Suppose you have a particle in a box, and there may or may not be a Dirac delta potential somewhere in the box. How could one detect whether or not the potential is present?
Furthermore: If there's more than one way to do this, what's the easiest or "cheapest"? How long will it take? Can it be done with certainty, or will there be some chance that you're wrong?
The energy spectrum of the particle will depend on whether or not there's a potential, so if there's a good way to distinguish between two different energy spectra, that would answer my question.
Answer
This answer is true in principal, but since it is rather idealized and simple, it may not be what you're looking for.
The quintessential method for figuring out the existence and location of a delta potential in the well is to prepare an ensemble of identical wells with identical particles and measure some observable on all of them. Now that you have this large amount of statistical data, compare the distribution derived from experiment to the theoretical probability distribution. By this method, you can determine not only the location of the delta potential but also its strength (to within some statistical error).
If you are happy to leave the box behind for a moment, you could also determine the presence of a delta potential with a scattering experiment. Send in a bunch of (free) particles to interact with this possible potential and record the statistics for how they get deflected. Compare with theory.
These methods work for just about any potential, actually, but if you don't know what the form of the potential is before doing the experiment (and therefore don't have a single theoretical prediction to compare it against), it can be very difficult to reverse engineer the form of the potential. But on the upside, you're probably doing some pretty interesting science!
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