All aboard! The next section of our journey... we delve into the strange quantum world of particles, waves, and fields. Pack a blanket, a thermos, and some sandwiches, it’s a bit of a head wrecker !
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The Fragile Sea Quantum Series
In Q01 we looked at sources, interpretations of the standard model of particle physics, and the establishment of scientific theories.
In Q02 it’s time to travel into the wonderful, strange, world of particles, waves, and fields, on the road to quantum computing.
Let’s go!
Particles, waves, and fields
The standard model identifies 17 elementary particles [1] , though there are more not on the list (explained below), and other conventions on how they are grouped or counted [2] , [3] . Elementary particles have no known inner composition; it isn’t known if they can be split further into sub particles (The correct term is ‘subatomic particles’).
There are many more composite particles in the zoo, over 200, made up of combinations of the elementary particles, many of which are highly unstable and exist for extremely short periods, often less than a millionth of a second. The more unstable particles are “typically produced in high-energy collisions within particle accelerators or through interactions with cosmic rays” [4].
Each particle, elementary and composite, belongs to one of two groups with certain characteristics separated by an internal property called spin (see below). Informally, fermions build matter and bosons mediate forces, approximating to "matter particles keep us moving", and "force particles keep our atoms from collapsing".
In addition, every type of particle of "ordinary" matter is associated with an antiparticle “with the same mass but with opposite physical charges (such as electric charge)” [5] . Antimatter is a substance made of antiparticles bound together [6].
If a particle is elementary, it’s mirror antiparticle is also elementary but not counted in the standard model’s 17. Also, there are six types of quarks (called "flavors") counted in the 17, each in three ‘colors’ that are not counted (red, green, and blue). Color has nothing to do with visible color - it's an abstract property related to the strong nuclear force [7].
‘Color’ and ‘flavor’ are typically formalised in standard texts using American English.
Briefly, of the 17 fundamental counted particles, 12 are fermions and 5 are bosons. They are explained and visualised well in this Quanta Magazine article by Matt von Hippel [8], and also in this: New map of all particles and forces, also published by Quanta Magazine [9].
Particles ain’t particles
The quantum realm is a roiling world alive in phenomena that confound rational sense. The idea of electrons as little balls rotating around a nucleus may have been helpful in earlier stages but is misleading. Quantum particles (electrons, photons, etc.,) are field disturbances or excitations that manifest as localized detections when measured [10], [11]; their position and momentum can only be predicted as probabilities in diffuse clouds, and described mathematically by wave functions.
Louis de Broglie was awarded the Nobel Prize in 1929 for his work proposing that every moving particle is associated with a wavelength, and that every quantum entity can be described as both a particle and a wave, now known as wave/particle duality [12].
The wave function describes probabilities of where the particle (or excitation) may be and where it might be heading (its momentum), though these two properties cannot be measured at the same time. Measuring one makes the other indeterminate, a central plank of the Copenhagen Interpretation and the Heisenberg Uncertainty Principle we met in Q01 (also see below).
The term ‘orbital’ was repurposed semantically in quantum theory where electron positions are defined by a probability cloud rather than a set path. The name ‘atomic orbit’or 'orbital' stuck however, and now refers to a spatial region described by a wave function, not a path taken by an electron as a tiny ball. The electron is diffuse across the orbital, existing as a probability wave without a classical trajectory [13].
Take the simplest atom, hydrogen, with only one electron and one proton in the nucleus. There’s a lot of space in there! If I stand in a field holding a tennis ball, and assume it is the hydrogen nucleus, the edge of the electron’s cloud – its probable excitation location, is around 6.7km (4.1 miles) away, and the electron is roughly 100,000 time smaller than the tennis ball nucleus [14].
Try to imagine a unit of energy 100,000 times smaller than a tennis ball, 6.7km away, but bound to the tennis ball by the electromagnetic force that binds the negatively charged electron to the positively charged proton in the nucleus.
A particle can transition between energy levels
There are defined energy levels (or shells) at fixed distances from the nucleus, within which are specific spatial distributions (orbitals) where electrons are most likely to be found, based on some limitations and rules summarized below.
By absorbing the right amount of energy (usually from a photon), an electron can transition from its current energy level to a higher energy level (‘excitation’), or, if the electron loses energy (normally by emitting a photon), to a lower energy level. This is how hydrogen emits light [15]; multiple photons can be expelled or absorbed, either simultaneouslty or in rapid succession.
Photons are spontaneously created from the transition energy and in total, match the difference between the two energy levels involved in the transition, according to the law of the conservation of energy which holds that energy cannot be created or destroyed, only converted from one form to another [16].
