what individual particles are doing, not only because they cannot be pinpointed precisely but also because they are doing what they are doing within a geometry of space we can't possibly visualize. In fact, they themselves may BE the geometry we can't visualize.
The electron (like all other particles) is a wave function. Does it even have a physical reality? If it doesn't, doesn't this mean that the universe, composed of particles of matter and energy, is nothing more than a collection of mathematical formulas and probabilities - wave functions? Or, does it mean that there is an underlying fundamental physical reality that we do not have the conceptual tools to experience?
Philosophical Implications of the Electron as a Mathematical Particle
Are We Looking In the Right Place?
When faced with the questions - what fundamentally IS matter and energy, and what ARE these particles the search into their deepest nature leads first to Max Tegmark, a very well known American cosmologist, who holds to the idea that reality at its heart is purely mathematical. In fact, he goes further to postulate that all possible mathematical structures exist in the physical universe and that we, as self-aware mathematical substructures ourselves, perceive ourselves living in a physically real word. The physical world, therefore, is nothing more than a human perception or a human interpretation of a purely mathematical reality.
Is Tegmark right or not? And must reality either be physical or mathematical, or can other possible explanations exist? I recently read a book called The Quantum and the Lotus, where this troubling question is turned on its head by two scientifically trained men who are also intimate with Zen Buddhism. This book inspired me to wonder if HOW we think about quantum mechanics might be part of our problem with understanding it.
Do we think that if we can peer into the workings of the universe carefully and closely enough, we will be able, at some point, to see how it all fits together? Will we have one unified logically consistent theory of everything? Tellingly, I think, Wikpedia has two entrees for the theory of everything - one scientific (the previous link) and one philosophical. Both are worth a close look.
The electron works within special relativity. The next logical step for some is to get the Dirac equation to commute with general relativity so that we can describe the electron and other particles at the quantum level living within the gravitational fabric of curved space-time. Others look for a purely quantum gravitational reworking of space-time or for a string theory version of gravity. Either way, this largely mathematical approach assumes that a logical unified theory exists for all observable phenomena.
Other investigators are peering deeply into the very substance of space-time. One tactic is to look for extra (real vector) dimensions in space-time, as predicted by string theory. Another tactic may be to look within the dimensions of complex space. Fractal theory is beginning to make fascinating strides toward understanding what goes on in the hidden world of particles. L. Marek-Crnjac presents a brief history of the development of fractal space-time theory, which serves as a good introduction to it. Again I suggest a look at my series of articles called The Fractal Universe. Can this approach describe physics at the very large scale, and what should we look for as evidence?
How Are We Looking?
It is almost as if we are not hearing what the Dirac equation has been telling us for decades. The electron confronts us with mind-shaking consequences that we cannot ignore.
For example, what happens to locality, the principle that states an object is influenced directly only by its immediate surroundings? Locality is the benchmark of classical physics. Interestingly, Newton's classical laws of gravitation, formulated in terms of "action at a distance," violated this principle and Newton himself knew it. He was certain that there must be some kind of mediating material to transmit gravitational force. Later, Einstein successfully reformulated gravity into his theory of general relativity. It obeys the principle of locality because it evokes a stretchy and bendable four-dimensional space-time fabric that pervades the universe. He was never a fan of "spooky action at a distance." Ironically, Einstein himself contributed significantly to the then-new theory of quantum mechanics and he famously stated that this theory must be incomplete somehow. There must be one or more hidden variables that would restore its locality. The idea that a particle such as an electron is really a vortex-like structure implies, I think, some kind of turbulent medium pervading the universe, an idea that is at heart localist. The quantum mechanical electron is non-local. Over the past few decades, a series of many experiments, called Bell test experiments, have convinced most theorists that the precept of locality is no longer valid. Experiments focused on the quantum entanglement of various kinds of particles convincingly tell us that the universe is nonlocal. The non-locality inherent in quantum mechanics brings up many questions about how time works. It has been shown that two separated entangled electrons, for example, would have to communicate faster than the speed of light. Either this, or particles in the quantum world do not experience the constraints of time and space that we perceive. It seems to me that the double slit experiments tell us that the latter is more likely.
Realism is the belief that reality is independent of our measurements and conceptions of it. The universe, therefore, exists independently of us. With realism comes the promise that, given careful enough observation and measurement, we will eventually come to a full understanding of this reality. In the sciences, we are trained in a manner that is both localist and realist. Objects, energies and forces are thought of as out there separate from us. Systems can be isolated, identified and manipulated. When some of us get into quantum physics, however, this approach is not only no longer useful but it can be a significant hindrance to our ability to understand the nature of the universe. Einstein and even Feynman and Tegmark are realists. Einstein was a physical realist. Feynman and Tegmark can be described as mathematical realists, in that a mathematical reality exists independent of us. The looming question: Is the electron real? An enormous collection of data tells us that the electron is not real in the physical sense in that it does not exist independent of our observations of it. Is it real in the mathematical sense? Is the Dirac equation the reality? Is the potentiality of the electron a peek into a deeper and far more obscure reality than we can imagine?
What of the possibility that, while physical experiment and mathematical theory offer us a framework in which to explore the workings of the universe, neither make contact with the true bedrock of reality? Calling reality bedrock or even calling it a word is, in the strictest sense, realist. Here we begin to see just how difficult it would be to be truly non-realist in our approach to quantum mechanics, where reality is no-concept. This may seem extreme but the point I wish to make here is to be cautious of where we hang our hat, or where we attach to one form of realism or another. Is our attachment necessary? Is it valid? Are we aware of our assumptions when we investigate the secretive quantum universe?
Quantum mechanics proves that particles obey neither locality nor (at least physical) realism. The electron, by its very nature, is neither here nor there. It is not strictly bounded by location or momentum. Its reality is blurred into the realities of other particles. There is no physically real electron hidden somewhere. The electron cloud is both empty and full - a startling and peculiar state of being that is foreign to most of us scientific minds, but is intimately understood by those, for example, well versed in Zen Buddhism. To understand this state of being, we will need to transcend the way we think. Deductive reasoning, so practically useful and sharply honed in the minds of physicists, gets us partway to understanding the electron.
We need to spend some time feeling out the uncomfortable space of the Dirac electron, to acquaint ourselves with its seemingly paradoxical nature, and dare I say it as a scientist, acquaint ourselves with our own natures. Are we getting in the way of ourselves in our quest for the "real" electron?
Next, check out Interpretations of Quantum Mechanics.
