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# Are metallic/ionic bonds weaker than covalent bonds?

In mineralogy class, I was taught that metallic and ionic bonds are weaker than covalent bonds and that's why quartz and diamond have such a high hardness value. However, in organic chemistry class, I learned that covalent bonds are weaker than metallic and ionic bonds, thus organic substances have a much lower melting point than that of metals and ionic compounds.

What am I getting wrong? Are ionic and metallic bonds weaker than covalent bonds or not?

• You're basically comparing an intermolecular force with an intramolecular force. – Mehrdad May 25 '14 at 7:46
• The problem with this "strength" is not well defined, and your teachers are using it in different sense. When bond dissociation energies are compared, ionic bonds can be pretty strong, to0: opentextbc.ca/chemistry/chapter/… – Greg Sep 17 '17 at 5:04
• There is an additional confusion here as compounds consisting of covalent molecules don't make crystals based on covenant bonding: the crystals are formed from weaker dispersion forces (van der Waals forces) between the discrete molecules. This are definitely weaker than ionic bonds or the covalent bonds in network solids like diamond where every atom has a covalent bond to other atoms. – matt_black May 29 '18 at 15:06

Quartz and diamond are stronger substances because their molecules form network covalent structures. These structures form a lattice-like structure, much the same as ionic compounds.

This molecular network is also the reason that diamond and quartz form a crystalline structures, just like you'd see in ionic substances such as NaCl. Some other structures you might want to look into are Graphite and Graphene, which are both allotropes of carbon (allotropes are, simply put, different molecular arrangements of an element).

The network structure combines to make the substance stronger than normal covalent bonded substances.

So to answer your question, substances with standard covalent bonds seem to be weaker than those with ionic bonds because the ionic bonds tend to form a lattice structure, that makes them much stronger. You can see this in the fact that the boiling points of ionic salts are much higher than that of a covalent substance like water. However, when covalent bonds form network covalent structures, atoms combine to form a singular macromolecule that is much stronger than singular covalent bonds.

• Your explanation doesn't seem to explain what you are saying. You explain using the fact that ionic compounds form lattice structures in the solid state but you then write that the boiling points of ionic salts are higher. In the liquid state, ionic bonds have already been broken. So when you mention about the boiling point of ionic salts, there is no relation to strength of the ionic bond. – Tan Yong Boon Aug 12 '17 at 12:22
• @TanYongBoon, I used boiling points because I felt they are most comparable to the true energy required to break the individual ionic bonds (vs. melting points, at least). We tend to measure strength of ionic interactions by lattice energy, which is defined as the energy required per mol of ionic substance to convert the solid to constituent gaseous ions. This number, though, accounts for both sublimation energy AND individual bond enthalpy. Using boiling points, then, at least allows us to avoid comparing melting enthalpies and more directly compare bond enthalpies. – Shafter Oct 18 '18 at 8:12
• @TanYongBoon of course the lattice nature of ionic compounds necessarily complicates the comparison, though. – Shafter Oct 18 '18 at 8:13

What you learned in your mineralogy class was correct; bond strength decrease in the following order covalent > ionic > metallic. The reasoning for this is as follows. In covalent bonds such as those in methane and oxygen, the valence electrons are shared between the atoms involved in the bond and they (the electrons) spend most of their time in the region between the nuclei involved in the bond; this makes for a strong bond. In ionic materials such as sodium chloride, the electrons are donated from one (the electropositive) atom to the other (the electronegative) atom in order for the atoms to achieve a filled shell structure. The ionic atoms are attracted to one another through electrostatic attraction and the crystal lattices that are formed. The bonds formed through electrostatic attraction are not as strong as those formed from covalent sharing of electrons. Finally, in metals the outermost electrons are donated or "pooled" in the band structure that exists in metals. The electrons are free to travel great distances (hence the conductivity of metals) and serve as a glue to hold all of the positively charged metal nuclei together. So in the case of metals, there are no significant metal-metal bonds and these bonds are therefor the weakest.

• As discussed in a different question, I respectfully disagree that ionic bonds are not as strong as covalent ones. – Geoff Hutchison Dec 18 '14 at 23:11
• In the above-referenced question you note that the largest covalent bond strength is 945 kJ/mol in $\ce{N2}$. While the ionic bond strength in $\ce{LiF}$, which should represent one of the stronger ionic bonds (see here) is only 577 kJ/mol. So at least when comparing the stronger covalent and ionic bonds, covalent bonds are stronger. Maybe you could provide BDE data for more average covalent and ionic bonds as well and we can see which are stronger in that realm. – ron Dec 19 '14 at 16:15
• I also included some comments on other covalent bonds. But I think this is a fool's errand. The great insight of Pauling was that any bond with different atoms will have at least some ionic/electrostatic component. Indeed an article I referenced in my answer suggests many bonds are strongly covalent and strongly ionic. – Geoff Hutchison Dec 19 '14 at 16:40

It depends, because for covalent there are two types of bonds, network or molecular, or as I have also heard it be called, polar covalent and non polar covalent. But, network covalent consists of a vast network among the atoms and each one are connected, and they mostly made up of one element.

Take a diamond for example, it is only made up of carbon, but since the atoms are connected to each other and have not bonding between molecules, like something such as salt, which is an ionic bond, it is harder to break. However, if it were a molecular covalent bond, then the whole story is different, because they tend to be very weak bonds and easily broken like sugar or otherwise known as glucose, sucrose doesn't matter, it is still a covalent molecular bond because they have molecules while as a diamond is technically one big molecule.

But, since sugar has multiple the bond between each other molecule is weaker than the bonds between the elements themselves, then it is really weak.

This turns out to be a nonsense question. Chemical bonds run the entire gamut from very strong to very weak as evidenced by the amount of energy required to break them. Trying to say that bonds which are either ionic or covalent are stronger is a big mistake, starting with the fact that "ionic" and "covalent" are merely the hypothetical extremes of the bonding continuum and can be considered "ideal" bonds. Real bonds lie along the continuum and have characteristics of both of the ideal bond types. Therefore, your original question has no place in the study of chemistry.

• It is a returning issue due to the fact that basic courses (and the teachers of that) pretend to simply order bonds in terms of their strength. Add to this the fact that ionic bonds loose their strength in water (a working horse case for most if not all Chemists) and then you run into the fact that even here many argue that a covalent bond is in itself stronger than an ionic one. – Alchimista Sep 17 '17 at 15:27

Ionic and metallic bonds are weaker than covalent bonds. This is correct, it is why covalent crystal is much harder than ionic and metallic crystal/polycrystal.

The second statement is wrong because firstly melting point is not proportional to the strength of chemical bond. There are more factors such as flexibility of molecules. Boiling point is more proportional instead.

More important, the inter-particle forces to be compared among the organic compound vs ionic vs metallic compound is NOT among covalent bond vs ionic bond vs metallic bond. It is among intermolecular force (dipole-dipole, H-bond, Van der waals) vs ionic bond vs metallic bond. And the first one is much weaker than the second and third for sure. So the boiling point of organic compounds is much lower.

Don't confuse the strength of bonds with the strength of the forces that hold crystalline solids together

There is a reason why the lessons you learned from chemistry are different from the lessons learned in mineralogy: they are not talking about the same things.

The problem is that mineralogy the bonds they talk about are the bonds that hold crystals together but in in chemistry what is often talked about are the bonds that hold the molecules together not the crystals made from the molecules.

This distinction is important. The vast majority of crystals in chemistry consist of discrete molecules held together by weaker intermolecular forces (sometimes called van Der Waals bonds). These are fairly weak compared to covalent bonds and result in crystals that are weak and have low melting points. So a chemist might look at compounds where the bonding in the molecules is covalent and make the generalisation that they typically form crystals much weaker than metals or ionic compounds. But that is because the bonds that make the crystals are not covalent.

A mineralogist will mostly look at compounds which are not made of discrete molecules but are made from ionic lattices or covalent networks (or both). There are no molecules of diamond, the crystal is a network solid held together by an (almost) infinite array of C-C covalent bonds, as silica is held together by an infinite array of O-Si-O bonds. Other minerals are a mix of the two with many silicates containing, for example, sheets of O-Si-O structures with a variety of ions in between. So, to a mineralogist, covalent bonds look strong compared to other types of bond. Ionic bonds are strong but not as strong as purely covalent network solids.

And the problem is further complicate by definitions of strength that are too narrow. Are metals stronger or weaker than diamond-like structures? It depends what you mean by strength. Diamond is harder than any metal but it is also more brittle. If resilience to being struck with a sharp object matters, choose a ductile metal object over a diamond any day. This happens because the crystal structure in some metals can absorb energy by reorganising crystal defects rather than by shattering bonds (pretty much the only option in silica or diamond). So, in one sense, metals are stronger than covalent solids.

The overall lesson is to be careful about definitions. There is no good generalisation of crystal strength based on bond types. Be careful whether you are talking about the bonds within the components of the crystal (molecules) or the bonds that hold those components together (many "covalent" compounds consist of crystals where molecules are held together by much weaker forces). Don't forget that many minerals have both ionic and covalent bonding. And be specific about what you mean by "strength" (eg harness are resilience to impact are not the same thing).

I am not sure what is the consensus that has been arrived at by chemists around the world but I would just like to offer my two cents' worth on the issue. This question has always been a question which my teachers would always tackle when they teach chemical bonding and their answer has always been the same:

It is not fair to make a comparison since these bonds are ultimately very variable in terms of strength.

I do agree with that but allow me to provide my perspective to this issue.

The strength of covalent bonds in simple molecular substances (as well as those in giant network structures) can be easily determined. Thus, the bond energies of most covalent bonds are known well and they can be easily used for such comparisons of bond energy. However, the strength of ionic bonds and metallic bonds is not so clear-cut.

By definition, the ionic bond is the electrostatic force of attraction between positively and negatively charged ions in an ionic lattice while the metallic bond is the electrostatic force of attraction between the positively charged metal ions and the surrounding electrons.

