Quantizable nature of the physical world

[Kherubin]

In lieu of my previous question, (https://www.physicsforums.com/showthread.php?t=494393), which seemed a bit tricky to answer (!), I will simply ask the deeper physical question that lies at its base.

That questions is whether there are any physical phenomena (temperature, pressure, concentration) which cannot ultimately be quantized?

Any thoughts on this matter?

Thanks
Kherubin

 

[alxm]

Temperature, pressure and concentration are all statistical properties. So they're not quantized in the sense that there's any fundamental discrete unit for them, for the simple reason that they become increasingly meaningless with smaller system sizes. There's no real meaning to speaking of the 'temperature of an atom' for instance. (Or as I heard one professor put it: "Temperature is a thing you measure with a thermometer.")

Quantum mechanics doesn't say everything is quantized. Rather, certain properties under certain conditions are quantized. (And in different ways) An electron bound to an atom can only be in certain energy states, and certain angular momentum states, for instance. But a free electron can have any value for energy, momentum and angular momentum. Things which are quantized aren't (always) quantized such that there's a fundamental "unit" which governs anything - i.e. the electron in an atom can have only certain values for its energy, but those numbers aren't the same in different atoms, or discrete multiples of some number.

To address your original question though, the number of chemical species isn't infinite. Because not just any arrangement of atoms constitutes a compound, only stable ones do. Which you can give a strict mathematical definition as an arrangement of atoms that has a (local) minimum in energy. Or in other words, where the forces on the atoms are zero (otherwise it'd be in the process of coming together or falling apart). At absolute zero, and ignoring that there isn't an infinite amount of stuff around, it may be possible in theory to have a countably-infinite number of molecules, since they could be infinitely large. (But not an uncountably-infinite set, which you'd have if every arrangement of atoms could be considered a compound) So it's "discrete" in that sense - the same way the integers are.

In reality you don't have an infinite amount of stuff and you can't achieve absolute zero, so the number of real compounds is finite. But so large it's practically endless from the human perspective anyway. In reality we're more limited by what can be synthesized than what can be imagined.

 

[SpectraCat]

Temperature, pressure and concentration are all statistical properties. So they're not quantized in the sense that there's any fundamental discrete unit for them, for the simple reason that they become increasingly meaningless with smaller system sizes. There's no real meaning to speaking of the 'temperature of an atom' for instance. (Or as I heard one professor put it: "Temperature is a thing you measure with a thermometer.")

Quantum mechanics doesn't say everything is quantized. Rather, certain properties under certain conditions are quantized. (And in different ways) An electron bound to an atom can only be in certain energy states, and certain angular momentum states, for instance. But a free electron can have any value for energy, momentum and angular momentum. Things which are quantized aren't (always) quantized such that there's a fundamental "unit" which governs anything - i.e. the electron in an atom can have only certain values for its energy, but those numbers aren't the same in different atoms, or discrete multiples of some number.

To address your original question though, the number of chemical species isn't infinite. Because not just any arrangement of atoms constitutes a compound, only stable ones do. Which you can give a strict mathematical definition as an arrangement of atoms that has a (local) minimum in energy. Or in other words, where the forces on the atoms are zero (otherwise it'd be in the process of coming together or falling apart). At absolute zero, and ignoring that there isn't an infinite amount of stuff around, it may be possible in theory to have a countably-infinite number of molecules, since they could be infinitely large. (But not an uncountably-infinite set, which you'd have if every arrangement of atoms could be considered a compound) So it's "discrete" in that sense - the same way the integers are.

In reality you don't have an infinite amount of stuff and you can't achieve absolute zero, so the number of real compounds is finite. But so large it's practically endless from the human perspective anyway. In reality we're more limited by what can be synthesized than what can be imagined.

If you count any molecule that is stable and can hypothetically be synthesized based on the established rules of chemistry, then there are a countably infinite number of stable molecules at room temperature. In fact, there are a countably infinite number of stable hydrocarbons at room temperature. In fact, there are a countably infinite number of stable polyaromatic hydrocarbons (PAH's) at room temperature. The network of carbon atoms in a PAH is extremely stable, and you can generate a "new" PAH molecule by simply fusing another aromatic ring anywhere in the structure.

