# What Energy Maximizes Neutron Trapping in a Finite Square Well?

• atlantic
In summary, the question asks for the most advantageous energy for a free neutron to have in order to be trapped by a finite square well potential. The probability of emitting a photon and decreasing its energy is proportional to the integral of the square of the wavefunction over time and space. The goal is to find a value for the initial energy that maximizes this integral.
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.

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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").

## 1. What is the difference between a free particle and a bound particle?

A free particle is an object that is not subject to any external forces and can move freely in space, while a bound particle is confined to a specific region due to the presence of an external force or potential. In other words, a free particle has infinite potential energy while a bound particle has finite potential energy.

## 2. How does a particle become bound?

A particle can become bound by interacting with an external force or potential. This interaction can result in the particle being confined to a specific region, typically in the form of an energy well or potential barrier. The particle's motion is then limited by the boundaries of this region.

## 3. What are some examples of bound particles?

Bound particles can include electrons within an atom, protons and neutrons within a nucleus, and molecules within a solid or liquid. These particles are bound by electrostatic forces, nuclear forces, or intermolecular forces, respectively.

## 4. Can a bound particle become a free particle?

In order for a bound particle to become a free particle, it would need to overcome the external force or potential that is confining it. This can occur through energy input, such as heating or ionization, which provides the particle with enough energy to break free from its bound state.

## 5. What are the implications of a particle transitioning from a free to a bound state?

The transition from a free to a bound state can have significant implications in various fields of science, including quantum mechanics, chemistry, and materials science. It can affect the behavior and properties of particles, and understanding this transition is essential in studying systems at the atomic and molecular level.

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