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**Free particle --> bound particle**

## Homework Statement

A free neutron meets a finite square well of depth [itex]V_{0}[/itex], 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 [itex]\int^{t_{1}}_{t_{0}}\int^{a}_{-a} |\Psi(x,t)|^{2} dx dt[/itex]. Where [itex]t_{1}-t_{0}[/itex] 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 [itex]E_{0}[/itex], the energy of the photon is [itex]E_{p}[/itex]

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

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