Now perhaps you can argue that once the PAH framework gets beyond a certain size, you stop considering them as different molecules, and start thinking of them as just different sized chunks of hydrogen-atom terminated graphene sheets. Fair enough, but there are still a countably infinite number of them ;).

 

[tom.stoer]

That questions is whether there are any physical phenomena (temperature, pressure, concentration) which cannot ultimately be quantized?

Quantum mechnanics allows one to extract classical and thermodynamical properties from quantum states (Hilbert space states or generalized states = density operators in statistical mechanics).

Looking at quantum mechanics it is not clear if time can be quantized; in standard QM time is an external, continuous parameter.

Looking at the problem of constructing a quantum theory of gravity it is not completely clear if we will succeed in quantizing space and time (but there are quite promising approaches).

 

[Kherubin]

Thank you all for your expansive and comprehensive replies.

What would happen to the number of potential chemical molecules should we (hypothetically!) develop a technique capable of stabilizing metastable molecules?

Would this then make the number of molecules uncountably infinite?

 

[Kherubin]

My original point was simply that if their are, strictly speaking, 'continuous' aspects of the physical world, for example, temperature, you could, in theory at leat, produce an infinite number of chemical compounds.

By way of example, you could conduct one synthesis procedure at 865.0000003K and another at 865.0000004K, and, again, in theory, the products of these two separate synthesis procedures would be different. There, of course, is a point reached, in practical terms, where it is in principle, impossible to differentiate between these two products on functional grounds, and therefore the genuine concept of infinity is meaningless in this regard, however, it remains in theory.

 

[SpectraCat]

My original point was simply that if their are, strictly speaking, 'continuous' aspects of the physical world, for example, temperature, you could, in theory at leat, produce an infinite number of chemical compounds.

I guess you are asking about uncountable infinities here? I think I already showed that there are countably infinite numbers of molecules, or at least of "allowed" molecular structures.

By way of example, you could conduct one synthesis procedure at 865.0000003K and another at 865.0000004K, and, again, in theory, the products of these two separate synthesis procedures would be different. There, of course, is a point reached, in practical terms, where it is in principle, impossible to differentiate between these two products on functional grounds, and therefore the genuine concept of infinity is meaningless in this regard, however, it remains in theory.

Well, now you are getting into semantics ... if you consider every possible value for a bond length, then there are an uncountable infinity of possible structures for H2, because the bond length is a real number. That is really no different than what you are doing in you statement above ... if we only allow one chemical mechanism (which is what I believe you intended), then the only difference between the synthetic products at different temperatures will be the average values of the structural parameters (i.e. bond-lengths and bond-angles). If you want to call those different molecules, then fine, but that is just semantics. At that point you have essentially assumed the characteristics you want by changing the definition of what a molecule is.

 

[Kherubin]

Thank you for your replies.

I'm simply interested in whether or not the number of chemical species can be infinite (countably or otherwise) without relying upon the construction of infinitely-sized molecules.

Is my point about stabilizing the metastable correct?

Thanks again for your time,
Kherubin

 

[SpectraCat]

Thank you for your replies.

I'm simply interested in whether or not the number of chemical species can be infinite (countably or otherwise) without relying upon the construction of infinitely-sized molecules.

Is my point about stabilizing the metastable correct?

Thanks again for your time,
Kherubin

Well, I think I answered that in my last post. Unless you change the definitions of what we understand to be molecules, then I think you must resort to infinitely-sized molecule to get infinite numbers of molecules. Maybe I didn't understand what you meant by "stabilizing metastable molecules", but it sounded like you just wanted to say that an H2 molecule with a bond length of 0.75 Angstroms is a different molecule than one with a bond length of 0.76 Angstroms, and so on. If I did understand correctly then:

1) You cannot do that in general (at least not without using external fields) .. H2 molecules are bound in a single well potential, so there is no driving force to stabilize them at different structures.