When an electron transitions between energy levels, its wave function shape changes, but there’s no classical orbital motion along a path between levels. The electron does not "travel" from one orbital to another like moving through space or orbiting a nucleus; instead, its quantum state changes instantaneously [17].
Most discussions of quantum absorption and emission refer specifically to electrons in atoms absorbing or emitting photons [18]. For composite particles, there are additional transition factors which can be quite complex, for example, not just changes in electronic states but also in how atoms move relative to each other (vibrations) and their overall orientation (rotations).
Spin and orbital angular momentum
All particles, elementary and composite, have a universal property called spin angular momentum. With electrons, for example, described as “not due to the literal spinning of the electron about its own axis, but an intrinsic property of the electron” [19]. Spin can have a projection such as spin-up or spin-down, critical states for quantum computing we’ll meet later.
Particles can also exhibit another property called orbital angular momentum (OAM) [20], depending on their state and momentum. OAM is due to the motion of a particle (or a field) in relation to an axis, normally for an electron with respect to the nucleus, as distinct from spin angular momentum.
But electrons do not orbit the nucleus like planets around the sun. Atomic orbitals describe regions of space within fixed energy levels where there is a certain probability of finding the electron, and, as the definition implies, at that probable location the particle has an angular momentum with respect to an axis, normally the nucleus. This serves as the reference point for angular measurements.
OAM is quantized - only certain discrete values are allowed [13], and perhaps the word ‘orbital’ should be dropped from the definition. It is inherent in the mathematics of quantum mechanics, regardless of any intuitive path, as noted by Hu et al 2025: “as modern technologies increasingly rely on tailored quantum state manipulation, developing methods to control electron OAM in neutral atomic systems has become a critical area of research” [21] – see also [22], [23].
More on orbitals
The Pauli exclusion principle applies to fermions only, (matter particles) and states that no two fermions can occupy the same quantum state simultaneously in an atom. Though a bit complicated, in practice it means that (for example) each orbital around a nucleus, where an electron is likely to be found, can hold a maximum of two electrons but they must have opposite spins [24]. No two electrons in an atom can share the same set of four quantum numbers (energy level, orbital shape, orientation, and spin), and the quantum state each electron occupies in an atom has unique properties [25].
There can be more than one orbital at each energy level, and the distinct patterns for where an electron might be found vary mathematically and visually [26], as follows:
‘s’ orbitals are perfectly spherical and can hold up to two electrons in each orbital, starting from the first energy level. They are not described by a trajectory circling a nucleus, but, as with all other electrons, their position and momentum are described by a probability distribution.
‘p’ orbitals look like dumbbells with two lobes (a figure of eight), that appear from the second energy level, with three p orbitals per energy level for a total of six electrons.
‘d’ orbitals have four types that are cloverleaf in shape (with four lobes), and one type with a doughnut shape ring around the middle and two lobes. They appear from the third energy level, with five d orbitals per energy level for a total of ten electrons.
The graphic representations in different rotation planes are pictured well here [27].
Orbital levels are filled from lowest to highest energy, even for atoms with less electrons than can fill energy levels (like hydrogen with only one electron). Particles in bound systems – e.g., atoms, up to large everyday objects and beyond to stars, (limited to a certain size and mass), always seek to occupy the lowest available energy state, thereby also tending towards trying to achieve maximum stability [13] [28], [29].
Galaxies and beyond are more complex; energy minimisation still applies, but gravitational dynamics, dark matter interactions, and other cosmic phenomena come into play [30].
The fundamental reason why there are different elements in the periodic table is because fermions (specifically, electrons) occupy quantized atomic orbitals according to the Pauli exclusion principle [31]. Transmutations between elements in the periodic table do not necessarily require high energies - natural transmutations are common in nature, especially in elements found in the decay chains of uranium, thorium, and potassium. Some transmutations arise internally from radioactive decay, others externally from nuclear reactions, for example, elements heavier than iron, such as gold and lead, require processes and energies that can occur in supernovae [32] (Wikipedia).
Bosons are different
Bosons have no such limitations with orbitals; they can happily bunch together, and any number of identical bosons can occupy the same quantum state together. This can lead to behaviour such as superfluidity (frictionless flow), at very low temperatures [33], and lasers (we’ll discuss how lasers utilise photons of the same energy, phase, and direction in Q05).
