I wanted to pose a question here (also posted to r/physics a bit earlier) and was hoping the you all might be able to help.

I was re-reading a book by the philosopher Hilary Putnam titled Naturalism, Realism, and Normativity and came across a footnote within a chapter that focused on semantics.** To give some clarity, I will quote the context within the chapter and then cite the footnote immediately below that (where there is emphasis in the original text it is Putnam's):

Of course, there are natural kinds that we are all pretty good at identifying without relying on experts, for example, water. But it would be wrong to think that the extension of the word "water" is fixed simply by the superficial properties of water--that is, that "water" means "liquid that quenches thirst, has such and such a taste,...and so on." And it would be equally mistaken to say that "water" means "$\def\w{\rm H_2O}\w $." It is true that most educated people now know (or think they know) that $\w $ is the chemical formula for water, but educated speakers and uneducated speakers do not speak different languages. (Moreover, "water is $\w $" is not precisely true!21) Water has the same meaning in the mouth of an English-speaking scientific ignoramus and the mouth of an English-speaking chemist; they both know the meaning of "water" in English in the sense of possessing the sort of linguistic competence that every person who masters the language is linguistically obliged to have a connection with that word. They refer to the same liquid when they use the word...

And the footnote:

  1. Normal water is actually a quantum mechanical superposition of $\w $, $\rm H_4O_2$, $\rm H_6O_3$...Very little (if any) water is simply $\w $

Now I just want to be clear that my question relates to the footnote that seems to promote the idea that every water molecule (or any other molecule?) is in a superposition state that consists of integer multiples of the base molecular formula that we know...going to infinity. I do not mean that water itself can very likely be composed of many different isotopes of Hydrogen (including Deuterium and Tritium) and Oxygen (16-17-18), which is tackled pretty cogently in this essay.

I've scoured the internet looking for other references to this with no luck. I feel like my handle on physics concepts is decently advanced and I've read enough about superposition to at least think I would have run across the concept couched in terms similar to those Putnam would make. I would have reached out to ask him myself but unfortunately he passed away shortly after the book was published. I have massive respect for him as a philosopher, so I'm going to give him the benefit of the doubt on this...but could anyone here point me in the right direction as to whether he is actually correct or whether he may have misspoken or taken some liberty here?

Any help is greatly appreciated!

**As I'm sure many of you know, an understanding of how the words and symbols we use come to acquire their meaning (whether through reference/correspondence with something "out there" in reality, common usage in a community of speakers, etc.) is one of the most important questions asked in philosophy.

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    $\begingroup$ I'm voting to close this question as off-topic because this is more like a Chemistry SE question as strictly speaking $H_2O$ and $H_4O_2$ would be different compounds, not different states of $H_2O$. $\endgroup$ – StephenG Mar 6 '18 at 1:39
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    $\begingroup$ I think Putnam thought about protonation where water turns into OH$^-$ and H$_3$O$^+$, and water dimers and clusters. I know at least some books in the 70s referred to non-H$_2$O water structures as a fact. $\endgroup$ – Anders Sandberg Mar 6 '18 at 1:39
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    $\begingroup$ @DavidWhite Putnam was a giant and not outside his field when discussing science at all, particularly mathematical logic and mathematics; see also here. His work was instrumental in the solution of Hilbert's tenth problem. Your comment may have some validity for chemistry, but as a blanket statement it's not always right at all. $\endgroup$ – WetSavannaAnimal Mar 6 '18 at 1:54
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    $\begingroup$ Why would you tag this question with "quantum-field-theory"?? $\endgroup$ – Prof. Legolasov Mar 6 '18 at 4:18
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    $\begingroup$ @WetSavannaAnimalakaRodVance being a great philosopher and mathematician does not make you a great physicist or chemist. $\endgroup$ – RonJohn Mar 6 '18 at 4:20

I'm not sure I would have phrased it exactly that way, but I think his statement is by-and-large defensible. The crux of the issue is that, in liquid water, there is no sharp line between intramolecular and intermolecular H-O bonding. The intramolecular H-O bond is closer than the intermolecular H-O hydrogen bond, but it's a difference of degree not kind, and in molecular simulations you'll frequently see smooth transitions where an H gradually detaches from one O while attaching to another O.

