According to quantum mechanics, is it true that if you throw a ball against a wall enough times eventually it would pass straight through the wall?

One of the things about probabilities and physics is that there are probabilities that are so tiny that they are truly indistinguishable from zero in any physical sense.

                                  

If you phrased your question this way: "Is it true that the probability that a macroscopic ball might tunnel through a macroscopic wall is non-zero?" the answer would be yes. As a purely mathematical exercise, you could calculate the probability that all the atoms and molecules of the ball are simultaneously in such a state with respect to all the atoms and molecules of the wall that this happens.

But will it ever happen? To that, the answer is no. Not in a year; not in a million years; not in the lifetime of the universe. Not even if you turned all matter in the universe into balls and walls and used the entire lifetime of the universe just to perform this experiment everywhere. Not even if you used a multitude of universes, say, as many universes as there are atoms in this one. And then some.

In other words: never.

This is the thing with exponentially vanishing probabilities. From a purely mathematical perspective, they are not zero. But from a physical perspective, they approach zero so rapidly that they are truly indistinguishable from zero in any physical experiment.

The writer Douglas Adams understood this. This is why, in the Hitchhikers Guide to the Galaxy, the spaceship Heart of Gold needed an Infinite Improbability Drive to accomplish things like conjuring up sperm whales and petunias in mid-air...

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