# How do physicists know wave function extends to infinity?

Is this proven that wave functions extend to infinity, or is it just a theory? What makes them assume that it extends to infinity?

## Answers and Replies

atyy
Science Advisor
Physically, the wave function is not generally assumed to "extend to infinity", because it is difficult to make sense of the "wave function of the universe".

However, it is mathematically convenient, and matches experiments to have models in which the wave function extends to infinity, or in which the results are observed "from infinity". Even in classical physics, a small conducting sheet can be treated as infinitely large for many purposes.

At a more formal level, one has to remember that the wave function is not a field on spacetime, but something in Hilbert space, so it does not "extend to infinity" in the usual spacetime sense.

Physically, the wave function is not generally assumed to "extend to infinity", because it is difficult to make sense of the "wave function of the universe".

However, it is mathematically convenient, and matches experiments to have models in which the wave function extends to infinity, or in which the results are observed "from infinity". Even in classical physics, a small conducting sheet can be treated as infinitely large for many purposes.

At a more formal level, one has to remember that the wave function is not a field on spacetime, but something in Hilbert space, so it does not "extend to infinity" in the usual spacetime sense.
So your saying, mathematically the wave function extends to infinity. But physically it cannot extend to infinity, correct?

atyy
Science Advisor
So your saying, mathematically the wave function extends to infinity. But physically it cannot extend to infinity, correct?

Yes.

atyy
Science Advisor
I should add that there are probably many subtleties I'm leaving out. In my answers above, I'm using Copenhagen, which is standard quantum mechanics. To discuss the subtleties, it is probably easiest to do it in Bohmian Mechanics just to have an interpretation in which the wave function of the universe makes sense, but I shall leave that for another time (or for Demystifier).

Nugatory
Mentor
Is this proven that wave functions extend to infinity, or is it just a theory? What makes them assume that it extends to infinity?

We assume that it extends to infinity because:
- The wave function is a solution to Schrodinger's equation, and it can be proven (by math) that many solutions of that equation never quite go to zero no matter how far away you go.
- Experiments show that, to the limits of experimental accuracy, these solutions accurately describe the real world everywhere that can and have checked - we can't check at arbitrary distances from earth, nor can we tell the difference between "wave function is zero" and "wave function is so small that the difference between it and zero is less than the limits of experimental accuracy".

There's nothing special about quantum mechanics here. Classical mechanics has its famous inverse-square laws that you will have learned in high school: Newton's law of gravity ##F=Gm_1m_2/r^2##; and Coulomb's electrostatic law ##F=CQ_1Q_2/r^2##. Not only do both of these suggest that the field extends out to infinity, but the inverse square fields fall off with distance much less than than the declining exponentials of the solutions to Schrodinger's equation.

Yes.
Sorry if I might be asking repetitive questions. But in all Ψ ontic interpretations of qm where the wave function is real( MWI, de brig lie, penrose, etc), they physically do not extend to infinity?

Nugatory
Mentor
Sorry if I might be asking repetitive questions. But in all Ψ ontic interpretations of qm where the wave function is real( MWI, de brig lie, penrose, etc), they physically do not extend to infinity?

OK, this thread is done.