In an ionic lattice, there are so many ions interacting electrostatically with each other. How then can the strength of the ionic bond be determined? The idea of lattice energy could be used but comparisons made using lattice energy would only be of any sense when we are comparing between ionic lattice. It cannot be used to compare with covalent bonds!

Consider the bond dissociation energy of $\ce {H-H}$ and the lattice energy of sodium chloride. The BDE of $\ce {H-H}$ is $\ce {+ 436 kJ/mol}$ while the lattice energy of $\ce {NaCl}$ is $\ce {+ 786 kJ/mol}$. Both are in terms of "per mole of something". But that "something" is different in each case. In the case of hydrogen, that "something" would be the $\ce {H-H}$ bond but in the case of the ionic compound $\ce {NaCl}$, that "something" is the $\ce {NaCl}$ formula unit. And that is not the same as "per mole of ionic bonds between $\ce {Na^+}$ and $\ce {Cl^-}$". The ionic bond strength is not so easily determined because each ion is in an electrostatic environment which is influenced by all the other ions around it. The same idea can be applied to metallic bonds.

In essence, my take is that there is no basis of fair comparison between metallic, ionic and covalent bonds in terms of their bond strengths.


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# What is the area under the standard normal curve between z = 0 and z = 3?

Sep 27, 2017

$= 0.4987$

#### Explanation:

we want

$P \left(0 < Z < 3\right)$

$= P \left(Z < 3\right) - P \left(Z < 0\right)$

from tables

$= 0.9987 - 0.5$

$= 0.4987$

format: plain


text:
# Fluid Tests Hint at Concrete Quantum Reality

Courtesy of John Bush

A droplet bouncing on the surface of a liquid has been found to exhibit many quantum-like properties, including double-slit interference, tunneling and energy quantization.

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For nearly a century, “reality” has been a murky concept. The laws of quantum physics seem to suggest that particles spend much of their time in a ghostly state, lacking even basic properties such as a definite location and instead existing everywhere and nowhere at once. Only when a particle is measured does it suddenly materialize, appearing to pick its position as if by a roll of the dice.

This idea that nature is inherently probabilistic — that particles have no hard properties, only likelihoods, until they are observed — is directly implied by the standard equations of quantum mechanics. But now a set of surprising experiments with fluids has revived old skepticism about that worldview. The bizarre results are fueling interest in an almost forgotten version of quantum mechanics, one that never gave up the idea of a single, concrete reality.

The experiments involve an oil droplet that bounces along the surface of a liquid. The droplet gently sloshes the liquid with every bounce. At the same time, ripples from past bounces affect its course. The droplet’s interaction with its own ripples, which form what’s known as a pilot wave, causes it to exhibit behaviors previously thought to be peculiar to elementary particles — including behaviors seen as evidence that these particles are spread through space like waves, without any specific location, until they are measured.

Particles at the quantum scale seem to do things that human-scale objects do not do. They can tunnel through barriers, spontaneously arise or annihilate, and occupy discrete energy levels. This new body of research reveals that oil droplets, when guided by pilot waves, also exhibit these quantum-like features.

To some researchers, the experiments suggest that quantum objects are as definite as droplets, and that they too are guided by pilot waves — in this case, fluid-like undulations in space and time. These arguments have injected new life into a deterministic (as opposed to probabilistic) theory of the microscopic world first proposed, and rejected, at the birth of quantum mechanics.

“This is a classical system that exhibits behavior that people previously thought was exclusive to the quantum realm, and we can say why,” said John Bush, a professor of applied mathematics at the Massachusetts Institute of Technology who has led several recent bouncing-droplet experiments. “The more things we understand and can provide a physical rationale for, the more difficult it will be to defend the ‘quantum mechanics is magic’ perspective.”

Magical Measurements

The orthodox view of quantum mechanics, known as the “Copenhagen interpretation” after the home city of Danish physicist Niels Bohr, one of its architects, holds that particles play out all possible realities simultaneously. Each particle is represented by a “probability wave” weighting these various possibilities, and the wave collapses to a definite state only when the particle is measured. The equations of quantum mechanics do not address how a particle’s properties solidify at the moment of measurement, or how, at such moments, reality picks which form to take. But the calculations work. As Seth Lloyd, a quantum physicist at MIT, put it, “Quantum mechanics is just counterintuitive and we just have to suck it up.”

Bottom: Akira Tonomura/Creative Commons

When light illuminates a pair of slits in a screen (top), the two overlapping wavefronts cooperate in some places and cancel out in between, producing an interference pattern. The pattern appears even when particles are shot toward the screen one by one (bottom), as if each particle passes through both slits at once, like a wave.

A classic experiment in quantum mechanics that seems to demonstrate the probabilistic nature of reality involves a beam of particles (such as electrons) propelled one by one toward a pair of slits in a screen. When no one keeps track of each electron’s trajectory, it seems to pass through both slits simultaneously. In time, the electron beam creates a wavelike interference pattern of bright and dark stripes on the other side of the screen. But when a detector is placed in front of one of the slits, its measurement causes the particles to lose their wavelike omnipresence, collapse into definite states, and travel through one slit or the other. The interference pattern vanishes. The great 20th-century physicist Richard Feynman said that this double-slit experiment “has in it the heart of quantum mechanics,” and “is impossible, absolutely impossible, to explain in any classical way.”

Some physicists now disagree. “Quantum mechanics is very successful; nobody’s claiming that it’s wrong,” said Paul Milewski, a professor of mathematics at the University of Bath in England who has devised computer models of bouncing-droplet dynamics. “What we believe is that there may be, in fact, some more fundamental reason why [quantum mechanics] looks the way it does.”

Riding Waves

The idea that pilot waves might explain the peculiarities of particles dates back to the early days of quantum mechanics. The French physicist Louis de Broglie presented the earliest version of pilot-wave theory at the 1927 Solvay Conference in Brussels, a famous gathering of the founders of the field. As de Broglie explained that day to Bohr, Albert Einstein, Erwin Schrödinger, Werner Heisenberg and two dozen other celebrated physicists, pilot-wave theory made all the same predictions as the probabilistic formulation of quantum mechanics (which wouldn’t be referred to as the “Copenhagen” interpretation until the 1950s), but without the ghostliness or mysterious collapse.

The probabilistic version, championed by Bohr, involves a single equation that represents likely and unlikely locations of particles as peaks and troughs of a wave. Bohr interpreted this probability-wave equation as a complete definition of the particle. But de Broglie urged his colleagues to use two equations: one describing a real, physical wave, and another tying the trajectory of an actual, concrete particle to the variables in that wave equation, as if the particle interacts with and is propelled by the wave rather than being defined by it.

For example, consider the double-slit experiment. In de Broglie’s pilot-wave picture, each electron passes through just one of the two slits, but is influenced by a pilot wave that splits and travels through both slits. Like flotsam in a current, the particle is drawn to the places where the two wavefronts cooperate, and does not go where they cancel out.

De Broglie could not predict the exact place where an individual particle would end up — just like Bohr’s version of events, pilot-wave theory predicts only the statistical distribution of outcomes, or the bright and dark stripes — but the two men interpreted this shortcoming differently. Bohr claimed that particles don’t have definite trajectories; de Broglie argued that they do, but that we can’t measure each particle’s initial position well enough to deduce its exact path.

In principle, however, the pilot-wave theory is deterministic: The future evolves dynamically from the past, so that, if the exact state of all the particles in the universe were known at a given instant, their states at all future times could be calculated.

At the Solvay conference, Einstein objected to a probabilistic universe, quipping, “God does not play dice,” but he seemed ambivalent about de Broglie’s alternative. Bohr told Einstein to “stop telling God what to do,” and (for reasons that remain in dispute) he won the day. By 1932, when the Hungarian-American mathematician John von Neumann claimed to have proven that the probabilistic wave equation in quantum mechanics could have no “hidden variables” (that is, missing components, such as de Broglie’s particle with its well-defined trajectory), pilot-wave theory was so poorly regarded that most physicists believed von Neumann’s proof without even reading a translation.

At the fifth Solvay Conference, a 1927 meeting of the founders of quantum mechanics, Louis de Broglie (middle row, third from right) argued for a deterministic formulation of quantum mechanics called pilot-wave theory. But a probabilistic version of the theory championed by Niels Bohr (middle row, far right) won the day.

More than 30 years would pass before von Neumann’s proof was shown to be false, but by then the damage was done. The physicist David Bohm resurrected pilot-wave theory in a modified form in 1952, with Einstein’s encouragement, and made clear that it did work, but it never caught on. (The theory is also known as de Broglie-Bohm theory, or Bohmian mechanics.)

Later, the Northern Irish physicist John Stewart Bell went on to prove a seminal theorem that many physicists today misinterpret as rendering hidden variables impossible. But Bell supported pilot-wave theory. He was the one who pointed out the flaws in von Neumann’s original proof. And in 1986 he wrote that pilot-wave theory “seems to me so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored.”

The neglect continues. A century down the line, the standard, probabilistic formulation of quantum mechanics has been combined with Einstein’s theory of special relativity and developed into the Standard Model, an elaborate and precise description of most of the particles and forces in the universe. Acclimating to the weirdness of quantum mechanics has become a physicists’ rite of passage. The old, deterministic alternative is not mentioned in most textbooks; most people in the field haven’t heard of it. Sheldon Goldstein, a professor of mathematics, physics and philosophy at Rutgers University and a supporter of pilot-wave theory, blames the “preposterous” neglect of the theory on “decades of indoctrination.” At this stage, Goldstein and several others noted, researchers risk their careers by questioning quantum orthodoxy.

A Quantum Drop

Yves Couder et al.

When a droplet bounces along the surface of a liquid toward a pair of openings in a barrier, it passes randomly through one opening or the other while its “pilot wave,” or the ripples on the liquid’s surface, passes through both. After many repeat runs, a quantum-like interference pattern appears in the distribution of droplet trajectories.

Now at last, pilot-wave theory may be experiencing a minor comeback — at least, among fluid dynamicists. “I wish that the people who were developing quantum mechanics at the beginning of last century had access to these experiments,” Milewski said. “Because then the whole history of quantum mechanics might be different.”

