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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.

Qmechanic
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daunpunk
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2 Answers2

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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!

Geoffrey
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To give a litte more vivid view one can consider a more concrete way to meassure. There are two basic ways comming up in my mind:

  1. Use the position operator $\hat{x}$. As the energy eigenfunctions of a system with and without a delta potential vary [1] one con employ this to check for such a impurity. So you would first meassure the energy and thus force the wavefunction to collaps into an energy eigenstates. Having noted down the meassured energy you can then meassure the position of the particle and by that probe the wavefunction. Repeating this process you get a meaningfull statistic like Geoffrey described.
  2. Another, more fundamentally QM-way would be to use the effect of the presense of a delta potential indirectly: In the plain box $\left[ \hat{H}, \hat{p} \right]=0$. This is directly connected to the translational invariance [2] of the potential. As now the eigenvalue spectrum of both operators is non degenerate meassuring one won't effect the meassurment of the other. Having a impurity in this potential will break this translational symmetry and therefor one could proceed as follows: First you meassure the energy and again force the system in an energy eigensate. Then one meassures the momentum and yet again the enegy. If the second meassurement of the energy results in a different enegy than the first you know for sure that your potential is not translational invariant.

These two possibilities could be worked out in more detail and then a estimation for the propabilities could be givin. Both these ideas though will never tell you for sure that there are no delta potentials in your box (the argument only works the other way around).

I hope this helps to clarify how such a meassurement could be done "in reality".

[1]: https://arxiv.org/abs/1001.0311 Analysis of the energy eigensates of the box with a delta potential. [2]: this connection is called the Noether Theoreme

Urb
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Hagadol
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