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Wikipedia's article on macroscopic quantum tunneling says

Quantum phenomena are generally classified as macroscopic when the quantum states are occupied by a large number of particles (typically Avogadro's number) or the quantum states involved are macroscopic in size (up to km size in superconducting wires).

To comply with copyright laws, the following is an edited paraphrase of this reference, pp6-7.

http://assets.cambridge.org/97805218/00020/sample/9780521800020ws.pdf

The term “dynamical degrees of freedom” should be used carefully. Imagine a baseball moving through a wall without being compressed. Certainly, this phenomenon can be called a macroscopic tunneling; since the ball is a collection of atoms, the number of degrees of freedom is comparable to the number of atoms.

Macroscopic tunneling depends on the number of microscopic degrees of freedom like the positions of constituent atoms. Collective degrees of freedom are superior: they are singled out by rearranging the microscopic ones.

Are there any circumstances under which the ball could pass through the wall via macroscopic quantum tunneling, or is this wishful thinking?

Community
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Mr. Davis
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  • "Macroscopic quantum phenomena" is just a catch all term for anything effect that involves large things behaving quantum mechanically, so you really need to be a bit more specific before we can answer your question. i.e. which macroscopic quantum phenomenon are you talking about. So for example the Josephson effect is an example of a macroscopic object (specifically the superconducting current) tunnelling across a barrier, but it would not make sense to talk about using this effect to tunnel one of your students through a wall. – By Symmetry Apr 17 '17 at 20:22

2 Answers2

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Are there any circumstances under which the ball could pass through the wall via macroscopic quantum tunneling...or, is this wishful thinking,

Quantum mechanics and tunneling solutions are dependent on there being a unique wave function describing the system. Wave functions have amplitude and phases, and their complex conjugate squared will give the probability density function for the particular problem.

This simple example illustrates the possibilities:

tunnelinbarrier

So your question asks really: is there a single wavefunction describing a ball hitting a wall so that the quantum mechanical calculations would give a probability for the ball to pass the wall.

Evidently the ball is composed of ~$10^{23}$ molecules and the wall too. Each molecule individually would have its quantum mechanical wavefunction , if single. One wave function theoretically describes the whole ball, including all the huge number of variables needed to write it down for the ensemble of molecules.

The density matrix formalism treats the many body quantum mechanical problem. The nutshell I have retained is that for macroscopic objects, as the ball, the quantum mechanical phases are lost because the off diagonal elements become very small and one ends up with a classical macroscopic body, except in cases as superconductivity, where a macroscopic quantum mechanical solution can be defined. In your ball example, the probabilities calculated are essentially zero, and one is back to classical mechanics.

anna v
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  • Sorry, Anna, all molecules always have a single wave function and separate wave functions of individual particles is a rudimentary layman's misunderstanding how QM works. – Luboš Motl Dec 13 '19 at 04:42
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    @LubošMotl thanks for the correction. – anna v Dec 13 '19 at 05:19
  • You're welcome. At some moment as a teenager, I also imagined separate waves - this is what the popularization wants to do, a part of the realism propaganda. And it's true to the extent that for completely uncorrelated particles, the wave function is just the tensor product of one-particle or subsystems "wave functions". But at soon as any interaction between them takes place, they're entangled, an entangled wave function is generic, and it must always be a single function of all the relevant coordinates. – Luboš Motl Dec 14 '19 at 09:12
  • At what scale the quantum mechanical phases are to be considered as "lost"? Is there a threshold? – Pavel Borisov Jul 23 '22 at 05:04
  • @PavelBorisov It will depend on the particular problem. have a look at the density matrix link, https://en.wikipedia.org/wiki/Density_matrix – anna v Jul 23 '22 at 05:18
  • It is too complicated for a layman like me to grasp the concept of the density matrix. What is the "parameter" according to which phases are lost? In other words: when quantum tunnelling becomes completely impossible? – Pavel Borisov Jul 23 '22 at 05:21
  • @PavelBorisov when the density matrix becomes diagonal the phases are lost. It IS complicated, that is why graduate studies are needed to get numbers, and it depends on the given problem under study. For example for the ball in the question because of the enormous number of parameters , the plases are lost., – anna v Jul 23 '22 at 07:04
  • So, is the following statement correct? "According to the standard axioms of QM and the minimal interpretation approach, the quantum tunnelling of a ball through a wall is COMPLETELY impossible" – Pavel Borisov Jul 23 '22 at 07:40
  • @PavelBorisov the correct term would be "the probability of a ball quantum tunneling through a wall is effectively zero". One always talks with orders of magnitude . – anna v Jul 23 '22 at 14:02
  • What does "effectively" mean here? For all practical purposes? Something else? If we demand a talk with orders of magnitude... "effectively" seems too vague for such a context... – Pavel Borisov Jul 23 '22 at 15:36
  • Strictly speaking, the quantum tunnelling of a macroscopic object is a theoretical predictions (the probability is so small we will never see it). However, I’m not sure everybody would agree it’s even theoretical possible. You agree? – Pavel Borisov Jul 24 '22 at 17:29
  • Could you please elaborate further? – Pavel Borisov Jul 30 '22 at 07:36
  • "effectively" means that if one went to the trouble to caluclate the minuscule probability of tunneling of macroscopic bodies, the number would be so small that the universe would end by the time one event could be seen. – anna v Jul 30 '22 at 08:42
  • this answer by lubos givea an estimate of the probability https://physics.stackexchange.com/questions/34092/probability-quantum-physics-and-why-cant-it-does-it-apply-to-macroscale-eve/34099#34099 – anna v Jul 30 '22 at 09:29
  • "the number would be so small that the universe would end by the time one event could be seen" The problem with such claims is that they are not scientific... – Pavel Borisov Aug 01 '22 at 14:41
  • @anna Pointing out that the probability is so small that the event is unlikely to happen within the estimated lifetime of the universe... that is copping out. It is not scientific: you can't prove it, neither disprove it. In that sense, some statements of quantum mechanics are rather mathematical statements, not science.... – Pavel Borisov Aug 01 '22 at 14:41
  • We cant experimentally make distinction between IMPOSSIBLE event and event with non-zero probability... – Pavel Borisov Aug 01 '22 at 14:43
  • One can hold that quantum tunnelling works for atoms, molecules and even for macroscopic objects (like balls, cats, humans etc.). However, that's merely a speculation/assumption within the mathematical framework of quantum mechanics. – Pavel Borisov Aug 01 '22 at 14:44
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The ball can indeed tunnel through the wall, but a proper description of that astronomically rare event would require a lot of nontrivial analysis. A single wavefunction description is not going to be adequate here, due to decoherence. If you condition on an astronomically low probability event going to happen, then that opens a can of worms containing events that are astronomically more likely but still astronomically rare. A good example is the analysis given in this article about a system spontaneously fluctuating to lower entropy states. While it's clear that an ice cube that has melted in a hot cup of tea can spontaneously unmelt and reappear, what is not clear is how in practice that would happen.

The formalism used in that article may be useful for this problem too, it uses the symmetric two state formulation of quantum mechanics, which allows you to impose the astronomically rare thing having happened as a future boundary condition.

Count Iblis
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