Fermions can also form superfluids and become superconductive, but it is rarer and more difficult. They must first pair up (‘Cooper pairs’) to form composite bosons which can then condense. Since spin states determine whether a particle is a fermion or a boson, they remain as individual fermions, but their composite spin state can equal a bosonic spin state, thus enabling the potential for superconductivity and superfluidity [34]. Bosons cannot become superconductive on their own, a fermionic pair has to be formed first [35].
Bosons and Fermions can be massless or massive [36], elementary or composite.
The Higgs and fundamental forces
To the sharp eye, five elementary bosonic particles which carry force, and four fundamental forces don’t add up. Well, this is subtle and can be confusing.
First the force carriers:
1. The electromagnetic force by the photon,
2. The strong force by the gluon,
3. The weak force by three bosons: W+, W−, and Z, (that totals 5 already), and
4. Gravity by the undiscovered graviton, which is not part of the standard model.
The fifth elementary boson is the Higgs, which does not mediate a force in the traditional sense. It was predicted by Peter Higgs and other physicists in 1964 [37], and discovered in 2012. The Higgs is responsible for generating mass for most elementary particles in the standard model, however most of the mass for composite particles arises from the strong nuclear force, being the mass generated from energy E = mc2 [38], [39].
By convention, it seems that W+ and W- are not split strictly in the counting, and gravitons are still hypothetical and not included in the standard model. So, “most presentations list the Higgs as the fifth boson after the four force carriers”, without noting the graviton [40]:
1. Photon
2. Gluon
3. W⁺ boson and W⁻ boson
4. Z boson
5. Higgs boson
And the Higgs is not listed of course in the Force table, where all three weak force bosons W+, W-, and Z, are listed together
1. Electromagnetic – Photon
2. Strong – Gluons
3. Weak – W+, W-, Z
4. Gravity – Graviton
It’s a bit hard to unpick otherwise when reading different tomes.
As a fascinating aside on the Higgs, an article by Matt von Hippel in Quanta Magazine discusses the conjecture that the Higgs might not be at its lowest energy state: “in that case, the Higgs field should eventually tunnel to that state, or “decay.” This decay would start in one place and then spread, a spherical bubble growing at the speed of light, transforming the universe… atoms would collapse” [41]. He assures us that this will not happen soon.
Ions
When more than two electrons ‘want’ to occupy the same orbital, the Pauli exclusion principle forces the extra electron (s) into available higher energy orbitals. If enough energy is applied from external sources, one or more electrons may be knocked out of the bond with the nucleus, forcing the atom to become a positively charged ion, known as a cation.
For elementary particles not bound (e.g., to an atom), such as an electron or photon, well-defined energy levels do not normally apply in the same way as when bound [42].
If the atom instead gains electrons it forms a negatively charged ion called an anion. Some solids exist in ionic form, table salt for example, zinc sulfide, and potassium chloride [43]. Ionic solids don’t conduct electricity but will do so if melted or dissolved in water.
Stranger properties still, of particles
Matt Strassler, theoretical physicist, educator, and Associate at Harvard, describes particles in his book Waves in an Impossible Sea in an illuminating way. He writes “We might say that a photon…(and other particles) is a particulate wave - a wave that, much like our naive idea of an elementary particle, cannot be disassembled into smaller pieces. Almost by definition, it has extraordinarily strange properties. Like any water or sound wave, a photon can spread out, even across a large room. Yet if it is absorbed into the walls of the room, a single atom, located at one microscopic spot somewhere on the room’s walls, will take it in wholesale, swallowing it in one gulp” [44].
Fields are real
Fields occupy the universe, everywhere, even in space. It is widely accepted that they are experimentally proven, though of course they can’t be seen. One strong indication, predicted in 1948 and observed in 1997, is the Casimir effect which exhibits an attractive force between closely placed metallic plates in a vacuum and arises from quantum field fluctuations in empty space [45].
In the vacuum of space, fields have certain properties. Paraoanu 2014 notes that: “quantum physics allows us to detach properties from objects. This has consequences: one does not need pre-existing real objects to create actual properties, and indeed under certain perturbations the quantum vacuum produces observable effects such as energy shifts and (the) creation of particles…” [82].
Fields – unusual insights
Matt Strassler also has deep insights on fields. His book Waves in an Impossible Sea is closely aligned with this link [46]. In the book, when he gets to the subject of fields, he notes “These conundrums and paradoxes are profound and troubling… The concept of fields will help us address them… But don’t expect comprehensive, satisfying resolutions to all these puzzles. I don’t have them. No one does. With that, we come to the most difficult section of this book. It was challenging to write, as I knew it would be, and it is challenging to read. It introduces concepts that are unfamiliar, eerie, and slippery even for physicists. One thing I’m sure of: before I became an expert myself, I would have needed to read this material twice. The subject is full of strangeness, and if you have trouble making sense of it, remember that physicists do, too” [44]. It’s a terrific book.