A neat demonstration of this fact is when you mix D2O with H2O; in a very short amount of time the liquid will be 50% HDO molecules!

So in the mess of liquid water, you can't neatly & perfectly split up all the atoms into H2O triplets. And if you try to define some criterion like "if an H+O are closer than distance X, I'll say they're part of the same molecule", you'll find that your criterion is kinda arbitrary, and more importantly you'll end up dividing the liquid into a mixture of H2O, OH, H3O, H3O2, H4O2, H5O2, etc. "molecules" at any given instant of time.

[One shouldn't take this too far though--at any instant, there are plenty of unambiguous tightly-bonded H2O triplets.]

(I don't have any good references off the top of my head, but I used to do research in the physical chemistry of water, and these are my vague memories from reading papers and attending technical lectures back then.)

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    $\begingroup$ I can appreciate that hydrogen bonding in liquid water is significant and complex, but the Gibbs free energy of formation for water is-237.14 kJ/Mole, which is over five times the heat of vaporization, 40.65kJ/ mole. Surely that's enough difference to distinguish them? $\endgroup$ – Charles E. Grant Mar 6 '18 at 4:40
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    $\begingroup$ @CharlesE.Grant - The free energy of formation is the difference between OH bonds vs O2 and H2 bonds, right? Well, OH bonds are definitely preferred. I'm not sure I follow how that's related though. I was saying that there is a low energetic cost to having an H which is equally shared between two different O's, or an O which is sorta-bonded to three different H's, etc., as opposed to discrete persistent H2O triplets. $\endgroup$ – Steve Byrnes Mar 6 '18 at 13:32
  • $\begingroup$ What era were you doing water research in? $\endgroup$ – WetSavannaAnimal Mar 6 '18 at 22:42
  • $\begingroup$ "The intramolecular H-O bond is closer than the intermolecular H-O hydrogen bond, but it's a difference of degree not kind." I don't understand this statement: intramolecular bonds are covalent, intermolecular bonds are hydrogen bonds, so they are qualitatively (very) different. Then maybe it's not easy to tell one from the other just by looking at a "snapshot", but this is another story. $\endgroup$ – valerio Mar 6 '18 at 23:23
  • $\begingroup$ "you'll frequently see smooth transitions where an H gradually detaches from one O while attaching to another O" Not if the bond is covalent, no. This can only happen if the bond is an hydrogen bond. $\endgroup$ – valerio Mar 6 '18 at 23:27

I think it's tosh. A very small fraction of liquid $\rm H_2O$ will spontaneously decompose and recombine with very small amount of $\rm H_3O^+ $ (hydronium ion) and $\rm OH^-$ (hydroxyl) ions. Weak hydrogen bonds may form between neighboring water molecules, but that doesn't really count as $\rm H_4O_2$ or higher order molecules.

While I have a great deal of respect for the study of philosophy, I think they can go far amiss when they try to invoke technicalities of the sciences in their arguments. Feynman's experience when discussing the nature of "essential objects" in a philosophy seminar is relevant.