The experiments began a decade ago, when Yves Couder and colleagues at Paris Diderot University discovered that vibrating a silicon oil bath up and down at a particular frequency can induce a droplet to bounce along the surface. The droplet’s path, they found, was guided by the slanted contours of the liquid’s surface generated from the droplet’s own bounces — a mutual particle-wave interaction analogous to de Broglie’s pilot-wave concept.

In a groundbreaking experiment, the Paris researchers used the droplet setup to demonstrate single- and double-slit interference. They discovered that when a droplet bounces toward a pair of openings in a damlike barrier, it passes through only one slit or the other, while the pilot wave passes through both. Repeated trials show that the overlapping wavefronts of the pilot wave steer the droplets to certain places and never to locations in between — an apparent replication of the interference pattern in the quantum double-slit experiment that Feynman described as “impossible … to explain in any classical way.” And just as measuring the trajectories of particles seems to “collapse” their simultaneous realities, disturbing the pilot wave in the bouncing-droplet experiment destroys the interference pattern.

Droplets can also seem to “tunnel” through barriers, orbit each other in stable “bound states,” and exhibit properties analogous to quantum spin and electromagnetic attraction. When confined to circular areas called corrals, they form concentric rings analogous to the standing waves generated by electrons in quantum corrals. They even annihilate with subsurface bubbles, an effect reminiscent of the mutual destruction of matter and antimatter particles.

Daniel Harris and John Bush

Video: The pilot-wave dynamics of walking droplets.

In each test, the droplet wends a chaotic path that, over time, builds up the same statistical distribution in the fluid system as that expected of particles at the quantum scale. But rather than resulting from indefiniteness or a lack of reality, these quantum-like effects are driven, according to the researchers, by “path memory.” Every bounce of the droplet leaves a mark in the form of ripples, and these ripples chaotically but deterministically influence the droplet’s future bounces and lead to quantum-like statistical outcomes. The more path memory a given fluid exhibits — that is, the less its ripples dissipate — the crisper and more quantum-like the statistics become. “Memory generates chaos, which we need to get the right probabilities,” Couder explained. “We see path memory clearly in our system. It doesn’t necessarily mean it exists in quantum objects, it just suggests it would be possible.”

The quantum statistics are apparent even when the droplets are subjected to external forces. In one recent test, Couder and his colleagues placed a magnet at the center of their oil bath and observed a magnetic ferrofluid droplet. Like an electron occupying fixed energy levels around a nucleus, the bouncing droplet adopted a discrete set of stable orbits around the magnet, each characterized by a set energy level and angular momentum. The “quantization” of these properties into discrete packets is usually understood as a defining feature of the quantum realm.

Harris et al., PRL (2013)

As a droplet wends a chaotic path around the liquid’s surface, it gradually builds up quantum-like statistics.

If space and time behave like a superfluid, or a fluid that experiences no dissipation at all, then path memory could conceivably give rise to the strange quantum phenomenon of entanglement — what Einstein referred to as “spooky action at a distance.” When two particles become entangled, a measurement of the state of one instantly affects that of the other. The entanglement holds even if the two particles are light-years apart.

In standard quantum mechanics, the effect is rationalized as the instantaneous collapse of the particles’ joint probability wave. But in the pilot-wave version of events, an interaction between two particles in a superfluid universe sets them on paths that stay correlated forever because the interaction permanently affects the contours of the superfluid. “As the particles move along, they feel the wave field generated by them in the past and all other particles in the past,” Bush explained. In other words, the ubiquity of the pilot wave “provides a mechanism for accounting for these nonlocal correlations.” Yet an experimental test of droplet entanglement remains a distant goal.

Subatomic Realities

Many of the fluid dynamicists involved in or familiar with the new research have become convinced that there is a classical, fluid explanation of quantum mechanics. “I think it’s all too much of a coincidence,” said Bush, who led a June workshop on the topic in Rio de Janeiro and is writing a review paper on the experiments for the Annual Review of Fluid Mechanics.

Quantum physicists tend to consider the findings less significant. After all, the fluid research does not provide direct evidence that pilot waves propel particles at the quantum scale. And a surprising analogy between electrons and oil droplets does not yield new and better calculations. “Personally, I think it has little to do with quantum mechanics,” said Gerard ’t Hooft, a Nobel Prize-winning particle physicist at Utrecht University in the Netherlands. He believes quantum theory is incomplete but dislikes pilot-wave theory.

Many working quantum physicists question the value of rebuilding their highly successful Standard Model from scratch. “I think the experiments are very clever and mind-expanding,” said Frank Wilczek, a professor of physics at MIT and a Nobel laureate, “but they take you only a few steps along what would have to be a very long road, going from a hypothetical classical underlying theory to the successful use of quantum mechanics as we know it.”

“This really is a very striking and visible manifestation of the pilot-wave phenomenon,” Lloyd said. “It’s mind-blowing — but it’s not going to replace actual quantum mechanics anytime soon.”

In its current, immature state, the pilot-wave formulation of quantum mechanics only describes simple interactions between matter and electromagnetic fields, according to David Wallace, a philosopher of physics at the University of Oxford in England, and cannot even capture the physics of an ordinary light bulb. “It is not by itself capable of representing very much physics,” Wallace said. “In my own view, this is the most severe problem for the theory, though, to be fair, it remains an active research area.”

Pilot-wave theory has the reputation of being more cumbersome than standard quantum mechanics. Some researchers said that the theory has trouble dealing with identical particles, and that it becomes unwieldy when describing multiparticle interactions. They also claimed that it combines less elegantly with special relativity. But other specialists in quantum mechanics disagreed or said the approach is simply under-researched. It may just be a matter of effort to recast the predictions of quantum mechanics in the pilot-wave language, said Anthony Leggett, a professor of physics at the University of Illinois, Urbana-Champaign, and a Nobel laureate. “Whether one thinks this is worth a lot of time and effort is a matter of personal taste,” he added. “Personally, I don’t.”

Courtesy of John Bush

Attendees of Hydrodynamic Quantum Analogs IV, a meeting held June 2-6 in Rio de Janeiro. The conference organizer, John Bush, a professor of applied mathematics at MIT, is pictured at left.

On the other hand, as Bohm argued in his 1952 paper, an alternative formulation of quantum mechanics might make the same predictions as the standard version at the quantum scale, but differ when it comes to smaller scales of nature. In the search for a unified theory of physics at all scales, “we could easily be kept on the wrong track for a long time by restricting ourselves to the usual interpretation of quantum theory,” Bohm wrote.

Some enthusiasts think the fluid approach could indeed be the key to resolving the long-standing conflict between quantum mechanics and Einstein’s theory of gravity, which clash at infinitesimal scales.

“The possibility exists that we can look for a unified theory of the Standard Model and gravity in terms of an underlying, superfluid substrate of reality,” said Ross Anderson, a computer scientist and mathematician at the University of Cambridge in England, and the co-author of a recent paper on the fluid-quantum analogy. In the future, Anderson and his collaborators plan to study the behavior of “rotons” (particle-like excitations) in superfluid helium as an even closer analog of this possible “superfluid model of reality.”

But at present, these connections with quantum gravity are speculative, and for young researchers, risky ideas. Bush, Couder and the other fluid dynamicists hope that their demonstrations of a growing number of quantum-like phenomena will make a deterministic, fluid picture of quantum mechanics increasingly convincing.

“With physicists it’s such a controversial thing, and people are pretty noncommittal at this stage,” Bush said. “We’re just forging ahead, and time will tell. The truth wins out in the end.”

• Mr. Raymond Kenneth Petry says:

And then what are “pilot waves”—Is space full of ‘irretrievable’ SHO E₀-state energy…?

• Eruvaer says:

Why isn’t it pointed out in the article that the de Broglie-Bohm theory is explicitly nonlocal? I think this is the biggest conceptual drawback of the theory and the reason that most physicists preferred (and still prefer) traditional quantum mechanics. As stated by Bell’s theorem one has to abandon either locality or reality (or accept some form of determinism) and it seems that for most physicists (me included) giving up on reality is the favored choice.
And regarding “[…] an apparent replication of the interference pattern in the quantum double-slit experiment that Feynman described as “impossible … to explain in any classical way.””: while the experiments on reproducing double-slit results in a classical way are definitely interesting on their own, one must not infer from this that full quantum mechanics can therefore be described “in any classical way” (which in my opinion is suggested by the article) since this is forbidden via Bell’s theorem.

• We, at dotwave.org, are a group of independant researchers, replicating couder’s experiment, both in silicium and in silico:)

We have two long term goals :
– reproduction and automation of Couder’s single particle double slit diffraction experiment. Indeed, that experiment has never been reproduced, and it lacks a proper statitistic. It has been made “by hand” and it took many month to obtain a simple 250 droplet statistics. Plans of the setup are ready : mobile droplet generator, optical path measurement etc…
– motion simulation in a powerfull cloud calculator. Following bush’s integro differential equation, we obtained interesting results while placing the droplet in a harmonic field : reproducting Couder’reported trajectories… and more to come with several droplets..

Thank you for that very nice article, and let the quest go on !

• Roger says:

that sounded kind of similar to the ripples in the liquid (caused by the bouncing droplet) interacting with each other and feeding back to the droplet to guide its motion. I was just curious if these things were connected at all. Again, I’m an amateur and know nothing about this but just wanted to throw it out there since they sounded similar. Thanks!

Roger

The FQXi blog mentioned a paper by T. Singh at the Tata Institute that talked about “trace dynamics” and said:

o The quantum gravitational field’ produced by the quantum mechanical particles’ feeds back on itself.

• This field also affects the motion of the very particles’ which are producing it. As a result, the equation of motion’ of the particles is a nonlinear equation [the equivalent of a nonlinear Schrodinger equation]. The nonlinearity is important only near the Planck mass/energy scale.

