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What is the upper limit of length for quantum effects to take place? Zooming down through millimetres and micrometres and nanometres and so on, when can I be said to have entered the domain of quantum mechanics? Please state the order of magnitude.

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There exists a rule of thumb for knowing when quantum mechanical dynamics has to be involved in the calculations, it is called the Heisenberg Uncertainty Principle (HUP).

$$\sigma_x \sigma_p \leq \hbar / 2 \, .$$

where $\hbar$ is a very small number $1.05 \times 10^{-34} \, \text{Joule}\cdot\text{second}$

The quantity $\sigma_x$ is the uncertainty in position, $x$, and $\sigma_y$ is the uncertainty in the momentum $p$. One is in the classical regime when the inequality holds because the measurement uncertainties are large with respect to the bound, and $\hbar$ can be considered as zero. When the product of the measurement uncertainties becomes of the same order as $\hbar$ one has to use quantum mechanical solutions, because the uncertainty is not longer controlled statistically, but it is controlled by the solution of the quantum mechanical boundary value problem at hand.

This happens because the HUP is the result of the operator formulation of quantum mechanics with commutation and anticommutation relationships which in effect define the HUP.

This means that quantum mechanical solutions can become necessary for macroscopic situations too, as for superconductivity and superfluidity, so it is not possible to define definite regimes unless the specific problem is examined. Definitely going down to nanometers constrains the momenta but it is not possible to give general limits.

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There is no upper limit in size to the applicability of quantum mechanics. This is illustrated by systems such as superconducting magnets as mentioned in comments on your question. Also, classical physics is just an approximation to quantum physics under certain circumstances when a system is interacting with the environment:

https://arxiv.org/abs/1412.5206.

So-called classical systems are all governed by quantum mechanics.

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  • $\begingroup$ "There is no upper limit in size to the applicability of quantum mechanics." That is a rather unscientific statement. Can you say we've tested quantum mechanics on the scale of, say, 100 miles? How about a planet? A solar system? It would be better to cite the largest physical system which has demonstrated a coherent quantum effect. Also, superconductors are a shakey example because they sit in their ground state. See comments on OP. $\endgroup$ – DanielSank Jul 30 '16 at 8:59
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    $\begingroup$ @DanielSank It is anti-scientific to restrict the applicability of a theory without a specific explanation of why its applicability should be limited. Rather, you should assume it is applicable until you have evidence to the contrary. The reason you do that is that you can't lose by doing it. Either the theory is correct in which case you understand stuff better, or it is wrong and you can find out it's wrong and you can look for a better theory. $\endgroup$ – alanf Jul 30 '16 at 9:42
  • $\begingroup$ I think it's misleading to say there's no upper limit for QM when the "largest" experiment we've done is two entangled photons separated by about one large building. We have no idea what happens if the number of excited degrees of freedom gets larger than around 10. I understand your point and I agree, I just disagree with the way the answer is worded; it seems too assertive, particularly since inexperienced readers might take interest in it. $\endgroup$ – DanielSank Jul 30 '16 at 17:51

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