quantum mechanics - Does the wave function always asymptotically approach zero?

I'm new to quantum physics (and to this site), so please bear with me.

I know that quantum mechanics allows particles to appear in regions that are classically forbidden; for example, an electron might pass through a potential barrier even though its energy is classically too low. In fact its wave function never decays to zero, meaning there is a non-zero probability of finding it very far away.

But I've seen a lot of people take quantum tunneling and the uncertainty principle to their logical extremes and say that, for instance, it's possible in theory for a human being to walk right through a concrete wall (though the probability of this happening is of course so close to zero so as to be negligible). I don't necessarily question that such things are possible, but I want to know what the limitations are. Naively one might claim that "anything" is possible: if we assume that every particle has a non-zero wave function (almost) everywhere, then any configuration of the particles of the universe is possible, and that leads to many ridiculous scenarios indeed. They will all come to pass, given infinite time.

However, this relies on the assumption that any particle can appear anywhere. I'd like to know if this is true.

Does the wave function always approach zero asymptotically, for any particle, at large distances?

Answer

From a pure mathematical point of view the answer is negative. As you probably know, wavefunctions are all of the functions $\psi$ from, say, $R$ to $C$ such that $|\psi(x)|^2$ has finite (Lebesgue) integral, namely $\psi$ belongs to the Hilbert space $L^2(R)$. One can simply construct functions that belong to $L^2(R)$ and that oscillate with larger and larger oscillations as soon as $|x|\to\infty$ but the oscillations are supported in smaller and smaller sets in order to preserve the $L^2$ condition. (It is possible to arrange everything in order to keep the normalization $\int |\psi(x)|^2 dx =1$.) These wavefunctions do not vanish asymptotically. From the physical viewpoint it seems however very difficult to prepare a system in such a state, even if I do not know any impossibility proof.