2) However, if you *did* somehow do that (for example by modifying the potential using external fields), then you would indeed have found a way to generate an infinite number of such states of H2, but they would not qualify as different molecules using standard definitions.

3) All of this discussion/analysis is in the context of molecules having well-defined structures in the first place, which they do not .. at least not in the quantum mechanical sense. All that is well-defined is the average structure, and that is temperature dependent. Since temperature is a continuous variable, there are an uncountably infinite number of such average structures for any molecule.

Hope that helps answer your question.

 

[maverick_starstrider]

In lieu of my previous question, (https://www.physicsforums.com/showthread.php?t=494393), which seemed a bit tricky to answer (!), I will simply ask the deeper physical question that lies at its base.

That questions is whether there are any physical phenomena (temperature, pressure, concentration) which cannot ultimately be quantized?

Any thoughts on this matter?

Thanks
Kherubin

I posted the same response on your near identical thread but I would post the same here:

Well David Bohm wasn't so active during the rise of the computer. As a computationalist I can say that, metaphysics aside, not every equation can be quantized, even in theory, and produce identical results to the continuous case. Simply google "numerical stability" and I'm sure you'll find thousands of discussions about how truncating or discretizing various partial differential equations will lead to them exploding with instability. Ultimately this is an artifact of non-linearity and chaos theory. Take any of a whole host of mathematical functions and impose hard limits on the precision of initial conditions (and time steps forward) and you will find that its behavior will wildly diverge from the continuous case. Thus, even if only through that lens alone, I believe we can discount the possibility that all relations in nature can be discretized (no matter how small the discrete packet) without producing drastically different results.

 

[maverick_starstrider]

On a side note I don't understand this discussion at all. There is most definitely not an infinite number (countable or otherwise) of stable compounds. The number of stable compounds is indeed quite finite. The periodic table stops and even if we allow for the most elaborate and exotic of conditions it will always have a finite size. In addition, infinitely sized molecules cannot exist, we might trick ourselves into thinking that a given polymer chain could potentially go on forever or the like but that is simply because we are thinking in a system of sticks and balls, which are really just ball-parked rules of thumbs. To take things to a ridiculous extreme, imagine a molecule so long that its mass could cause a significant gravitational collapse, that guy... he's a not-a going to be stable. Thus, molecules cannot be of infinite size, the number of constituents can also not be infinite. The number of combinations is finite.

As for the rest, what is a metastable molecule? What does that even mean? Either, in the absence of external forces, the structure is maintainable as time goes to infinity (or pragmatically, a large number) or it isn't. Where is there a meta? I apologize if that's a chemistry term that I am unfamiliar with. There are clearly finite many molecules, especially if you want to apply any level of realism to your thought. Even if you rewrote the dictionary and defined molecules of slightly differing Hamiltonians (i.e. a perturbative external field) as "different" molecules you still would never arrive at an UNCOUNTABLY infinite (i.e aleph-null) number of combinations because the number of constituents would still be finite.

 

[Kherubin]

Thank you for all your thoughts.

I posted the same response on your near identical thread but I would post the same here

Interesting that you would think of the two threads as identical. I had never considered it this way.

This entire discussion was based upon the musings of the Philosopher of Chemistry, Joachim Schummer:

We have no reason at all to assume that the realm of possible substances is limited. If we take that seriously, we must assess the finite growth of chemical knowledge against the background of an infinity of possible knowledge. An infinite realm of possible substances corresponds to an infinite amount of possible knowledge that we not yet have. To be sure, the fast increase of our chemical knowledge decreases our lack of knowledge in a certain sense. But that does not matter. Mathematics forces us to accept that a finite decrease of an infinite amount does not affect the infinity at all. As a consequence, whatever the rates of growth of chemical knowledge will be, that does not change the fact that our knowledge gap is infinite and will remain infinite in the future. -- Coping with the Growth of Chemical Knowledge

I was just interested to see if others agreed or disagreed with him.

Thanks for your time,
Kherubin