• HT says:

@RKP: Some “ether”-like other dimension of energy seems much, much more plausible to me than infinite “worlds” supposedly spinning-off from every quantum measurement, just to placate one preferred interpretation of the equations…

• Natalie says:

@Eruvaer:

Sheldon Goldstein, professor of mathematics, physics and philosophy at Rutgers University and a supporter of de Broglie-Bohm theory, responded to your comment as follows:

“What [Eruvaer] expresses is a common complaint against Bohmian mechanics. This complaint is misguided. While it is true that Bohmian mechanics in nonlocal, what Bell showed is that that nonlocality is intrinsic to quantum theory itself—to its very predictions—and can’t be eliminated while retaining those predictions. Thus if Bohmian mechanics were local it would have to be wrong. And the nonlocality of standard quantum mechanics is quite explicit: it arises from the combination of entangled states and collapse upon measurement.

This very issue has recently been discussed in a paper by Tim Maudlin, What Bell Did, to appear in a special journal issue devoted to Bell.”

Best,

Natalie Wolchover

• Tim Maudlin says:

Although I appreciate this article, I’m not sure that an average reader could quite understand the exact situation here. The pilot wave theory needs no help or support from experiments like these: it is a mathematically perfectly well-defined theory (in the non-Relativistic domain) that provably makes all the same predictions as the standard quantum formalism while also solving the measurement problem. What the oil-drop experiments provide is a tangible partial analog of the pilot-wave picture, but restricted to single-paricle phenomena (that is, this sort of experiment cannot reproduce the sort of phenomena that depend on entanglement). That is because only in the case of a single particle does the wave function have the same mathematical form (a scalar function over space) as do the waves in the oil. Once two particles are involved, the fact that the wave function is defined over the configuration space of the system rather than over physical space becomes crucial, and the (partial) analogy to the oil-drops fails.

It is, of course, very nice to bring attention to the pilot-wave approach, and these experiments can given one a sort of visceral sense of how it works in some (single particle) experiments. But if over-generalized, the picture can also be somewhat misleading.

To second the point about non-locality made above: yes, of course the pilot-wave theory is non-local. It had better be if it is to recover the predictions of quantum theory. That was what Bell proved. Einstein, of course, insisted on the obvious non-locality of the standard (Copenhagen) understanding of quantum theory: that is what the EPR paper was all about. Einstein hoped that a different approach could avoid the non-locality (“spooky-action-at-a-distance”) in the standard approach. Bell showed it can’t be done, so non-locality cannot be considered a defect of a theory. It is just the opposite: a local theory must be defective: it cannot make the right (experimentally verified) prediction of violation of Bell’s inequality for distant systems.

• mloufek says:

A wonderfully comprehensive article about core issues in QM. Thanks, Natalie.

• Roger says:

Ms. Wolchover and Dr. Maudlin,

Ms. Wolchover: Thanks for writing a great article! Dr. Maudlin: Thanks for the extra information, which was very helpful! My comment is that even if pilot wave theory is perfectly defined mathematically and doesn’t need any help, the bouncing droplet experiments seem to bring up some possible physical mechanisms for how the quantum weirdness stuff may come about. I think one problem with physics is that they seem to emphasize the mathematics over the physical mechanisms. For instance, in addition to the quantum weirdness stuff, I don’t think there’s any physical mechanism yet for why a positive and a negative charge attract. How does the exchange of photons cause attraction in some cases and repulsion in other cases? In my field, biochem., physical mechanisms are very important in figuring out how things work.

Thanks.

Roger

• Kenneth Udut says:

I’ve been a fan of Pilot Waves for many years. We’ve made great progress in the Copenhagen and other popular interpretations of Quantum Mechanics, such as many worlds, but a lot of these are based on a perceived need to distance ourselves from Maxwell – a perceived “break” with the past, as all old formulas get their new (but not necessarily improved) Quantum Mechanical equivalent.

As we get closer to making practical devices using quantum level technologies, having a proper understanding of quantum mechanics is crucial for us to continue.

The analoges of electricity, pressure, behavior of fluids, etc – all lining up nicely with each other in practical engineering fields (a water computer is no more difficult than an computer of electrons; space is the main issue) – all are primarily expressed in waves, and rightly so.

Excess emphasis on the perceived “spookiness” of quantum level interactions does help fund the almost religious mystique of all things “quantum”; it has become a magic word in many circles…. but it simply means “counting how much/how many”.

The idea of separate little things we can count is *very* useful and helpful – and indeed, for many things, we can ‘count’ and act “as if” things are isolated from other things.

but they’re not.

The complex interaction of all things is obvious from the very nature of the beginnings of the Universe; we were once all one and there is no “nothing” inbetween things, even though we are often taught that in school. There is a “something” in the “middle” of things – there always is.

This doesn’t mean the path will be easy; we may still depend on a particle zoo view of things simply because so much wonderful work and research has been done in that area – and there is no reason to abandon it all.

But an understanding of the *context* within which this “apparent” particles are living in – – as expressions of waves – not “clouds” of statistics… raising math to a level that Pythagoras’s followers would fully recognize as their religion – will most definitely benefit future research and technologies, in my opinion.

Kenneth Udut
Naples, Florida USA

• James R. Johnston says:

Natalie and Tim, Thanks: good comments.
I am interested in the “special journal issue devoted to Bell”.
Could you provide info on that?
Thanks,
Jim

• Daniel Maksuta says:

It would be interesting to see if they could do this experiment while also making the sphere spin at a high angular velocity.

• I think its pretty cool that the author will actually go and get a response to a comment from the physicists under discussion and then post it. That said, there are good reasons why this idea is a minority interpretation of QM. And needless to say none of these experiments constitute evidence regarding the underlying nature of quantum mechanics.

Eruvaer’s complaint was a meaningful and important one: non-local causation is a serious matter in light of special relativity, and Goldstein’s response is typical of pilot wave supporters, but not satisfactory. What quantum mechanical experiments require is non-local correlation, but this is emphatically not the same thing as non-local causation. When the principles of quantum mechanics are taken seriously they imply the former but not the latter, due to the simple fact that such correlations can only be verified in any one observer’s future light-cone. Definite properties of quantum systems are only obtained when we measure them, regardless of whether or not an entangled partner is being acted upon some distance away. So this particular defence is not a persuasive one: causal influences acting at space-like separations are deeply in conflict with special relativity, because such pairs of events have no well-defined ordering. The argument that experiments say otherwise might have carried weight if these entanglement experiments weren’t already well-described by standard quantum mechanics, in which all causal influences propagate strictly into the future lightcone (at least, that is, if we agree that the postulates must apply to observers as well, rather than cutting off quantum mechanics at the boundary of a would-be “single objective classical reality”).

Its not hard to sympathize with the desire to want a viable picture of a totally concrete “classical reality”, and if the pilot wave theory could really do the job just as well as standard quantum mechanics, it might be very persuasive. However this theory requires the introduction of a massive unobservable apparatus in order to achieve these properties, so there is a major complexity cost to achieving the outcome. Standard quantum mechanics, despite many counterintuitive consequences, is at least highly economical in its principles. This difference is particularly clear when relativity is involved: in many ways standard quantum mechanics fits with special relativity like hand-in-glove into quantum field theory, whereas the pilot wave versions are exponentially more complicated and rube-goldberg-like. They seem to clash dramatically with the symmetry principles that are foundational to our understanding (for example, what are the true “classical states” of a spin-1/2 electron? Only proper quantum mechanics seems to permit such a 2-state system to exist while preserving rotational symmetry, since it treats all bases as equally fundamental).

If these theories could ever be developed to be anywhere near as parsimonious and harmonious with what we know, then many physicists would justifiably reexamine them. For the moment though that motivation just doesn’t seem to be there.

• Jerry Lisantti says:

As usual a well written article, thank you.
That quantum mechanics has a variety of very different interpretations is interesting. Does the variety of answers to the question of what quantum mechanics means on a fundamental level mean something is wrong with quantum mechanics? The success of the application of quantum mechanics is well known and justifies the response of “shut up and calculate”, but does having various interpretations imply that the theory on a fundamental level is not understood, and does this mean that something is wrong with quantum mechanics? Shouldn’t a fundamental theory be self explanatory?

• Robert Fowler says:

To my non-scientist mind, this sounds like it takes some of the bizarreness out of quantum theory. If that’s so, I would welcome it if only to shut up the woo hucksters like Deepka Chopra.

• Tim Hoefer says:

I didn’t see where the new theory classically explained the effect in the double-slit experiment that happened when an attempt was made to measure which slit the particle passed through. Isn’t that arguably the majority of the “mystery” behind the experiment’s results? What did I miss (seriously)?

• I agree with Tim Maudlin that it is unclear yet how the Couder experiments can be related to quantum mechanical nonlocality. Having published about 20 papers in recent years on a “subquantum” approach to QM making use of an analogy to Couder’s bouncing droplets, our group recently visited Yves Couder and Emmanuel Fort in Paris, and we agreed that this issue of nonlocality is an open one w.r.t. (in fact, any) fluid mechanics approaches.

In our model, we consider “particles” to be bouncers, i.e. oscillators, embedded in the medium of the non-empty “vacuum”. Said medium is characterized by wave-like oscillations as well as stochastic fluctuations, and the bouncers are dynamically coupled to the wave-like oscillations of the medium, just as in the case of the Couder group experiments. The important point of our approach is given by the circumstance that the frequencies and relative phases relating different parts of the medium in an experimental setup are essentially determined by the geometry of the latter, just as the size of the bath container, as well as its form, determines the waves and their behaviors in the Couder group experiments. This is actually the topic of emergence: the creation of ordered structures of a medium, given that there is a constant throughput of energy to maintain them. Now, where could that energy related to the maintenance of structures in the vacuum come from? One possible answer: the universe! As we live in an expanding universe that potentially also contains something called “dark energy”, any constrained spatial area of it, like an experimental setup constrained by its boundaries, may have a flux of energy through it that we can (directly or indirectly) observe as oscillations and fluctuations of the vacuum, whose quantitative features are co-determined by the boundaries. (Note that in this case one also has the option that nonlocality could be “explained” … admittedly by putting it in “by hand”, so to say, but if the world is truly nonlocal, it would not make much sense to try to reconstruct it on a purely local basis, would it not?)

