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Assume I place a tea cup on a table (say, about a centimetre thick). Quantum mechanics tells us that the wave function for the nuclei and electrons of the cup is not zero below the table (while being terribly small, though).

I'm constantly measuring that system. So, with a very small (but not zero) probability, the particles should be tunnelling through the table. Afterwards, measuring them will cause the wave function to collapse and the particles falling to the floor. So, I'd assume that if I leave the cup long enough on the table, it will eventually tunnel through the table. Completely.

Obviously, I wouldn't be around to witness it, given the small probability even a single particle has to fall through the table. But I'm wondering if it is truly a matter of (a lot of) time, for this to happen, or if it is truly impossible for the cup to gradually (!) make its way through the table. Assuming the cup, table, floor, building, earth, solar system, galaxy, and universe would survive an arbitrary timespan, what would happen?

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For the same reason that macroscopic objects do not display quantum mechanical behavior, except in very special setups, as in superconductivity etc. Macroscopic systems in general, due to Avogadro's number which is something like 10^23 molecules per mole, cannot be described by one quantum mechanical wave function .They are an incoherent superposition of this large number of individual wave functions.

Thus one can solve for one particle in a potential well, and that particle has a probability of tunneling through that potential well, but ~10^23 potential wells for ~10^23 particles ensure the total incoherence of all these wave functions, and the impossibility of a teacup tunneling whole through the table ( considering the table another ~10^23 potential wells).

Now googling I found electron tunneling in protein crystals, which shows some effects of tunneling measurable macroscopically.

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I explicitly thought of the cup gradually tunnelling through the table, molecule by molecule. Not as a whole. Single molecules on the bottom of the cup being separated from the rest of the cup and falling through the table. Does that make more sense? – bitmask Aug 13 '12 at 12:40
If it were not a table with incoherent molecules forming it, but a single crystal, for example, i.e. some thing that could have a coherent wave function a single molecule could tunnel through. The totality of ~10^23 would take unphysical time. Experimental fact: muon beams hitting crystals tunnel through the main axis of the crystal without losing energy. for example. – anna v Aug 13 '12 at 13:24
i wonder if a Bose-Einstein condensate (being a coherent state) stands greater chance of tunneling macroscopically through a potential – user56771 Sep 12 '12 at 22:25
@user56771 It will depend on the setup, imo – anna v Sep 13 '12 at 3:23

I would assume that since each individual molecule has a non-zero chance of falling through the table, that the 10^23 per mole, and however many make up the teacup, would still have a non-zero chance, at least mathematically.

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The chance to tunnel is non-zero for each individual particle, but the cup is made up of a large amount of particles. They will not all tunnel through the table at the same time.

Furthermore, they will not all unanimously tunnel together in the same direction (down, through the table).

And the 1cm-thick table is very thick (relatively speaking). It's even more unlikely that a single particle (much less, and entire cup) will tunnel through all the billions of particles in that 1cm. That's a very long distance.

I would like to mention that just because something has a non-zero chance of happening, doesn't mean it will happen given enough time. The universe may suffer heat death prior to it happening, or the feat may be so contrived that it simply will never happen in practice, even if the universe could last forever -- like for example, I could say there is a mathematically non-zero chance of the tea-cup turning into a yellow school-bus.

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Please re-read the question carefully. – bitmask Jan 10 '13 at 21:56
@bitmask Aside from your introduction, your only question is "what would happen?" [if the system lasted for an arbitrary timeframe]. Since that's not exactly the most specific question, I believe I explained multiple different reasons why the the entire teacup would not go through the table. I did mention in one part that the system would not last forever (which I know departs from your scenario), but that doesn't mean that the rest of the answer does not address your scenario. – Lakey Jan 10 '13 at 22:40
I was thinking about gradual tunnelling. I'm specifically not asking about whether or not the cup could tunnel through the table at the same time which seems to be the core of your answer. – bitmask Jan 10 '13 at 23:17

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