Homework #4 discussion related to non-local states

Zack: Today in office hours we had an interesting discussion on the integral in problem 4 (see above image) and the nature of the non-local (free) states that are involved in some of this week's HW (the t-dependent problems). Among other things, we discussed how to think about and deal with the non-local states in problems 4 and 6? (They just sort of drift away.)
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Kelsey: To continue the problem, what the above jpg basically means is that there is some probability equal to (c_1)^2 that the particle is in the ground state.

The excited states are unbound, so we don't know yet how to actually calculate their probabilities, nor do we really know much about them, except that their wavefunctions are not localized--they extend through all of space --so in any particular region their amplitude is small. [infinitely small]. So basically, the particle is either localized around the delta function, the likelihood of which is determined by (c_1)^2, or it is basically distributed through all space. [Similarly, in problem #6, the part of the initial wavefunction Psi(x,0) that is in the well that disappears at t=0 becomes essentially a superposition of unbound states. Then that part of the wave-function, after t=0, sort of spreads out and "disappears" very quickly (after t=0). The particle is therefore either localized in the ground state of the remaining well or distributed over space. I think.]

Since the probability of the particle being in the ground state, and therefore localized, is a function of c_1, which is itself a function of Psi(x, 0), the likelihood that the particle is localized is highest if the delta well is placed close to a peak of the original wavefunction Psi(x,0).

As the exponential constant k increases, so that the groundstate wavefunction of the delta potential is itself becoming concentrated in a spike around x_0, like a delta function, the probability (c_1)^2 would then seem to become an expression for the strength of the original function Psi(x, 0) at x_0, which determines the probability that the particle be found in a very small region around x_0. [ And the width of that region seems to be 4/k.]

I'm pretty sure that that is a summary of everything that was covered at office hours, but I didn't write everything down, so I might have gotten some stuff wrong.