Anyway, with this ansatz we were able to describe exactly various features of QM without using the latter’s formalism in any way. In fact, we just published in Annals of Physics a paper where we explain Born’s rule and derive the basic postulate of the Bohmian “guiding equation”, which is synonymous to the QM probability density current. Actually, we show that by employing concepts of emergence, one gets rid of the necessity of a configuration space description and obtains the QM results in real space. For a more didactical paper on this issue, see the free access conference contribution http://iopscience.iop.org/1742-6596/504/1/012006 . The formalism turns out to be stunningly simple. In fact it is so simple that we have developed a protocol for computer simulations which enables us to demonstrate via a combination of classical wave mechanics and diffusion (i.e., of the bouncer hopping in a Brownian-type manner) features such as the decay of the Gaussian wavepacket, interference at the double slit or at n slits, respectively (Talbot carpets, etc.), the Aharonov-Bohm effect, and so on. Our results show perfect agreement with the figures obtained in Bohmian mechanics, but with an important difference: our model explains the outcomes classically (i.e. does not rely on the QM wave function formalism as Bohmian mechanics), and provides trajectories only as average ones (i.e. averaged over much more complicated sub-quantum Brownian motions), whereas the Bohmian trajectories are considered as the “real” ones in their model.

You can look up results of our simulations in our papers, all of which can be found on the arXiv, or via our webpage http://www.nonlinearstudies.at . Note also that we have been organizing a conference series on “Emergent QM”, the most recent one being http://www.emqm13.org/ .

• HT says:

@TH: There’s a very short bit in the article addressing that, re: “…disturbing the pilot wave in the bouncing-droplet experiment destroys the interference pattern.” Measuring the particle automatically disturbs the “pilot waves” (whether classical or “other-dimensional”), but that does not necessarily mean that the particle had no trajectory, just a previously unknown one. Epistemologically unknown, chaotic trajectories would still be real trajectories.

• Vitaly Golovach says:

I find that “…disturbing the pilot wave in the bouncing-droplet experiment destroys the interference pattern.” is not the right thing to do. One should “disturb” (i.e. measure) the oil droplet and not the wave. Thus, weakly measuring the oil droplet at a slit may still preserve the interference pattern, but also provide sufficient information to tell with a good confidence that the particle has (or has not) passed through that slit. Therefore, the analogy between the oil droplet experiment and quantum mechanics fails at this point, as suspected by Tim Hoefer in the post above.

• HT says:

@VG: Correct, although my understanding is that, at the macro level, the droplet bouncing is actually co-creating its own pilot wave, so it’s unclear (to me at least) what a “weak” or minimal threshold would be for measuring aspects of the droplet which would not in turn disturb or modify the wave patterns. Simply observing the interference pattern develop from 2 slits & the macro walking-droplets is already possible, now, though, I take it. I could be wrong about the imagined applicability to a quantum level, but if a pilot wave model is eventually somehow demonstrated to have an “ontological” reality for the quantum level (and/or beyond it), then the unifying aspect of the “superfluid substrate of reality” (phrase used in article) is presumably what stops the interference pattern once measurement of a single slit starts, since presumably the pilot waves are in turn affected by the modified setup (?). Yet that doesn’t make the trajectories any less actual.

• Pete says:

Excellent article Natalie.

I’m all about the realist interpretations of QM (Bohmian mechanics, Many-Worlds, etc.) and think that anything else doesn’t make sense at all. I mean how can physicists talk about there being no “reality” when one of the primary assumptions any working scientist makes is that there is an independent external reality to begin with? This seems entirely self-defeating.

• TerraHertz says:

What intrigues me, is the apparent avoidance of discussion of *what* medium would be carrying such pilot waves, and what the particles must be composed of in order to interact with the pilot waves. The pilot waves cannot be electromagnetic, but something more fundamental.
Is their nature just being left unstated (in the article) because it’s too obvious? Or is it currently an unknown, being neglected in the discussion because it’s unknown?

It occurs to me that there’s a strong association between the Pilot-Wave ideas, and another whole branch of alternate (and often rather woo-woo) theorizing going by the name of Scalar Waves. And that perhaps *both* particles and Pilot Waves may be manifestations of topologies of disturbance in an underlying Scalar medium.

• Bill says:

@Roger
I believe you are thinking in terms of classical particles being exchanged; this is one source of confusion. The mechanism you seek for the explanation of attraction and repulsion by (virtual) photons depends on the “quantum weirdness.” (The photon was the first “quantum” particle) I believe Feynman’s popular book, “QED: The Strange Theory of Light and Matter” may offer some answers.

• Christophe Galland says:

TerraHertz said “What intrigues me, is the apparent avoidance of discussion of *what* medium would be carrying such pilot waves, and what the particles must be composed of in order to interact with the pilot waves. The pilot waves cannot be electromagnetic, but something more fundamental.”

Does this imply that we know what medium is carrying electromagnetic waves? The aether hypothesis was unanimously rejected after we found that the speed of light was frame-independent, but nothing came to replace it as a medium for EM waves. Everybody is happy with the (quantum) vacuum. The quantum vacuum being arguably anything but a dull, “empty” vacuum I can imagine it sustaining some sort of pilot wave (see Gerhard Groessing comments above for related ideas).

• Bruce C. Mitchell says:

There have long been documented macro scale demonstrations of quantum effects. This study of pilot wave mechanics seems to fall under that category. This is just my opinion, but this application seems to assume at the heart of quantum mechanics is the measurement problem. If you believe the measurement problem is an artifact of the deeper situation, that reality is interactive, and those interactions at their edges are indeterminate, then the fluid dynamics described here are simply a macro example of interactive reality playing out in the new situation.

As an aside, the chaotic path of the oil droplet in the magnetic field does not seem, on the surface, to be analogous to the states of electrons around an atom. The probability cloud notion, describing electrons’ locations around the nucleus of an Atom, is that the nature/architecture of the situation determines the locus of existence of electron behavior. Mere chaotic behavior of a droplet in a situation created by the interaction of magnetic fields and fluid waves is a drunkards walk situation, replacing gravity with a magnetic field. It does not seem to be analogous to the stability of our material world world based on the preferred architecture of The atom’s probability clouds.

From a bird,s eye view , are their any connection between the guiding pilot wave and the
U- bit ???
Are all of this search feels the necessity of ” something ” acting as a pattern generating ? , remember Bohm,s ” pattern generating active information field ” ?
M. N.

• Robert Matthews says:

Congratulations on such a great piece of science writing. Intriguing, clear, balanced, well-referenced.

• Tim Maudlin: the droplet experiments do indeed allow you to visualise a pilot wave in the configuration space of two or more particles. In our paper quoted in the above article, Why bouncing droplets are a pretty good model of quantum mechanics, we show that the standing wave created by the droplets bouncing on the vibrating bath is modulated with an
analogue of the quantum mechanical wavefunction \psi; where there are two droplets it’s a function of the position and momentum of both of them. In fact you can see \psi with your naked eye in the pictures of the diffraction experiments. Even although this is only a two-dimensional analogue of quantum mechanics, it could be really helpful as a teaching aid, as it can get across the idea of configuration space and the wavefunction in an intuitive and physically realistic way.

Roger: you’re right that the droplet experiments ask many novel questions. And there is an analogue of the Coulomb force: droplets bouncing in phase attract each other, while droplets bouncing antiphase repel. This phenomenon been known since 1875 in fluid mechanics, and is called the secondary Bjerknes force; it’s used in ultrasonic degassing equipment. If you have oil with a lot of small air bubbles, you can put an ultrasonic transducer in it and the bubbles will expand and contract in phase. This gives rise to an inverse square force of attraction between them; they merge, coalesce and rise to the surface, removing the air from the oil.

Cliff: spin-half behaviour also emerges in the droplet experiments, where there are two droplets orbiting each other. This involves an analogue of the magnetic force, and gives rise to some cute behaviour such as droplets playing hopscotch as they walk along the edge of the tray. For details see our paper.

• Fred Krogh says:

I have wondered why John Cramer’s cosmic handshake (google it) is discussed so little. But perhaps expressing it in terms of pilot waves make it more palatable. The backwards in time information can be considered as part of the pilot wave. I like the analogy of solving a boundary value problem using finite differences. One gets a global solution, but each of the equations that make up the sytem one solves only involves local information. Thus the backwards in time business can be handled in terms of equations using only local information, albeit some of the local information may be on either side of (but arbitrarly close to) the time at a given point.

And if you are still reading, here is an idea for an experiment, that might have a small chance of showing some difference between what quantum theory predicts, and what some pilot wave theory might predict.

The usual double slits, but one slit close to the north pole of a bar magnet and the other slit close to the south pole. Electrons are passed one a a time through a filter that determines the spin on the electron as either up or down prior to passing through the slits. Get the diffraction pattern for these electrons keeping different statistics for the up and down cases. At the same time take statistics on the spins of the electrons when they are detected. This will require separated runs. Once again statistics are separate for the up and down cases.

But that first spin can be at any angle, say a, and the spin detector after passing though the second detector can also be at any angle, b. Thus you accumulate statistics for what is detected in terms of the diffraction patterns, and what is seen for spins at angle b for different choices of angle a.

The unlikely hope is that something shows up differently than expected as there is more of an effect from the slit that the electron actually passes through, that would not be indicated by the usual quantum mechanics calculations.

P.S. I really don’t have much of an idea of what I’m talking about.

• Richard says:

How has no one mentioned the aether. Why use Superfluid instead of aether? The term aether is well established in the literature as an all encompassing fluid-like substance which all things exist in. The word aether has been in use for thousands of years and much thought has been invested into it by brilliant and dedicated minds.

Superfluid is a fake!