  • $\begingroup$ How strong does a bond have to be to count? Hydrogen bonds are a key structural component of DNA. Does that mean the double helix is a complex of two DNA molecules? $\endgroup$ – Mark H Mar 6 '18 at 4:11
  • $\begingroup$ @MarkH to my mind it depends on the environment and the number of bonds involved. In liquid water two $\rm H_2O$ moieties would be linked by a single hydrogen bond and the bond between any two moieties would be transitory. In biological conditions although the pairing of any single base pair may be transitory but the overall coupling of two strands of DNA is quite stable. $\endgroup$ – Charles E. Grant Mar 6 '18 at 4:25
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    $\begingroup$ @CharlesE.Grant I think that you have read too much into Putnam’s footnote. In “shorthand” Putnam was referring to the formation of open structure clusters of hydrogen molecules in liquid water, an idea which can be used to explain the anomalous expansion of water? $\endgroup$ – Farcher Mar 6 '18 at 7:09
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    $\begingroup$ Every link to "Surely you're joking, Mr Feynmann" should be tagged with "serious time trap" or something, like they do with TvTropes on the movies SE. I swear I lose half an hour everytime $\endgroup$ – Nico Mar 6 '18 at 7:53

I am not sure I will call this an "answer" so much as a lengthy comment that nonetheless points to a plausible answer. I think I have an idea as to what is going on here, based on intuition I realized from an earlier discussion I had once on this forum.

The trick is that what's taught in those chem books is in some strict sense a 'lie' when it comes to things like this involving quantum mechanics. To understand what we mean, we should consider a rather simpler example, namely that of the question of atomic orbitals and atoms heavier than hydrogen. As we all know, the usual set of orbitals $1s$, $2s$, $2p$, etc. are all cute names for the energy eigenstates of hydrogen or a hydrogen-like atom, described by a solution to the Schrodinger equation with a Hamiltonian (in position basis)

$$\hat{H} = \frac{\hat{p}^2}{2m_e} - \frac{1}{4\pi \epsilon_0} \frac{e^2}{r^2}$$

where as usual $\hat{p} = -i\hbar \nabla$ is the momentum operator, $e$ is the charge on the electron and $m_e$ is the electron mass. The first term is the kinetic term, the second is the Coulomb's law term for the potential about the nucleus. The solutions to this Schrodinger equation are the orbitals, and are good enough 3-dimensional wave functions $\psi(x, y, z)$ (or perhaps since we used $r$, $\psi(r, \theta, \phi)$).

However, when you go up one electron to Helium (He), you get a wave function that involves six coordinates - 3 for each electron: $\psi(r_1, \theta_1, \phi_1, r_2, \theta_2, \phi_2)$, and moreover the Hamiltonian contains a cross potential term for the repulsive interaction between the two electrons. When one tries to solve this, one finds out that the random variables for the individual electron positions are not statistically independent, that is, each electron cannot be described on its own as simply having a wave function that does not depend on the position of the other electron: depending on where one is measured to be, the probability to measure the other will be different. (Another way to say this is there is a non-zero correlation. Technically you have to use a somewhat modified notion of "independence" as the electrons are indistinguishable fermions, than the ordinary notion in classical statistics you'd see outside quantum mechanics or in other fields of science beyond physics, where "independent" means an antisymmetric combination) This is entanglement, and it means that in the most honest description we cannot speak of separate orbitals at all but must think about the holistic 6-dimensional wave function for the atom. Now this doesn't mean that you can't talk of the "atom" at all as a physical entity - you can describe roughly a shape by taking an "electron localization function" that is the probability to find ANY electron in some volume of space, and this is the "shape of the atom" in some sense (it looks like a blob), but on their own the electrons cannot honestly be assigned to separate orbitals, in other words it makes in the strictest sense no sense to talk of "both electrons are in the $1s$ orbital" etc. because those are simply not solutions of the Schrodinger equation.

Nonetheless, because we cannot solve this three-body problem exactly and analytically, we try to approximate it using various fictitious constructs, and one such way to do that IS to try and use a product of the hydrogen orbitals, suitably adjusted with various types of correction, and this works to a reasonable extent, such that the more corrections and bells and whistles you add to it the more accurate you can get it. You could think of these as being other "models" of the atom and different "pictures" beyond the orbital picture, but in the "exact" case it's none of these, it's a statistically correlated electron blob which you have to describe using a 6-dimensional wave for the actual probabilities.

