krum,
Short answerIn theory, no, wave functions tend to have non-zero values just about everywhere (with the exception of nodes), so they basically have the volume of the universe.
Practically, almost all wave functions we discuss have almost all of their probability in a very localized area. It's hard to ascribe an exact volume because there's no known fundamental cut-off.
Long answerWave functions have a value at all points in the universe, so you can say that wave functions have infinite volume. However wave functions tend to have near-zero values throughout most of the universe except in very confined locations.
Conceptually, I'd suggest thinking of the
Copenhagen interpretation of quantum mechanics, which asserts that the square of a wave function,
, describes the probability that a particle is in any particle location. And since the total probability that a particle exists
somewhere in the universe must be exactly 100% (by definition), then
.
Since the universe is huge, it's mathematically impossible for a wave function to have any significant value throughout a large portion of it.
Okay, so that's the Quantum Physics 101 answer. For a more theoretical discussion, it's worth considering if quantum mechanics is valid at the most extreme edges - which has
not been proven, nor is particularly likely to be true.
So one weird prediction of quantum mechanics is
quantum tunneling. We say that a particle is tunnelling when it's in a location that should require more energy than it has. Tunnelling was first predicted by the wave equations used in quantum mechanics, because they're still non-zero even when they don't have enough energy,
, to move into a region of higher potential,
:
.
If you suspect that tunnelling is just a mathematical artifact of the wave equations that might not represent an actual, real-life phenomena, you'd be in good company. It was pretty weird.
Just about two weeks back, researchers made headline news with the
report that they discovered water molecules tunnelling. In short, the researchers were looking at water in rocks, and they discovered that single water molecules could simeltaneously exist in multiple gaps in the rock.
Quantum Tunneling of Water in Beryl: A New State of the Water Molecule, Kolesnikov et al. (2016) wrote:ABSTRACTUsing neutron scattering and
ab initio simulations, we document the discovery of a new “quantum tunneling state” of the water molecule confined in 5 Å channels in the mineral beryl, characterized by extended proton and electron delocalization. We observed a number of peaks in the inelastic neutron scattering spectra that were uniquely assigned to water quantum tunneling. In addition, the water proton momentum distribution was measured with deep inelastic neutron scattering, which directly revealed coherent delocalization of the protons in the ground state.
Anyway, the point here is that we've now confirmed that wave functions can cause particles to exist in ways that don't make sense through the lens of classical physics, and we can see that wave functions continue to work even when their values are near-zero,
.
So is it technically possible that a particle here on Earth might somehow interact with a distant star where its wave function is technically non-zero? As far as I know (stressing that I'm not a theoretical physicist), that's an open question.
I'd suppose that, if particles
can't interact with distant objects via near-zero wave functions, then it might be appropriate to revise wave mechanics such that wave functions do have finite volumes.