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We all know that subatomic particles exhibit quantum behavior. I was wondering if there's a cutoff in size where we stop exhibiting such behavior.

From what I have read, it seems to me that we still see quantum effects up to the nanometer level.

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The classic experiment demonstrating quantum effects, the 2 slit experiment, has been preformed with subsequently larger and larger particles as our technology available to do it advanced. Originally, it was performed with electrons, which are just as much matter as any other matter, but are extremely small. The largest particle it has been demonstrated with are Buckminsterfullerene, which contain 60 Carbon atoms. For size comparison:

$$m_e = 5.485 x 10^{-4}u$$ $$m_{buckyball} = 720.642 u$$

There is a good reason that the experiment gets more difficult with increasing mass, and to be sure, the buckball experiment was quite an accomplishment. To the basics of quantum mechanics:

$$\Delta x\, \Delta p \ge \frac{\hbar}{2}$$

Alternatively, the de Broglie wavelength is:

$$\lambda = \frac{h}{p} = \frac{h}{m v}$$

I believe that in order to obtain the same wavelength with the same mass you have to decrease the velocity. The reason this could be problematic for such experiments is that it is hard to successfully create the conditions needed with a large and slower moving particle, such as needing a better vacuum.

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is there a minimum size of wavelength needed to observe such behavior? Why do we have to keep the wavelength the same? – Tuva May 23 '11 at 18:32
@Hydra Good question, and something I left out as my answer got longer. For the double slit experiment, take $d$ as the separation between slits, $L$ as the distance to the target sheet. Due to some arguments I won't make here (but could), the number of peaks you can see will be in the neighborhood of $\frac{d}{\lambda}$. So some minimum wavelength will be needed to have resolvable peaks, which is a tradeoff with the # of particles you shoot, which is a tradeoff with the purity of the vacuum/system you have. Increasing $d$ also conflicts with those limitations. It's about signal-to-noise. – AlanSE May 23 '11 at 18:47
very interesting. Where can I find how one calculates the number of peaks seen? By resolvable peaks, do you mean having enough "particles" to create such peaks? – Tuva May 23 '11 at 18:58
@Hydra The # of peaks is nearly $\frac{d}{\lambda}$. The number of particles has to do with how long you perform the experiment, but given that you may need a given number of particles/peak (in order to see it at all), you ideally want just a handful of clearly visible peaks. Reading the Wikipedia for the double slit experiment might be the most helpful, although it doesn't address the practical size limit very well. – AlanSE May 23 '11 at 19:13
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There is no known cutoff in "size" where systems stop exhibiting "quantum behavior". I put those words in quotes, because "size" can mean different things to different behavior, and you weren't explicit about what you mean by "quantum behavior".

Folks have seen double-slit interference with molecules made up of ~60 atoms (I think recent experiments may have increased this number). If you were to try to do the same thing with a baseball, as far as anyone knows there's no fundamental reason why it wouldn't work, but the experiment is far beyond our technical abilities (it would take a length of time greater than the age of the universe to do the experiment, nobody knows how to isolate the baseball from the environment for that length of time, etc. etc.).

Superconducting rings have been built with diameters of a few centimeters (probably bigger) and still show flux quantization, which is a quantum-mechanical effect due to the electron wavefunction. Those rings are definitely bigger than a few nanometers, and I'd consider it "quantum behavior".

Folks have shown quantum-mechanical entanglement between two light beams that were separated by kilometers (in EPR tests). A kilometer sounds like a big "size" to me, but maybe that's not what you meant?

Take your pick as to what's the biggest "size" here, but as far as anyone knows, there's no size cutoff to quantum mechanics. However, depending on the kind of experiment (2-slit interferometer, etc.) it definitely becomes more and more technically difficult to perform the experiment as the system is scaled up.

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+1 Hi @Anonymous Coward. Point taken on "size" comment. By "size", what I'm trying to understand is if there's a kind of Reynolds-like number (I realize that it's dimensionless) between what's Newtonian and Quantum. Apologies if this is too vague. – Tuva May 23 '11 at 18:43
Why exactly is the baseball example not technologically feasible? – Tuva May 23 '11 at 18:49
Re: the Reynolds-like cutoff: no, there isn't a known one, as described in the answer. – Anonymous Coward May 23 '11 at 22:06
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Re: the baseball example, if you want to make a double-slit for a baseball, the slits have to be a few cm wide (otherwise the baseball can't get through). To see interference fringes from slits separated by a few cm, you'll need the deBroglie wavelength of to be on the order of a few cm. To get that, the baseball has to be moving incredibly slowly. So slowly that it'll take longer than the age of the universe to go through the slits. Waiting that long isn't technologically feasible. I've glossed over some math, and there are other issues as well, but I'll skip those for now. – Anonymous Coward May 23 '11 at 22:07
Thanks Anonymous Coward! That's helpful. – Tuva May 23 '11 at 22:19
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There are definitely circumstances where we see quantum behaviors at rather large scales like, for example, in superconductors. In BCS theory cooper pairs are described by a macroscopic wave-function which, to my knowledge is valid for the bulk superconductor no matter how large it is.

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@C Earnets: Thanks. I'll take a look at BCS Theory. – Tuva May 23 '11 at 18:02
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Correct. In super conducting magnets: "The total length of the tape wires used was 32.2 km." from global-sei.com/tr/pdf/energy/65-09.pdf – anna v May 23 '11 at 18:32
The BCS field is a classical field in the classical field limit, so it is not really quantum in the appropriate interpretation, it's just a different classical limit than usual for electrons. – Ron Maimon Aug 2 '12 at 3:19

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