What happens if particle is measured outside finite well? A particle, which is in bound state (and eigenstate) of a finite well, has a small probability of being found just outside the well. If one happens to locate it there, then to the experimenter how will the particle manage to occupy that potential energy (greater than total energy)?
 A: Let's take a concrete example of the measurement process. Suppose I'm detecting the particle by firing high electrons at it. I can fire my electron beam towards the well and there will be some probability that my beam scatters off the particle. By measuring the trajectory of the scattered electron and tracing it back I can tell where the particle in the well was when the scattering event happened.
And what I'll find it that there is a small probability that the scattering event happened outside the well i.e. that the particle was outside the well when it collided with the electron.
But this doesn't mean that conservation of energy has been violated. During the scattering there will be some transfer of energy from the incoming electron to the particle. If the energy transfer is less than the well depth then the particle stays in the well, just in a higher energy bound state. If the energy transferred is greater than the well depth then the particle will be knocked completely out of the well and will head off towards infinity. In all cases when I add up the total energy before and after I find it's the same, and this is true whether the location of the scattering event was inside the well or outside it.
A: Please provide an experiment (at least a thought experiment that seems reasonable) that would "find" the particle outside the well without damaging the structures that made the well (I am aware that the well is not made of concrete).  As the above comment indicates after measurement the particle is NOT in an energy eigenstate.  In fact it will be in a position eigenstate, with wavefunction given by a Dirac delta.  To understand how the particle behaves after that you need to expand that state in the energy basis and let it evolve.  Since the position measurement interrupted the momentum it is not unreasonable (intuitively) that the energy is also interrupted.  H and p do not commute with x.  The probability to observe this is very small for a bound state but not impossible.  So rather than ask "how will the particle manage to occupy that potential energy", ask "What is the probability for the particle to be in a bound state after a position  measurement finds it outside the well".  That is a better question from a QM perspective.  And, if you are familiar with the mathematical machinery of QM you can write down the expression for this. 