Sean Carroll’s book Quanta and Fields is similarly absorbing. He notes: “In classical mechanics we distinguish between particles which have particular positions in space, and fields which take on values at every point in space. We can do the same thing in quantum mechanics. The quantum mechanical theory of fields is quite sensibly known as quantum field theory, or QFT for short. Some folks speak as though there was first “quantum mechanics“ and then it was superseded by “quantum field theory”. That's not accurate. QFT is part of quantum mechanics just applied to fields rather than particles. As of this writing, quantum field theory is the single best way we have of describing the universe at its deepest known level” [47].
Field chaos, and quantum or spacetime foam
The odd thing about fields is that at large scales they appear to be smooth and stable. Quantum effects can be measured there but the effects of quantum fluctuations are averaged out. At microscopic quantum scales, fields are dynamic and ‘roiling’ with rapid fluctuations, constantly "folding" and experiencing changes due to interaction, superposition (discussed in Q03), and the creation/annihilation of virtual particles. Fields are filled with quantum noise and unpredictable variations [48], [49].
John Wheeler introduced the term quantum foam in 1955, and developed it in papers and science narratives, including his autobiography written with Kenneth Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics, released in 1988 [50]. It is a theoretical concept only, useful for illustrative purposes and awaiting experimental evidence. He maintained that quantum foam (or spacetime foam) characterized the wild, roiling nature of spacetime at quantum scales, being fundamentally chaotic and turbulent. At these scales, the geometry of spacetime itself jostles and bubbles with energy, leading to constant creation and annihilation of virtual particles - a “foam” of fleeting, tiny entities and distortions [51], [52], [53].
Particles exhibit quantum uncertainty
Heisenberg’s uncertainty principle is central to all field theories, including the standard model and the Copenhagen Interpretation [54], [55]. As noted above, it’s a fundamental property of nature where certain pairs of physical properties cannot be measured at the same time – for example, the location and the momentum (how fast it is moving) of a quantum particle or entity. At the large scale of our observable existence, the effects are so small we don’t notice; at the very small quantum level, they’re significant.
But it’s not only a pair of physical properties that cannot be measured at the same time. The uncertainty principle generalizes to multiple properties, such as multiple spin directions. As the measurement accuracy increases for one, the precision for others decreases [56]. Quantum mechanics, Q field theory, and Q information science are also generally non-commutative, that is, changing the order of operations changes the outcome. Position and momentum are non-commutative, as are spin directions. Order matters in quantum - measuring one property disturbs another, which is unlike classical physics where most measurements are commutative (order does not matter) [57].
Uncertainty principle and wave function collapse
The uncertainty principle is often referred to as wave function collapse, but it’s not the same. Wave function collapse is about the observed behaviour, or what does happen when a particular property is measured.
Importantly, in quantum computing, because quantum bits (qubits) cannot have well-defined values for complementary properties simultaneously, the resulting probabilities allow quantum algorithms to explore solution spaces in parallel [58], in a manner not achievable in classical computing. We’ll discuss this in Q03.
How many fields are there?
Each elementary particle (electron, photon, quark, etc.) corresponds to its own quantum field that permeates all of space. These fields are real physical entities, not just mathematical abstractions [46]. The Standard Model of particle physics describes several distinct fundamental fields, for example, the electron field, quark field, etc., around two dozen. There are no more fields than elementary particles, and the standard model recognizes 17 elementary particles, however anti-particles and variants such as quarks with specific color states bump up the total.
The main point is, fields are more fundamental than particles, with particles viewed as their quantized excitations.
The wave function and entanglement
There is also the wave function. Most formal texts use two words, i.e., wave function, though there seems to be no hard convention, online it's expressed often as one word, i.e., wavefunction [59]. Every quantum system - such as a single particle or a large molecule, has a wave function describing its quantum state, including its possible position, momentum, etc., abiding by the uncertainty principle (i.e., momentum and location are uncertain and cannot be measured at the same time).
Entanglement is a quantum phenomenon where two or more particles in a group are found to have perfectly correlated physical properties when measured, even over long distances apart [60]. When multiple particles become entangled, they are described by a single joint wave function, which is the entire quantum state of the entangled particles [61].
It is probably the subject that has generated the most words in quantum physics, and the most angst; we'll discuss it more in Q03 and Q04 [62].
