Chance of “macro tunneling”?

We know that subatomic particles can and do tunnel through barriers, so it is theoretically "possible" somewhat that a grain of sand could tunnel through a paper, but Id like to get some perspective on it.

Can anybody give any sort of estimate of how long one would have to wait to expect to see a grain of sand tunnel through a sheet of paper? (for instance2 10 times the life of the universe")

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i would hope that such amplitude would depend strongly from the state of the sand grain: it should be different if the sand of grain is amorphous, random solid matter than say, actually a bose condensate –  lurscher Oct 28 '11 at 20:00
This problem is not sufficiently defined in that it is not given how high an energy barier the piece of paper provides to the grain of sand. In other words: how much energy would be required for the grain of sand to go in classical fashion through the piece paper? (For the other unknowns one can make reasonable assumptions.) –  Johannes Oct 28 '11 at 22:04
No, how long would you have to wait for the grain of sand to tunnel through the sheet of paper spontaneously (all particles quantum tunneling AT THE SAME TIME through the paper) I know it's infinitly (almost) unlikely, but I want a more elaborate answer than that. –  SchroedingersGhost Oct 29 '11 at 0:45

That is hard even to estimate correctly. For a grain of sand to tunnel through a sheet of paper the probability is so small not because of the tunnel barrier but because it would require the whole grain to move in one direction spontaneously. Starting with the Transmission probability (from Wikipedia) $$T = \frac{e^{-2\int_{x_1}^{x_2} dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}}{ \left( 1 + \frac{1}{4} e^{-2\int_{x_1}^{x_2} dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}} \right)^2}$$
Then you have to estimate the likelihood of all $$\approx 10^{23}$$ atoms doing that at the same time, so in the order of $$T^{10^{23}}$$ with $$T \ll1 .$$