So I am commenting that perhaps the same is what applies for water as well. Given two interacting water molecules, you can write up a gigantic Hamiltonian with six nuclei and 20 electrons and myriad cross terms such that the wave function for the whole thing is a 26-particle, 78-dimensional monstrosity at any given time, with none of its component particles truly statistically independent, just another blob effectively. Some states or approximate solutions of this may "look like" for some suitable definition (e.g. like the ELF above) two "separate" water molecules (that is, $2\ \mathrm{H}_2 \mathrm{O}$), others may look like them more joined together (that is, $\mathrm{H}_4 \mathrm{O}_2$), and a more "real life" configuration could be better approximated by summing together these in a superposition, but even that may not necessarily be the most accurate setup. If a 3 body problem is hard, a 26-body one is insane, much less a roughly $10^{24}$-body one for a full cup of water.

(FWIW if you want to talk about the fullest quantum solution for a hydrogen atom you will want to include in the wave function a freely-varying position for the nucleus. If you do this the solutions to Schrodinger's equation are a product of the usual electron-orbital wave functions and a free-particle wave function for the nucleus, and the latter is what allows say, a hydrogen atom to diffract through a two-slit matter-wave setup. Good luck finding this in a textbook talking the solution of the hydrogen atom btw. The physical reasoning is relatively simple: there is a probability distribution on the nucleus's position. Once you've found the nucleus by one measurement though, then the probability of the electron will be obtained by an orbital centered around that position at which you saw it.)

As said though I don't know if this is exactly what is going on, but it seems to be right. The original discussion I had with orbitals was at - pay attention to Emilio Pisanty's answer:

What does an orbital mean in atoms with multiple electrons? What do the orbitals of Helium look like?

and it makes sense the original asker cannot find information because in truth it is hard to find information even on the case of orbitals by searching terms like "orbitals are fake" or "orbitals are a fiction", you get only a literal handful of results - which suggests it's not generally something most are aware of, or at least this specific phrase is not used to describe the situation.

As I said I don't know if nor do I expect the above to be exactly the "truth", I'm just putting this out more to suggest the general direction of what's going on. Basically you have the true solution which is none of these things, really, but you can approximate it to various better and better extents with more complex approximative models that look like at one level separate individual molecules, then another as superpositions of multi-molecule forms, then another as something even more complex, etc. and also likewise with the detailing in the individual atoms themselves. FWIW the same kind of thing is also done with classical mechanics as well, e.g. describing orbits in a complex planetary system. You can to a first order treat them as separate elliptical orbits, then if you want to add more detail you can add things like precession modeled as certain (simplified) kinds of planet-planet interactions, etc. The "truest" solution is they're none of this stuff, but to get closer to it you can add in various details - it's a general feature of how physics is done.


Putnam is giving you an example of a word whose meaning is correctly understood by everybody, including the two year old toddler, and then pointing out that the word is used by experts to denote something entirely different, the water molecule. Yes, roughly speaking water consists of H2O molecules, but

  1. the meaning of that phrase is especially complicated in the case of water (contrary to "oxygen is O2" which is a much more simple case)
  2. a non-specialist does not, as a rule, even understand what a molecule is exactly (even if he/she thinks otherwise).

But since your question was about physics, the truth as of 2018 is still not very well understood. Some years ago there was a controversy where there was a quantitative theory of water in which it was modelled as a solid with phonons and people ridiculed it. Then, it turned out the theory was the one which best explained several puzzling phenomena regarding water, so that it gained serious acceptance. I think there is still no final settlement to these issues.

  • $\begingroup$ I don't doubt you last paragraph, but could you find a citation? $\endgroup$ – WetSavannaAnimal Mar 6 '18 at 22:43
  • $\begingroup$ You can read about it here, phys.org/news/2016-01-ice-like-phonons-liquid.html, but the original theory was developed by some russian theorists, many years ago and these guys are not citing them for some reason. $\endgroup$ – Kostas Mar 8 '18 at 7:37

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