• bryanjb says:

Very interesting presentation and comments, thanks.
As for the problem of an analog for measuring the droplet system compared to the quantum counterpart of the process, it occurs to me that most quantum measurements use interactions between particles that have mass / size in some neighborhood of similar proportions. Measuring the droplets in a system with analogous scales would require use particles proportional in size to the oil droplets themselves. Perhaps the analogy becomes weaker at this point, since the oil droplet is not held together with anything analogous to the ‘resilient’ nuclear weak force, but rather surface tension originating in Van der Waals forces.
I presume the oil drops and the pilot wave medium are visualized using interaction with photons, which are many orders of magnitude smaller and less energetic.

Wondering how non-local entanglement might be considered in a many particle open system oil droplet / pilot wave system? Wouldn’t the entangled pair’s respective pilot waves be overwhelmed by noise from the noise of all the other pilot wave / particles?

• stargene says:

Ms Wolchover states
“…But at present, these connections with quantum gravity are speculative, and for young researchers, risky ideas.” Sadly, this seems to be correct these days… young researchers
are often taught to avoid risky areas and thought. While this may have been true even
back in Einstein’s prodigious youth, I rather doubt it, or that it carried much weight
if true.

BTW, it occurs to me that the concept of C-Y manifolds may be able to accommodate
not only the notion of space-time being a superfluid, but the possibility that two or
more such ‘superfluids’, matching, say, the number of lepton-quark families, might
co-exist, in principle. This might allow a richer playing-field, so to speak, for the
care and feeding of particle physics.

So to sum up, this is interesting for philosophers but not for physicists re the interest groups seen? And the problem is that the simplest pilot wave theory is non-relativistic. [Maudlin]

• Emory Anderson says:

Well, I’ll have to dig into this. I don’t really understand how this is particularly different from the pilot wave in Shrodinger’s wave equation. That is also an invisible (though mathematical) wave that one then operates on (via math) to derive observables. It is in fact that wave that is responsible for quantum interference effects.

A “fluid”-like interpretation of Schrodinger’s pilot wave, however, doesn’t really explain how this pilot wave can move through all space/time without any other interactions, etc…

Given those quoted in the article (eg, Wilczek, etc…), I guess there must be something interesting here, but right now I’m not sure I see it.

• Hartmut Körner says:

So the fluid would be vacuum / non-excited quantum field(s), the Pilot Wave / ripples would be virtual particles, and the walkers / droplets would be actual particles / excited qantum fields?

Is that correct? How do virtual particles keep the “path memory”? Do they interact with each other?

P.S. I’m not a physicist, and english is my second language.

• Hartmut Körner says:

I have no idea what I’m talking about, but after using google all day I stumbled over a sentence at the end of the following work, that describes quantum fluctuations as “book-keeping-devices” – which sounds like the answer I was looking for: a way for having a “path memory” in the vacuum / zero point field?

http://www.math.univ-toulouse.fr/~bouclet/GDR-2012/mallick.pdf

• What a great Article and a great comment section.

The EPR class of experiments is an interesting goal for this framework and would be a significant result. The model non-local, but can have emergent non-local correlations. The model would then involve 2 particles creating their wavefield and establishing a correlation over long distances. Part of the ‘paradoxes’ arise because of the mental categories we use to describe the problem. For example, what makes the EPR experiments so magical is that wave collapse seems to happen at A and B even if A and B are separated cones of light. The answer to this apparent paradox in the wave/particle picture is that there is no wave collapse. There never was. Just intermittence between states (which is driven by memory which induces chaos as Couder likes to remind everyone). It is the intermittence that creates the statistics we observe. “There is no such thing as superposition, dead cats and wave collapse” is what I take away from this study.

• DirkH says:

I discovered de Broglie a few years back. I’d be delighted if it became the new standard interpretation as it explains the “weird” behaviour of quanta as emergent behaviour. As to problems with relativity: Oh well -astronomy has a problem with relativity as well – you can only compute movements of celestial bodies if you assume that gravity propagates instantaneously. You can’t compute an orbit without that! But “instantaneously” has NO MEANING in relativity because there’s no global clock!!! Hmmm….
… so if astronomy can play a wildcard, why can’t de Broglie?

• mpc755 says:

What ripples when galaxy clusters collide is what waves in a double slit experiment; the superfluid substrate.

Einstein’s gravitational wave is de Broglie wave of wave-particle duality; both are waves in the superfluid substrate.

The superfluid substrate displaced by the particles of matter which exist in it and move through it relates general relativity and quantum mechanics.

The Milky Way’s dark matter halo is not a clump of stuff traveling along with the Milky Way. The Milky Way is moving through an displacing the superfluid substrate.

The Milky Way’s halo is the state of displacement of the superfluid substrate.

• Chris Hill says:

This was a very interesting article that has provoked a couple of questions, which perhaps a physicist can answer:
1. What is the medium in which the pilot wave is propagating?
2. How does a quantum particle excite that medium? In the large scale analogy, the excitation occurs through bouncing of the droplet, with gravity providing the force. Clearly something else is required for quantum-scale particles.
Thanks

• book lover says:

what a great article!!Quanta magazine is absolutely fantastic……

• Andrew Moore says:

and then there was The Aether…… and finally reality starts to return to physics……. Waves can only exist in a medium and all gravity is a pressure differential and electrons are standing waves. Sanity at last.

• Mark Iverson says:

Suggestion for an experiment…

Make one of the slits with adjustable width, so that you could decrease the width in very small increments. *IF* there’s a real physical particle, it will have a definite physical dimension. At some point, the width of the slit will have been reduced to something smaller that the physical dimension of the particle and it will no longer go through… but the waves should; perhaps with a much diminished distance. Another variation on that theme would be to change the separation of the two slits; looking for a difference in behavior when the separation distance coincides with crests or troughs of the waves. These suggestions require that the changes in slit width and separation can be controlled accurately to some extremely small increments, say 10e-15m.

• Sam says:

The article mentions that van Neumann’s theorem about the impossibility of hidden variables had a mistaken proof. However, I understand that von Neumann’s (claimed) result was strengthened, with a correct proof, by the Kochen-Specker theorem. Is the Kochen-Specker Theorem still understood to show the impossibility of hidden variables? Is this a problem for the “pilot-wave” approach described in the article?

• Andrea Calaon says:

Well written! Really well done!

Hidden Variables are thought to have to major shortcomings (J.S. Bell “Showed”):
– Are NON LOCAL,
– Are CONTEXTUAL (which is by far the most important problem with them).

My contribution:
– They are non-local because all particles are intrinsically light-like and time as we know it is an illusion.
– They are actually NON contextual because the sole argument for contextuality is the formalism of spin, which, as unbelievable as it may seem, is WRONG. So that the Kochen-Specker theorem does not apply.

Sooner or later these things will emerge. Hehehe
Something more:
Hidden Variables is essentially the “correct explanation” for QM. Due to the time-less nature of the electromagnetic interaction (see also delayed choice experiments and Aharonov–Bohm) particles tell you that they know the future and their behaviour depends on the past of the universe …
To understand the Zitterbewegung and what happens inside hadrons you need only another ingredient … time … Search Study, Think, … you will find one day …

• Andrea Calaon says:

Sam, the theorem of Kochen and Specher is used to show that Hidden Variable theories are contextual, which is the most important shortcoming of these theories. The other problem is that the HV must be non-local theories (Bell).
However if the commonly accepted formalism of spin is wrong it can be that the Kochen and Specher theorem can not be applied and HV are NOT contextual.

• Andrew Paton says:

The 2 slit theory.

I’d say in mathematical terms, what you see is what you get.

Your passing an object through a 3 dimensional object that in affect has 6 edges, 2 sides each and one front opening each, the reflection from, in affect the 4 edges and 2 openings equates to 6 and the fact you have 2 slits means you cancel out one of these as they run into one another i.e. 6/2 = 3.

It would if this mathematical method could be applied to a multiple number of slits to se whether this is in fact the case.

Kind regards,

Andrew Paton

• Len Berman says:

Can you factor numbers with it?

• Steven Sagaert says:

A more recent alternative interpretation of QM that is in line with underlying classic reality is scale relativity (Nottale). In it’s early work it’s mostly Feynmannian in spirit (in the sense of path integrals) but in his latest papers the Bohmian aspect which was always there but in disguise is more explicit (see papers on macroscopic quantum fluids), he even introduces the Quantum potential from Bohm mechanics there. His latest papers on fluids is right in line with these experiments. In SR the pilot wave is replaced with a fractal spacetime which itself is a consequence of the principle of scale covariance.

Scale relativity goes further than BM because it also “derives” the fundamental equations of the standard model fields, so it’s not only for QM but also for QFT. Although the mathematical entities to derive the fields seems to come out of nowhere and not well founded in principles like symmetry (the standard approach to QFT).

Another even more fascinating model (although just like most interpretations of QM it doesn’t get any further than the Bohr atom so no advanced QM/QFT), is the fact that quantum theory itself might not be fundamental at all but rather already present in classic Einstein relativity but due to the fact that we do not fully mathematically exploit all its consequences by only using differential equations instead of the harder delay differential equations we do not see QM-like behavior in the classic relativistic equations.
See
* ON THE ORIGIN OF QUANTUM MECHANICS, Jaume Giné
* ON THE ORIGIN OF THE GRAVITATIONAL QUANTIZATION: THE TITIUS–BODE LAW, Jaume Giné.
* The electrodynamic 2-body problem and the origin of quantum mechanics1, raju
*Retarded gravitation theory, Raju

on a more general note: Sir Michael Attiyah seems also to think that delay differential equations might play an important role in physics:
*A Shifted View of Fundamental Physics, Attiyah & Moore

The model of the delay differential equations is right in line with the fluid model experiment where the droplet (=particle) interacts with its own echos from the past (via the fluid).

• Dima says:

Hmm… Is it viable to make a hydrodynamical reproduction (experimental, or at least in simulation) of the delayed choice quatum eraser?

• Zack Yezek says:

The upside of Bohm’s pilot waves is that they replace the magic wave function with a more normal wave (of ‘some’ kind). But they still don’t address, at least not directly, what for me is the far bigger mystery- what is the physics governing quantum “measurement”?

