Simplest case I can think of:
Consider 2 neighboring square energy wells. One of them is occupied with a particle at ground state and has length L, the other is unoccupied and has a length 2L/3. In this way, if the particle happens to tunnel over to the nearby well, its natural frequency at ground state will be maximally dissonant to that of its previous state. Please correct me if I'm mistaken but, I expect that in this situation, the probability of tunneling is much less than it would be if the 2 wells were common in length (all else constant). If this is the case, should we then also expect that the particle has a disproportionately increased probability of tunneling when hit with a photon having 1/3 of its ground state energy (increasing its frequency to the 1st harmonic of the neighboring energy well)?
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1$\begingroup$ What is the distance between the two wells? Also note that localisation to one of the boxes is a result of a position measurement. The wavefunction of the system exists throughout space (being zero at finite width infinite potentials). So we need to solve the entire Hamiltonian. That is both the boxes simultaneously. And so the single box energy states may no longer be eigenstates. $\endgroup$– Superfast JellyfishCommented Feb 10, 2020 at 4:13
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The tunneling between the two wells will be more affected by details you didn't mention in your description such as the potential barrier between them and the distance between them, but your point is not entirely off the mark. There are two ways to look at the problem:
1) (the 'correct' but slightly less useful way) the system is taken as a whole, and the Hamiltonian describes all its components - the two potential wells, the barrier etc. We can think of something like $$ H = \frac{p^2}{2m} + V_0 \theta(x-x_{1,\rm{left}})\theta(x_{1,\rm{right}}-x)+V_0 \theta(x-x_{2,\rm{left}})\theta(x_{2,\rm{right}}-x)$$ where $x_{i,\rm{right/left}}$ mark the positions of the edges of the wells. Now we can solve it and find a wave function, which will not be entirely localized in each of the wells, and ask about the probability that if the system is in an eigenstate that is mostly localized in one of the wells we would find it in the other well. However, I think that your intuition and way of describing the question is more like the second approach -->
2) (more useful and intuitive) Now we consider two infinite wells, separated. As the wells are infinite there is no possible tunneling process between them. Each eigenstate is completely localized within its own well (or can be chosen like this), and the overlap is zero. Now we 'artificially' introduce a tunneling amplitude between the two wells. This can be done, for example, if we say that there is a finite height of the potential between them and we want to treat it in perturbation theory. If the lengths of the wells is identical, then the tunneling matrix element in perturbation theory will be first-order, as they share an energy spectrum and the system is degenerate. If, on the other hand, they have incommensurate lengths, with different energy values, then the first order in perturbation theory will not connect states in both wells, and only the second order will do.