Question: This seems to me it must be trivial, but I have not been able to grasp it.

As I understand it, the wavefunction crucially depends on its immediate surroundings, whether it be a nucleus, a box, etc. Energy levels are quantized accordingly.

And yet, there is no reduction of state needed to gain this information. The wavefunction does not collapse, there is no interaction. The potential seems an open book to be read anonymously.

Can we detect a particle in the same manner, without interaction, simply by reading its effect on another wavefunction?

I'm curious to know what I'm getting wrong

**Edit after helpful comments:** more specifically, what I am getting at is interactions that are not measurements, like the effect of slits on an evolving wavefunction. The wavefunction evolves into a form that considers the shape of the slits, and this shape can be inferred from measurements.

How can it be that the wavefunction and the slits can interact without a reduction of state? Are we really getting information about a system without having to do a direct measurement of the system itself? And does this also work for getting information from quantum systems?

Answer: 
No, the wavefunction does not have extra knowledge of its surroundings. If $\psi(x\_0) = 0$ at some time, then the evolution of the wavefunction at that time does not depend on $V(x\_0)$ at all.

You're getting confused because you've only looked at stationary states, in particular the 'standing waves' that can get set up in a potential. But this has nothing to do with how a particle outside a stationary state evolves. If you put a particle in a box, its wavefunction will gradually spread out, totally unaware of the walls until it hits them.

As a classical analogy, a string makes a note when plucked. But that doesn't mean that each atom in the string knows where all the other atoms are, the relevant wave equation is local.
I would attempt an answer of this in the hope that I learn something from comments.

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> As I understand it, the wavefunction crucially depends on its immediate surroundings, whether it be a nucleus, a box, etc. Energy levels are quantized accordingly.
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No offence intended, but you are only talking about bound states when you say this, I guess you are ignoring free electrons for example, unconfined by any local potential. Is that correct?

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> And yet, there is no reduction of state needed to gain this information. The wavefunction does not collapse, there is no interaction. The potential seems an open book to be read anonymously.
> 
> 
> 

I don't follow your point here, sorry, we have not gained any information until we perform a measurement, otherwise we just have a series of equations with probably, many possible outcomes.

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> Can we detect a particle in the same manner, without interaction, simply by reading its effect on another wavefunction?
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> 
> 

I would back go your assumption above that, (I hope I understand you correctly), we can essentially discover something real, by looking **only** at the math behind it. We can't do this, we must interact in some fashion.

There are plenty of people on this site who know far, far more than I do, so hopefully they will tell us something we can both learn from, but we can't detect a particle without interacting with it. 

Entanglement may be what you mean by this last statement , but I am not, through ignorance of the subject, able to give you an answer on that.