Pilot waves don’t make Bell’s Inequality go away. They’re a hidden variable governing the state of the observed particles, so they must be non-local. But then how do you square them with special relativity? Existing entanglement experiments have already proven that any signal between the pairs, hidden or not, would need to transit way faster than c, if not instantaneously, to the lab observer.

The only way I see to logically square this circle is to have both advanced and retarded pilot waves- ones moving forward and backwards in time, respectively. Entanglement then becomes perfectly rational- the particles coordinate their states with exchanges of advanced & retarded pilot waves until they hit a state of equilibrium. Both types of wave move at c or slower, but to external observers the ‘transaction’ appears instantaneous because the net elapsed time of the interaction is 0 in their reference frame. Otherwise I just don’t see how you can have Pilot Waves without contradicting existing experimental results, short of imbuing them with the same sort of wave function magic they are intended to replace.

• Doug Corrigan says:

There seems to be a infinitely recurring cycle here. If you represent the quantum world with a classical fluid – then you have to understand that our understanding of how fluids work is based on quantum mechanics. The fluids they are using in their experiment, are themselves, the product of summed up quantum events that behave classically. So now, to get to first principles, you have to describe the physics of that new superfluid that fills space- But, how are you going to do that? To describe that fluid, you will have to use some form of physics – you will also need to use some form of physics to represent the interaction of the fluid bubble with the surface of the fluid. In a classical fluid, there are classical mechanical properties such as viscosity, surface tension, density, etc – but all of our understanding of these classical properties are built upon measuring a infinitely summed up quantized system, and all of the classical phenomenon emerge from this summation. So are we to believe that the universe is a fractal, where this superfluid that fills all of space, is itself, some form of medium that has its own set of classical laws that are the result of a summed up physics at even a smaller scale? And then if we peel back that layer to that smaller scale to understand what makes that superfluid work, do we repeat the process again by inventing another fluid that governs the particles behavior in the larger fluid? It seems like we are just working in circles and not getting anywhere.

• Maxim says:

There could be that pilot waves exist in quantum description just for the short time, thus the guiding particle dissipates and reemeages from the probability field, it looks compartible with the conventional description

• So, by analogy the photon doesn’t go through both slits, bit instead jumpst all over the place, sometimes going through one slide, sometimes through the other?

• France Čop says:

About nonlocality, look at this video. And Dr. Mardari has said, that particles and waves in his theory are on par with droplets.


text:
# Improper Integral Sinx/x^2 and similar with sinx

1. Jun 9, 2015

### mekise

1. The problem statement, all variables and given/known data

This is the improper integral of which I have to study the convergence.

∫[1,+∞] sinx/x2 dx

3. The attempt at a solution

I have tried to use the absolute convergence.

∫f(x)dx converges ⇔ ∫|f(x)|dx converges

but after i have observed that x^2 is always positive and the absolute value is only for the sinx, I do not how to move!

How to see the convergence or the divergence of sinx when you can not to approximate to x for x->0?

Thank you for the help!!

2. Jun 9, 2015

### Staff: Mentor

Try the p test, then the comparison test for absolute convergence.

Thanks
Bill

3. Jun 9, 2015

### ecastro

Try starting with the Squeeze Theorem.

4. Jun 9, 2015

### ecastro

Try starting with the Squeeze Theorem.

5. Jun 9, 2015

### mekise

I tried the Squeeze Theorem.

-1≤sinx≤+1 ⇒ -1/x2≤sinx/x2≤+1/x2

the two functions converge so my function converge!

Thank you very much!

6. Jun 9, 2015

### Staff: Mentor

I gave you a like because you are on the right track - but that is wrong - the squeeze test applied to functions that converge to the same thing.

Like I said use the p test on 1/x^2, note that the absolute value of your function is less than or equal to 1/x^2. Thus its absolute value converges by the comparison test, then apply the absolute convergence test.

Thanks
Bill

7. Jun 9, 2015

### Zondrina

You have already pointed this out:

$$\int_{1}^{\infty} \left| \frac{\text{sin}(x)}{x^2} \right| \space dx \leq \int_{1}^{\infty} \frac{1}{x^2} \space dx$$


text:
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text:
# Talk:Greek numerals

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## Attic numerals

I'm using a Unicode (utf8) MSIE browser and not all of the characters are displaying correctly. Does someone know what the symbols are and can they correct them to work with standard characters? comment by Special:Contributions/71.139.23.245, 21:22, 17 April 2006

The Attic numerals for 5, 50, 500, 5000, 50000 are not rendered correctly. I think images should be used until wider support for these characters is available. These images could eitehr replace the characters inline, or be a separate summary image of all the numerals.
—DIV (128.250.80.15 (talk) 00:37, 6 February 2009 (UTC))
I am duplicating the French ref for Linear A and B for the Attic numerals as well, because images of both are present in it. That ref has individual images of them, but they're probably covered by Copyright, so cannot be used here. — Joe Kress (talk) 03:23, 6 February 2009 (UTC)

## "extended by using obsolete letters"

From the article: This requires 27 letters, so the 24-letter Greek alphabet was extended by using three obsolete letters: digamma (ϝ, also used are stigma ϛ or στ) for 6, qoppa (ϟ) for 90, and sampi (ϡ) for 900. See Numerals: Stigma, Koppa, Sampi.

Isn't this somewhat reverse? I'd think the numeral system retained three letters that have since gone obsolete in Greek, no? —Muke Tever talk 01:42, 2 May 2006 (UTC)

The three characters were already obsolete in classical Greek as letters or were used only in restricted dialects or locations during classical times. See the reference cited in the article Numerals: Stigma, Koppa, Sampi and Greek alphabet#Obsolete letters. — Joe Kress 04:15, 15 May 2006 (UTC)

## numeral vs number

A number is an abstract entity that represents a count or measurement, whereas a numeral is a symbol or group of symbols that represents a number. Therefore, "numeral" is the correct title for this article.   Andreas   (T) 01:52, 18 May 2006 (UTC)

## modern Greek

The line "In modern Greek, upper case is preferred" is the first I knew this is still used. How widespread is it? Is it like Roman numerals in latin-alphabet countires, just a veneer of antiquity not used for serious maths or arithmetic? jnestorius(talk) 22:46, 26 October 2006 (UTC)

The Greek numerals are used in modern Greek texts approximately to the same extent and in the same context as Roman numerals are used in texts written with the Latin alphabet. For example, they are used exclusively for royalties, for chapters in books, etc. The use of Lower-case letters, however, is very rare. Roman numerals are rarely used.

Examples from the Greek WP and Wikisource:

Andreas   (T) 01:24, 27 October 2006 (UTC)

They are used generally as ordinal numbers. Names of kings, chapters in a book, but also for instance school classses. Thus, first grade would be Α' τάξη. Roman numerals are very rarely used in Greek, mostly in an academic context. Greek numerals are deffinitely not used for computations, excepting numerology, which seems to be a trend in Greek mystico-nationalistic circles. Note however that your average Greek would only easily recognise Greek numerals up to 6 (the number of grades in primary school), or in the best case 8 (the number of academic semesters). Anything higher than that would require some contemplation to recognise, and anything above 29 would cause much consternation even to your average philologist. Any knowledge of Roman numerals is even more limited, naturally. Another indication that Greek numerals are muchly a dead piece of tradition in Greece is that "Κωνσταντίνος Β'" would be literally pronounced as "Constantine B (the letter)" by most people, rather than "Constantine the Second". That is to say, they fully understand the intended meaning is "the second", but still pronounce the letter alphabetically rather than read it out as a numeral. Druworos 09:28, 28 September 2007 (UTC)
They are used in the 1904 Patriarchal Edition of the Greek New Testament - this is Koine, not modern Greek, but it is a widely-used, relatively modern edition of the New Testament that still uses these numbers. 24.211.251.205 (talk) 20:12, 23 April 2012 (UTC)

## Ambiguation?

"For example, 2006 is represented as ͵βϛʹ (2000 + 6)." How did they made a difference between e.g. 950 001 and 900 051? Was it both ͵ϡναʹ?129.206.26.17 10:39, 28 September 2007 (UTC) From the example given in the Greek version of this exact article, the answer seems to be that they simply use a space between the numerals. 950,001 would be ͵ϡν αʹ, while 900,051 would be ͵ϡ ναʹ. - Misha 216.254.12.114 (talk) 19:38, 6 December 2008 (UTC)

## Limitation of large numbers

I thought Ancient Greek did not have words for higher numbers than tens of thousands. This lead to profound difficulty to imagine large numbers. Archimedes eventually solved the problem by inventing exponentiation. Anyone who can verify this?

2009-03-10 Lena Synnerholm, Märsta, Sweden.

## Alternate name ref

[Moved from User talk:Joe Kress#Greek numerals.]
Why did you remove my reference? Languagehat (talk) 22:05, 21 June 2009 (UTC)

Your ref was from a Wikipedia mirror, Slider.com, which copied it from Wikipedia. The equivalent Wikipedia article is here as it appeared on 15 October 2005 before a merge tag was added in January 2006 and redirected to this article. The alternate name, Ionian numerals, should have been added to this article at that time. I removed your ref because a Wikpedia article cannot be a reference for another Wikipedia article as it is not a reliable source. — Joe Kress (talk) 23:43, 21 June 2009 (UTC)
Gotcha -- thanks, and now I know better! Languagehat (talk) 13:38, 22 June 2009 (UTC)

## Capital letters

Capital letters were certainly used as numbers in antiquity as well, for instance when coins were dated. Like on Seleucid coins.

