Wikipedia:Reference desk/Archives/Science/2018 April 7

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April 7

Infinite dimensionality of quantum state

WP's biography John von Neumann mentions that he "realized, in 1926, that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle."

Why is this? Is it just because the Rydberg formula has solutions for n=1,2,3...∞ etc. (added: or that some radiation has a continuous spectrum)? Is that supposed to be an actual physical prediction if n is very large, like a billion? Was the infinite dimensionality of von Neumann's Hilbert space formulation of QM always regarded as really physical? Or is it just a model of QM the way that calculus on real numbers models Newtonian mechanics, but not as far as to say that the Banach-Tarski paradox (a theorem about real numbers) really means we can physically duplicate the Sun. Motivation for question: the extremely large dimensionality of the states required for quantum computing to make any sense. Thanks. 173.228.123.121 (talk) 01:04, 7 April 2018 (UTC)

It's all rather speculative, but I think most leading physicists quickly realized that the uncertainty principle makes the question of infinite dimensionality irrelevant at a practical level -- it implies that the finer the level at which you try to probe the wave structure, the higher the energy you need. If the total energy of the universe is finite, then the effective dimensionality of the wave function must be finite as well -- but no conceivable experiment could ever test this. Looie496 (talk) 03:34, 7 April 2018 (UTC)

Pretty much spot on, although there are some open questions on how to self-consistently and covariantly measure energy and count states in curved space. Still, as far as I know, pretty much nobody believes the infinities you can encounter in those cases are real. When you encounter infinity in physics, it's usually either because (1) you manually substitute infinity for "some huge, unknown number N" so you can use integrals and continuum limits instead of annoying sums, or (2) you hit some mathematical artefact that can either be removed by renormalisation or indicates an inherent flaw in your theory (e.g. because you're hitting the limits of perturbation theory, as is the case when you naively add general relativity to quantum field theory).

That said, from a purely theoretical point of view you can always assume there are infinitely many states (and infinite energy) in a system, and then try to see how to mesh the Banach-Tarski theorem with quantum mechanics. It seems to me that this would violate the no-cloning theorem, for instance.

As for the number of states required for quantum computing, you really only need two-level one-particle Hamiltonians for that. If you want N arbitrarily entangled qubits, you'll get a combined Hilbert space of dimension 2^N (it's just the tensor product of all one-particle Hilbert spaces), but it's still safe to assume N will always be many orders of magnitude lower than the number of atoms in the universe. There are absolutely no infinities required for quantum computing. --Link ^(t•c•m) 10:29, 7 April 2018 (UTC)

I am not sure what the problem is? Neumann meant a functional space, which usually has an infinite dimension. A simple example is the wave function of a single particle in a free space. This space has infinite and in reality uncountable number of basis vectors: you can choose them as $\exp(i\mathbf {kr} )$, where vector $\mathbf {k}$ assumes any value. Ruslik_Zero 20:19, 7 April 2018 (UTC)

Oops, I just saw this and am thinking about it. I don't actually know any QM beyond looking at a few WP articles in the hope of understanding quantum computing, so I'm sure I'm missing something dumb. I'll look at the article about the wave function. Thanks. 173.228.123.166 (talk) 21:47, 7 April 2018 (UTC)

Added: the "suspicious" part isn't the infinite dimensionality per se (I think the equation you wrote just means that the wavefunction has a continuous Fourier transform). Rather, it's that tensoring the states of a combined system results in multiplying the dimensions, with no clue how the "bookkeeping" happens as an entangled state evolves. It's not like classical thermo where you can ascribe a fixed-sized position and momentum vector to each particle and let the rest take care of itself. 173.228.123.166 (talk) 23:38, 7 April 2018 (UTC)

Thanks. Yes it's the 2^N that's been bothering me. The mathematical justification for it that I've heard amounts to "well, 2^N is tiny compared to infinity, so stop worrying!". So I wondered if the "infinity" in the original theory had to do with (say) atomic spectra, where the number of lines has no inherent upper bound, but in actual observations it's maybe dozens rather than billions. If the model is not really infinite-dimensional, then there's some finite N where the theory starts breaking down, but where? As Looie496 says, do you need (exponentially) higher and higher energy to perform measurements on those N-cubit entangled states? 173.228.123.166 (talk) 20:46, 7 April 2018 (UTC)

Added: Ruslik0, are you saying further up that the wavefunction of a single particle can be any arbitrary function? I expected it was more constrained, something like a vibrating string. Thanks. 173.228.123.166 (talk) 18:05, 9 April 2018 (UTC)

Statics: Method of joints

With EF an obvious zero-force member in this problem,[1] how can EH also be a zero-force member, when the method of joints applied to joint E would give a horizontal component due to ED? Imagine Reason (talk) 05:17, 7 April 2018 (UTC)

i doubt EH is a zero force member. Looks like homework to me. Greglocock (talk) 07:07, 7 April 2018 (UTC)

I think the hoist at top is three points rigidly held at a fixed angle. Therefore it is only necessary that the forces in the x-axis on C and E together add up to zero... provided it doesn't start bending, anyway. Wnt (talk) 10:13, 7 April 2018 (UTC)

Yes, but shouldn't the sum of forces at every joint add up to zero? Imagine Reason (talk) 20:51, 7 April 2018 (UTC)

Since C, E, the pulleys etc. are all rigidly bound together in a fixed shape, they are all like one "joint", I think, provided at least that the torque is balanced. Wnt (talk) 23:16, 7 April 2018 (UTC)

Never mind. EF is a zero-force member, so EH must be also. Thank you all. Imagine Reason (talk) 21:10, 7 April 2018 (UTC)

Plastique: the Terminator corn syrup bomb

As long as there are The Terminator re-runs, people will continue to wonder: Is there really a way to make a bomb out of "mothballs, corn syrup, ammonia..."?

