# How does such strange microscopic behavior at the atomic level (quantum mechanics) lead to the macroscopic behavior at our level?

So, I'm only a high school student researching quantum physics, and I find it very interesting. However, there's one question that keeps nagging at me in the back of my head. How exactly do odd behaviors like quantum parallelism that occur on the atomic level lead to the behaviors that we consider normal at everyday sizes and scales? That is, what is it about having so many atoms together (classical physics) that makes them behave so very differently from the way a single atom behaves (quantum physics)?

Sorry if it seems like I don't know what I'm talking about... because I may not! So, if there are any misconceptions on my behalf, please tell me so I can actually learn something... :)

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I can recommend relatively simple sources--- Everett's PhD thesis, reprinted in "The Many Worlds Interpretation of Quantum Mechanics" edited by DeWitt is the original, and still essential. In conjunction with the philosophical discussion in "The Mind's Eye" by Dennett and Hofstadter, and an article by Douglas Hofstadter on the Many-Worlds interpretation reprinted in "Metamagical themas" (all these are completely accessible to any high school student versed in Dirac's book), you can understand the whole classical limit. Its mostly simple physics, only thorny philosophy. –  Ron Maimon Mar 2 '12 at 3:23
Tangentially, if you want to have a quite humorous account of what the macroscopic world would be like if these effects were visible, read The Adventures of Mr. Tompkins, the author, George Gamow, was a great physicist. –  recipriversexclusion Mar 2 '12 at 16:48

There is a phenomenon called decoherence in quantum mechanics which is largely responsible for this. Basically (the following is a simplification), all the strange behavior that occurs in QM tends to happen when the wavefunctions of different particles are in phase. Decoherence occurs when the phases are randomized, so there's no special correlation between different particles. In that case, the properties of the different particles tend to just combine the way we'd expect them to classically.

A decent (but very basic) analogy for this would be like having a bunch of identical cars whose drivers all turn their turn signals on at the very same time. The turn signals would be blinking together, so we'd say they are in phase. But on a real road, that's not the case at all; different drivers turn their turn signals on at different, pretty much random times. And besides that, there are many different models of car whose turn signals blink at different rates. For both those reasons, the turn signals on a real road are not in phase. That's kind of like decoherence.

The reason I bring this up is that I've posted an answer about it which you might be interested to read. The gist of that answer is that when you have a small system like a single particle, any interaction makes a big difference to the system's momentum. But the same interaction will make only a little difference to a system which contains a large number of particles with partially uncorrelated momenta, like a measuring device. Now, in the paragraphs above, I talked about phase, whereas my other answer talks about momentum, but the idea involved is similar in both cases.

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Thanks you so much for your answer and for your haste. Your simplification is greatly appreciated. I did research decoherence as it plays a big part in my narrower research topic of quantum mechanics and had an idea it had to do with the answer to my question. Thanks for clarifying! –  Brandon Mar 2 '12 at 3:25

Brandon, the simple truth is that you have just asked one of the hardest and least understood questions in all of physics. So, don't feel bad if you don't understand it very well, because, er... no one else really does either?

It's not that we can't model this stuff mathematically. Shoot, Richard Feynman's version of something called Quantum Electrodynamics (QED), which is sort of quantum mechanics merged with Einstein's theory of special relativity, is arguably the most accurately predictive theory in all of physics. (Or was; I haven't kept track lately.) The problem is that whenever we use such precise theories, we can't help but toss in a bit of everyday life in the mix, sort of like a salad in which we mix things more by taste than by precise rules.

So, for example, Feynman's QED theory is incredibly precise in predicting how an electron in one place and state (e.g., velocity) gets to some other place and state. However, to set up the electron in a real experiment -- to create the location and state you are describing in the QED problem setup -- you must use real-world equipment. And that is the fly in the ointment (or is it the secret ingredient in that salad?): The real-world setup for any physics problem is unavoidably embedded at some points in everyday physics concepts like "ordinary," or irreversible time. Once you toss something like ordinary time into the mix, all the nicely reversible properties of time at the atomic scale no longer apply, at least not for the experiment as a whole. Or stated a bit differently: Everyday physics seems to beget more everyday physics. That is the flaw you will find at some level in every single experiment looking at the physics of very tiny scales. It has to be that way, since otherwise how would we as large-scale creatures every find know about the result in the first place?

So, as the amazing physicist John Bell once said while mulling over pretty much the same question you just asked (he could never really answer it; that's how hard it is!), folks who do experimental physicists just sort of develop a "feel" for when you stop applying quantum physics and start applying everyday (or "classical") physics. Time is a very big part of the transition: If time is reversible, it's almost certainly quantum, and if it's not, it's probably better treated as everyday (or classical). Size is less reliable, but for most phenomena at ordinary temperatures, classical physics starts to kick in at roughly the size of a medium-sized molecule, say a buckyball. That metric is very unreliable overall, though, since things as ordinary as a reflection off of a piece of silver are deeply quantum events that cannot be modeled using only classical physics. Shoot, size is a deeply quantum phenomenon, and so is chemistry. Without quantum mechanics stepping in, we'd just be part of some huge big black whole, and so would not be having this conversation.

I'll end by recommending a book: Richard Feynman's "QED: The Strange Theory of Light and Matter." It's paperback, cheap, uses almost no math, yet provides profound and accurate insights into that very precise quantum theory I mentioned above. I won't say it will answer your question, but at least it will present the remarkably non-intuitive features of quantum mechanics about as sharply and starkly as possible.