A universal wave function?
Finally, there is also a mathematical construct of a universal wave function which may or may not be accepted in all quantum models but is part of the many worlds interpretation. The universal wave function in the MWI contains the quantum state of the entire universe including all particles and fields, in all worlds, even if we can only ‘see’ one world. It embodies all possible configurations and histories of the fields and particles, not just subsystems or single worlds like our own [63].
The many-worlds interpretation of quantum mechanics posits that all possible outcomes of quantum measurements are physically realized in parallel, non-interacting worlds. Instead of a wave function collapse, every quantum event causes the universe to branch into distinct histories, each taking place in its own world. We see only one outcome, but all possibilities exist simultaneously, resolving paradoxes like Schrödinger's cat, by treating each result as real in its respective world. This is a contested theory, considered difficult or impossible to test even if passionately defended by many adherents [64], [65].
The important point is that fields are real, whereas the universal wave function is a theoretical mathematical construct.
Some current educators and physicists
Apart from the Strassler and Carroll books mentioned above, Carlo Rovelli’s book Reality is Not What It Seems [66], and Jim Baggott’s Quantum Reality [67] are also exceptionally well written. The interesting fact is that all four of these educators are wide apart in their positions on quantum physics:
Baggott is a former physicist and philosophical skeptic who does not advocate any single model but analyses their strengths and weaknesses. He’s also a persuasive and compelling writer, I’ve appreciated his objective critical approach over many years [68].
Carroll defends the many worlds interpretation (MWI) first proposed by Hugh Everett III in 1957 as part of his doctoral thesis at Princeton University [69]. MWI is a theory one could easily get lost in and never resurface, it’s highly persuasive [70], [71], but also presents many problems [72]. Some of Carroll’s writing can be profound [73], [74].
Rovelli is an elegant writer, often poetic, and a longstanding advocate of his own relational quantum mechanics interpretation. He believes there are no observer-independent facts - all quantum events are relational, meaning reality itself is perspectival and contextual. He also rejects the concept of collapse and does not accept the MWI or hidden variables [75], [76], [77]. Hidden variables will be explained along with entanglement in Q03 and Q04.
Strassler supports the standard approach (the standard model and Copenhagen Interpretation) and analyses the limitations involved. His writing is attractive and provocative, sometimes humorous [78].
I admit the list is highly selective, there are plenty of other physicists and writers who are also persuasive in their expositions. Not all of them focus on interpretations but on aspects of pure physics and teasing out what fundamental reality is or might be.
John Gribbin lit the spark for me, and I return to his writings with fondness, and Richard Feynman’s work is almost like a guiding hand, albeit the fields of discovery are never static. In Q04, we’ll meet Paul Dirac, Werner Heisenberg, David Hilbert, Tim Maudlin, John Bell, David Deutsch, and Anton Zeilinger (plus others).
What I generally gravitate towards is the absence of arrogance, growing weary when I encounter intemperate takedowns of other contributors. A sense of humour is always appreciated, as are an awareness of the human element in the face of this extraordinary reality (Feynman was particularly good at that), and a sense of fair play. Look how far we’ve come!
Still, the percentage of researchers who believe different interpretations of quantum physics, and their interpersonal engagements and disagreements, make fascinating reading. Elizabeth Gibney has written up this report in Nature, referenced in Q01, which surveys researchers about their views on different interpretations [79].
The train... pulls in to the station...
I hope you've lasted the distance! And thanks, if you have arrived here with your mind still intact. It’s a suitable waypoint for a rest, and the boarding station for the next journey. Rest up!
In Q03, we’ll uncover classical and quantum computing concepts in prep for Q04, where we get to grips with strange quantum behaviour, superposition, entanglement, interference, superdense coding, decoherence, and teleportation.
In Q05 we’ll return to particles in everyday life, energy, and telecommunications.
It’s time to go, but first, it’s summer, a poem in the Japanese classical waka form [80]:
When summer moon’s
Light lightly
Shines
From the running waters
Haze arises!
Kanpyō no ōntoki kisai no miya uta’awase 38[81]
Till next time, then. Take care, Brent.
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Full disclosure:
I own all the books referenced in this post and unless noted otherwise in the post, I have read them cover to cover.
All the written words, apart from acknowledged quotes, are my own. Some phrases and definitions are common to many sources.
For around 35% of primary research, I have used perplexity.ai to check multiple sources and gain a sense of the accuracy of my own writing, but the words written come from developing my own understandings and concepts over many years. All the papers and most of the web sources were already known to me or found myself before or during writing, without using AI.
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