What Energy Maximizes Neutron Trapping in a Finite Square Well?

[atlantic]

Free particle --> bound particle

Homework Statement

A free neutron meets a finite square well of depth $V_{0}$, and width 2a centered around origo.

However, the probability that the neutron emits a photon when it meets the potential well, and thus decreasing its energy is proportional to the integral $\int^{t_{1}}_{t_{0}}\int^{a}_{-a} |\Psi(x,t)|^{2} dx dt$. Where $t_{1}-t_{0}$ is the time it takes the neutron to cross the well.

The question then is: "What energy is the most advantageous for the neutron to have, in order to be trapped by the potential well?"

The Attempt at a Solution

The initial energy is $E_{0}$, the energy of the photon is $E_{p}$

I'm guessing I have to find a value for $E_{0}$, so as to make the integral a large as possible.

 

[turin]

Is there some form of ψ here that you are neglecting to tell us (like it is the wavefunction of the neutron)? If ψ is independent of E (or E₀), then I don't see how it would make a difference (unless it's as silly as to realize to treat the neutron classically, so that t₁-t₀ depends inversely on the square root of E, which it may be, since it talks about "the time it takes the neutron to cross the well").