I think they should be added as well. Sponsianus (talk) 09:57, 19 September 2009 (UTC)

## Zero and seventy

If by Byzantine times zero was written exactly as omicron, and the old notation is still used to this day, how would one distinguish the omicron for 70 from omicron for zero? Wouldn't it be grounds for confusion? 75.85.81.0 (talk) 12:06, 3 November 2009 (UTC)

The Byzantine zero was only used when copying Hellenistic astronomical works with sexagesimal notation, where the largest allowable number in any sexagesimal position was 59 (νθʹ), thus omicron must mean zero, not seventy. Compare this with our decimal notation, where the largest allowable number is any decimal position is 9. — Joe Kress (talk) 02:40, 11 November 2009 (UTC)

## Spam watch

Be on the lookout for spamming on this page. I just edited the statement "It came from Mars." from the "History" section of the article. mast3rlinkx (talk) 22:38, 10 November 2009 (UTC)

That was vandalism, not spam. I had to revert further back because the same IP editor added Mars as a link in See also. — Joe Kress (talk) 02:40, 11 November 2009 (UTC)

## These are known as byzantine

In Greece, this type of numeration is called Byzantine numbering. Fix it. Kthxbye. 195.10.10.180 (talk) 16:51, 12 March 2010 (UTC)

Do you have a source for that? I thought they pre-dated the Byzantine era. -- Radagast3 (talk) 21:18, 12 March 2010 (UTC)

## Upper-case / Lower-case

Greek minuscule letters weren't invented until the 6th century CE, at the *earlierst*. It seems strange (incorrect) to use them for the whole article on ancient Greek numerals...64.134.229.227 (talk) 22:54, 11 July 2010 (UTC)

## some letters in TeX

The versions of TeX used by Wikipedia now include the archaic letters ${\displaystyle \mathrm {\stigma} \,}$ (stigma) and ${\displaystyle \mathrm {\koppa} \,}$ (koppa):

${\displaystyle \alpha \beta \gamma \delta \varepsilon \mathrm {\stigma} \zeta \eta \vartheta \iota \kappa \lambda \mu \nu \xi \mathrm {o} \pi \mathrm {\koppa} \varrho \sigma \tau \upsilon \varphi \chi \psi \omega \,}$

The precise appearance of these may depend on whether you're using mathJax or not.

I created the table below for use in the article titled Ptolemy's table of chords. It doesn't go beyond 100 because no numbers bigger than 180 are needed there.

${\displaystyle {\begin{array}{|rlr|rlr|rlr|}\hline \alpha &\mathrm {alpha} &1&\iota &\mathrm {iota} &10&\varrho &\mathrm {rho} &100\\\beta &\mathrm {beta} &2&\kappa &\mathrm {kappa} &20&&&\\\gamma &\mathrm {gamma} &3&\lambda &\mathrm {lambda} &30&&&\\\delta &\mathrm {delta} &4&\mu &\mathrm {mu} &40&&&\\\varepsilon &\mathrm {epsilon} &5&\nu &\mathrm {nu} &50&&&\\\mathrm {\stigma} &\mathrm {stigma\ (archaic)} &6&\xi &\mathrm {xi} &60&&&\\\zeta &\mathrm {zeta} &7&\mathrm {o} &\mathrm {omicron} &70&&&\\\eta &\mathrm {eta} &8&\pi &\mathrm {pi} &80&&&\\\vartheta &\mathrm {theta} &9&\mathrm {\koppa} &\mathrm {koppa\ (archaic)} &90&&&\\\hline \end{array}}}$

Michael Hardy (talk) 17:31, 18 October 2011 (UTC)

## Largest writable value?

What is the largest writable value using Greek numerals?

I am not an expert but using the rules as presented here, it appears that:

• ͵θʹ = 9,000
• ͵ϟʹ = 90,000
• ͵ϡʹ = 900,000

DMahalko (talk) 01:14, 1 March 2012 (UTC)

### GNLS stacking?

Does the "Greek Lower Numeral Sign" (GLNS) stack?

If GLNS stacking is permitted then:

• 9,000,000,000,000,000,000 = ͵͵͵͵͵͵θʹ

DMahalko (talk) 01:14, 1 March 2012 (UTC)

### GNLS, applied per character or group?

Also, from the syntax described in this article, it is unclear to me if the GLNS applied per letter over 900, or if it is applied to letters in groups of three. Does ͵ϡ͵ϟ͵θʹ or ͵ϡϟθʹ equal 999,000 ?

If GLNS sign stacking is not permitted then it looks like the largest single writable value (without the Myriad) is (assuming the GLNS is used per digit) -- ͵ϡ͵ϟ͵θϡϟθʹ = 999,999 DMahalko (talk) 01:14, 1 March 2012 (UTC)

There is no indication in the references that the GLNS was used beyond one extra digit - if it was there would be ambiguities and duplications. I propose to make the following change unless counter-evidence can be found:
To represent numbers from 1,000 to 9,999 the letters α-θ are reused to serve as thousands. A "left keraia" (Unicode U+0375, ‘Greek Lower Numeral Sign’) is put in front of thousands to distinguish them from the standard use. For example, 2011 is represented as ͵βιαʹ (2000 + 11).
It would then follows that beyond 10,000 the Myriad notation has to be used. As currently phrased there would be no way to indicate whether ͵ριαʹ represents 110,001 or 100,011 (an there would be redundant ways of expressing numbers from 10,000 to 999,999).--Keith Edkins ( Talk ) 19:07, 2 March 2012 (UTC)

## WP:ERA

Per WP:ERA, this edit established the page's usage as AD / BC and, pending a new consensus, that should be consistently maintained. Thanks. — LlywelynII 05:04, 1 November 2013 (UTC)

## M

The article on sampi is very well done and sourced and clearly states that the Greek letters placed over an M-looking symbol (really a variant san or sampi, not mu) are being multiplied by 1000, not 10 000. Accordingly, I've removed this text from the article:

To represent greater numbers, the Greeks also used the myriad from the old Attic numeral system in their notation. Its value is 10,000; the number of myriads was written above its symbol (M). For example,

${\displaystyle {\stackrel {\delta \phi \pi \beta }{\overline {\mathrm {M} }}}{\overline {\psi \theta }}=4582\times 10,000+709=45,820,709.\,}$
Other forms are also possible. When that didn't suffice, the "myriad myriad" (one hundred million, written ΜΜ) was used.

The source here was very good as well (Thomas Little Heath (1931). A Manual of Greek Mathematics. p. 18. ISBN 9780486154442. Retrieved 2 Mar 2011.... in fact, it contains additional information about alternate systems and specifics that should be added to clarify some of the questions above) but I'd really like some direct clarification of when M is 40, 900, 1000×X, or 10000×X rather than just putting everything together with no explanation at all.

Was the myriad M just written larger? Was the Upsilon sometimes included over it important? Was the sampi m (×1000s) only a regional thing or very limited in time? etc... — LlywelynII 09:49, 1 November 2013 (UTC)

## Symbols do not display

Although most of the non-Latin characters show properly there are a few, such as in the first sentence of the History Section of the main Article, that appear on my PC as little rectangles. Is this a problem with my setup, or with the way the text has been coded? --DStanB (talk) 15:51, 7 January 2015 (UTC)

Observation reproduced, here is what I see in the first sentence of the History Section:

An image of a text excerpt from Wikipedia that shows a possible error in text formatting.

Hristo kirov (talk) 10:02, 22 March 2016 (UTC)

## Isopsephy (Gematria)

I added... The art of assigning Greek letters also being thought of as numerals and therefore giving words/names/phrases a numeric sum that has meaning through being connected to words/names/phrases of similar sum is called isopsephy (gematria). - The Decoder 50.153.106.162 (talk) 18:15, 29 May 2015 (UTC)

## Ptolemy

According to Nobbe's edition, Ptolemy used a different system in his Geography. Apparently, rather than expressing the fractional part as multiples of 1/60, he composited the fractions from increasingly smaller partial fractions, like this:

λ̄ε 𐅵' γ' ιβ' = 30 + 5 + 1/2 + 1/3 + 1/12 = 35 + 55/60 — Preceding unsigned comment added by 80.114.146.117 (talk) 11:25, 19 June 2016 (UTC)


text:
# Raman Instrumentation for Confocal Raman Microscopy

• Olaf Hollricher
Chapter
Part of the Springer Series in Surface Sciences book series (SSSUR, volume 66)

## Abstract

Raman spectroscopy is a weak scattering technique in which typically less than one in a million excitation photons give rise to a single Raman photon. The situation is even more difficult in confocal Raman microscopy, as the detection volume is reduced by the small pinhole and thousands of spectra must be collected in a short period of time. Therefore, careful design of the excitation and detection optics, as well as the appropriate choice of spectrometer and detector is important for successful experiments. As detectors and instrumentation in this field continue to quickly evolve, a snap-shot of state of the art Raman equipment and instrumentation will be provided.

## References

1. 1.
C. Raman, Indian J. Phys. 2, 387 (1928)Google Scholar
2. 2.
T. Maiman. US Patent$$\sharp$$ 3353115 (1961). Apr 13Google Scholar
3. 3.
W. Boyle, G. Smith, Bell Syst. Tech. J. 49, 587 (1970)
4. 4.
S. Sze, Physics of Semiconductor Devices (1981), p. 407Google Scholar
5. 5.
I. Lewis, H. Edwards (eds.), Handbook of Raman Spectra (Dekker, 2001)Google Scholar
6. 6.
T. Dieing, O. Hollricher, Vib. Spectro. 48, 22 (2008)
7. 7.
U. Schmidt, S. Hild, O. Hollricher, Macromol. Symp. 230, 133 (2005)
8. 8.
U. Schmidt, W. Ibach, J. Mueller, O. Hollricher, SPIE Proc. 6616(1), 66160E (2007)


text:
FizeauV

2020-10-26

What is the value $\mathrm{cos}\left({72}^{\circ }\right)$? Write your answer in decimals.

Demi-Leigh Barrera

The formula:
$\mathrm{cos}\left(x\right)=\mathrm{sin}\left({90}^{\circ }-x\right)$
Substitute the given angle:
$\mathrm{cos}\left({72}^{\circ }\right)=\mathrm{sin}\left({90}^{\circ }-{72}^{\circ }\right)$
$\mathrm{cos}\left({72}^{\circ }\right)=\mathrm{sin}\left({18}^{\circ }\right)$
$=\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}$
$\approx 0.30901$

Do you have a similar question?