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In QM, take a particle in a bound state in $\mathbb{R}^n$ subject to a potential (need not be smooth and not necessarily bounded above, but is bounded from below, say, something that might roughly look like the harmonic oscillator with possible kinks).

Why is it impossible for a bound state's wave function to be zero outside of some suitable set? (i.e., it has to have some support, however small, anywhere in space) If it is not impossible, can you construct an explicit counter-example? When, generally, is it true that the wave function is strictly non-zero everywhere?

My experience with particle in a finite well says that there should always be a portion of the wave function that "drips" out of the well, i.e., there will always be quantum tunneling, but I can't quite justify this, let alone quantitatively.

What I could think of is: (1) the uncertainty principle: if the wave function is supported in a ball of radius $r$, then the kinetic energy is larger than $\frac{1}{r^2}$ in appropriate units. (2) The virial theorem which connects the kinetic energy and the potential. But those two still don't tell me enough.

Note: if you can give quantitive explanations it would be even better, i.e., estimate the distance of the wave function from zero given the distance from the potential's minimum.

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