Good luck!

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That is, what is it about having so many atoms together (classical physics) that makes them behave so very differently from the way a single atom behaves (quantum physics)?

This difference is still not completely understood and not because from a lack of trying. I disagree with Terry that we can model this in a mathematically correct way and I would even go so far as to say that it cannot be solved by a deeper understanding of the microscopics. David pointed out the decoherence, which explains some properties under the condition that there are no special correlations, but in real materials the interesting bits are often caused by those correlations.

We would not have things like magnetism, superconductivity, spintronics and other phenomena without correlations. Part of this area of physics is summed up by P.W. Anderson: "More is different" (Science 177, 4047, 1972).

A first approach is to these problems is the mean field theory. You model a single particle together with a mean field in which the particle is embedded. At the same time this field is caused by the sum of all other surrounding particles. This simple model explains some parts but often breaks down completely. One modern approach to those phenomena is renormalization group theory, where you try to model first a microscopic system and then try to scale this up repeatedly to understand the macroscopic behaviours.

The correlations are the key reason why quantum effects can be seen real life materials and why many particles can behave totally different then a single atom.

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Macroscopic objects have a large enough mass. Their uncertainity becomes negligible. Let's take a look at Heisenberg's equation: $$\Delta x\cdot\Delta p\geq \hbar/2$$ $p$ is momentum. If mass is large, the LHS becomes large as $\Delta p=m\Delta v$. Planck's constant has an order of $10^-34$, so the uncertainities in position and velocity can be pretty small once an observation is made. No uncertainity$\implies$ no quantum effects. The size of the wavefunction is tied down by uncertainity (one can also say that it is tied down by it's De Broglie wavelength, which comes out to be nearly the same thing)

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Can you clarify the first word of your post? "Mcroscopic" is missing a rather critical vowel! –  dotancohen Mar 2 '12 at 9:08

I don't know which level of abstractness and 'deepness' is appropriate, so I will restrict myself to the basics, e. g. "normal" non-relativistic, single-particle quantum mechanics.

In this case the state of the system (e. g. all the data you know about the particle) is described by the wavefunction $psi$ and the time evolution is given by the Schrödinger equation $i \hbar \frac{\partial}{\partial t} \psi = \hat{H} \psi$, which is more or less the quantum analog to Newtons famous $F = m \cdot a$. So now we ask the question, how we get the classical behaviour as an approximation to the quantum one. This limes is called "semiclassical approximation" and often described by $\hbar \rightarrow 0$, which means, that all characteristic (energy) scales are big compared to $\hbar$. Inserting the formal ansatz for the wavefunction $\psi = a \cdot e^{i/\hbar S}$ in the Schrödinger equation yields in zeroth order (=dropping all terms with $\hbar$) the Hamilton-Jacobi-equation $H(x, \frac{\partial S}{\partial x}) = E$. This equation is a (complicated) reformulation of Newtons law.

So the important point is: If the particles have high enough energy their quantum behavior is nearly equal to the classical one. (This is not only some mathematical trick, but really observed in our labs, e.g. the higher states of the hydrogen atom are only separated by a very small energy and so are nearly continuous instead of being discrete states).

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You asked, what is it about having so many atoms together (classical physics) that makes them behave so very differently from the way a single atom behaves (quantum physics)?

The answer to your question is the law of large numbers. It accounts for the fact that a large aggregate of atoms behaves much more regularly than tiny systems. Since almost everything accessible to our senses is composed of a truly huge number of atoms, typically $>10^{20}$ of them, our senses just notice the mean behavior, which is much more regular than the detailed behavior of individual atoms.

The law of large numbers is the most basic principle of probability theory, and it is the basis of all of statistics. In physics, the discipline of statistical mechanics treats the cooperative effects that follow from the law of large numbers applied to huge collections of atoms. In fact, it usually deals with the so-called thermodynamic limit, which is the idealization that the number of particles is infinite.

In the thermodynamic limit, the probabilistic predictions of the law of large numbers turn into certainties certainties, and the microscopic laws allow one to derive the macroscopic laws of thermodynamics. The latter govern the behavior of macroscopic bodies and fluids, giving rise to the laws governing hydrodynamics, elasticity theory, or phase equilibria. They also give the mass action law governing chemical reactions.

Of course, the thermodynamic limit is an approximation only, but for macroscopic objects an extremely good one. This is why one has to go to tiny scales to discover the less intuitive behavior of quantum systems.

If you want to read more along this lines, look at Part II: Statistical Mechanics of Classical and Quantum Mechanics via Lie algebras.

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Well, my answer to your question may be far too simple considering the sophistication of most of the explanations given so far. I hope it's correct, but its simplicity makes it suspect. The Heisenberg relation for the uncertainties of simultaneous position $x$ and momentum $p$ measurements on a quantum particle is $$\Delta x\Delta p>h$$ If you divide both sides by the (uncertain) momentum of the particle, you get a relation involving the fractional uncertainty in momentum $\Delta p/p$,
$$\Delta x(\frac{\Delta p}{p})>\frac{h}{p}$$ Now for a macroscopic object, $p = mv$, and $m$ is in the order of $10^{23}$ atomic masses. Since $h$ is already tiny, about$10^{-34}$, then $h/p$ ~ $10^{-57}$. This makes the product of the uncertainties on the left very small. If one of the uncertainties there were large, the other one would have to be extremely small for the right hand side to be $10^{-57}$. So must both be extremely small, and can be neglected compared with macroscopic sizes and momenta.

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