The question was considered back in the golden age of the internet and people thought it was an obvious put-on to avoid giving a real formula. And admittedly that is the likely answer. But ... can we rule out the notion that a puckish director might have asked a chemist for a partial list of ingredients that seemed completely unbelievable, thereby satisfying twin demands for verity and preservation of ignorance?

Anyway, the film says "it is nitroglycerine, mostly". Back then moth balls were naphthalene. It turns out that nitrated naphthalenes have been used in explosives in the past - three and four nitro groups make it explosive, five is extremely unstable, but even mononitronaphthalene was used widely in bombs, mixed with picric acid. [2] So one working hypothesis is that he had other sources of sulfuric acid (perhaps battery acid) and nitric acid. The corn syrup is more mysterious, but from that source "nitro-sugar" (nitrated sugar) actually is a thing, though it is too unstable to use; the Germans mostly used it in WWI as a glycerin substitute, in some proportion. Now the naphthalenes are generally said to have stabilizing effect, so is it conceivable that mixing them in with nitro-sugar could make something usable? And the ammonia might have been used to neutralize the acid after preparation and/or generate ammonium nitrate from the nitric acid.

So with a whole lot of handwaving, I feel like it might almost be possible to get from that scene to an actual composition, probably something obscure from WWI Germany. Does this seem plausible to you, or have I strained your suspension of disbelief? Wnt (talk) 16:33, 7 April 2018 (UTC)

You can make an explosive out of a great many "household chemicals", with the addition merely of a little concentrated nitric acid, or else high strength hydrogen peroxide. The point is though, that without that single ingredient, all you have is a sticky mess of potential fuel. Even if that mess involves some other nitrogen compound, like ammonia, as a piece of scriptwriter's camouflage.

Explosives are high energy compounds (that's by definition). They can be either mixtures of fuels and oxidisers (e.g. black powder) or they can be single compounds with an energetically strained bond(s), which you made with some reaction such as a nitration. But one of these needs a concentrated oxidiser, the other needs that nitrating agent, and neither of those are especially common. Andy Dingley (talk) 16:49, 7 April 2018 (UTC)

The Anarchist Cookbook explains how to make explosives, most of which use nitroglycerin as a base. It goes into the chemistry of it, ending with a list of household products that have the chemicals needed. Mothballs, corn syrup, and ammonia are listed. I can't confirm that Cameron had a copy, but it did come out at a time where it remains a distinct possibility that he'd own one (or have gone digging for one for research purposes).

Granted, it also contains some really, really, really stupid ideas. Like, Bananadine. It also recommends taking a cheap beer can, jaggedly cutting the top and bottom off, wrapping the middle in electrical tape, holding it out (the jagged edges perpendicular to your arm), and swinging it back and forth as a weapon. It recommends this in the face of martial law, like you're going to slice through riot gear (instead of just brushing your knuckles against it) and somehow not end up just annoyingly jabbing the rioters standing next to you. Ian.thomson (talk) 17:17, 7 April 2018 (UTC)

I remember hearing that the producers of Breaking Bad hired scientific advisers to give them meth-cooking descriptions that were accurate enough to make the scripts plausible, but (deliberately) not accurate enough to help anyone cook meth for real. 173.228.123.166 (talk) 20:52, 7 April 2018 (UTC)

They had a budget, though. The Terminator is kinda famous for doing so much on the fly and under the radar (the car chases scenes were technically kinda illegal, IIRC). Ian.thomson (talk) 20:58, 7 April 2018 (UTC)

That's not the worst of it! Back when there were people mooting the idea of changing the constitution to let Schwarzenegger run for president I would joke that the only way to prevent nuclear war (or at least a political mistake to rival current events) would be to get a .45 Longslide with laser sighting and an L.A. phone book and start looking up Sarah Connors... what made it so damned funny is it would have worked! I wonder how many refugee women that film has chased out of that town over the years. Wnt (talk) 16:24, 8 April 2018 (UTC)

There's never a bad reason to crack open a beer aboard a plane ... well, almost never, or at least sometimes. It's been ages since I went through that book; I certainly don't remember seeing anything about corn syrup. Do you recall what they claimed it could be used for? Wnt (talk) 23:13, 7 April 2018 (UTC)

This is the original book by William Powell, not the later updated internet doc that replaces some of the stupid ideas with novel ideas of varying utility and legality. I was skimming for my last response, but read the section in more detail this time. "Household substitutes," p. 122, mentions corn syrup as a substitute for glucose.

I'm not finding glucose explicitly mentioned in any of the recipes, but there are some that mention dextrin (though no explanation of how to make dextrin from glucose). Dextrin and naphthalene don't appear in any of the same recipes, nor in any of the recipes for or involving nitroglycerin. If Cameron was using the Anarchist's Cookbook as a guide, he appears to have been scrambling ingredients to discourage copycats.

Looking elsewhere for instructions that might cause my VPN provider to get emails asking "who was that fella doing searches on Jinn and pipebombs?", the only recipes I'm not immediately spotting any recipes that use glucose, and the only ones that use naphthalene are the ones in the Anarchist's Cookbook. Ian.thomson (talk) 00:08, 8 April 2018 (UTC)

The really dangerous stuff is red mercury. 173.228.123.166 (talk) 00:16, 8 April 2018 (UTC)

There's no mention of that in the Anarchist Cookbook, but they do explain how to supposedly make Mercury fulminate. Ian.thomson (talk) 00:42, 8 April 2018 (UTC)

Yep, the Anarchist Cookbook (1971) predated red mercury's discovery, or at least predated the secret getting out ;-). 173.228.123.166 (talk) 02:47, 8 April 2018 (UTC)