Assume that the interaction between the neutron and the proton that make up a deuteron can be represented by a square well potential with \(a=2.80 \times 10^{-18} \mathrm{~cm} .\) If the lowest \((l=0)\) energy level of the system is \(-2.18\) Mev (million electron-volts), calculate \(V_{0}\) in Mev to three significant figures. How does the answer compare with that which would be obtained from the approximate formula derived in Prob. 5 ?

Imagine a particle trapped inside a box with infinitely high walls. This is essentially the square well potential. The particle can only exist within this confined space and possess specific energy levels. The 'well' shape of the potential confines the particle to particular regions, much like trapping a ball in a box where it can only move within the box's walls.
An important feature of the square well is that the potential inside the well is constant and the same across its entire length. Outside, the potential becomes infinitely large, ensuring the particle can't escape. This simplified model helps us understand more complex quantum systems.
To calculate phenomena in a square well potential, we often use the Schrödinger equation. This equation helps us determine allowable energy levels and the behavior of particles within the potential.

In a square well potential, particles can only occupy certain discrete energy levels. These levels represent the allowed energies a particle can have while confined within the well. Think of these levels as the steps of a staircase—the particle can stand on one step or another but nowhere in between.
The fundamental formula to determine these energy levels is derived from solving the Schrödinger equation for the system. For a particle of mass \(m\) in a one-dimensional well of width \(a\):
$$E_n = \frac{\text{h}^2 \text{n}^2}{8m a^2} - V_0$$
Here, \(E_n\) represents the energy of the particle at the \(n\)th level, \(h\) is Planck's constant, and \(V_0\) is the potential depth. The integer \(n\) is the quantum number, determining the 'step' or energy level the particle occupies.
In our exercise, we look at the lowest energy level (\(l = 0\) and \(n = 1\)), leading us to the energy value required to calculate the potential depth.

In the realm of quantum mechanics, particle physics calculations are crucial to determine the characteristics and behavior of particles within a given potential. Let's recap the solved exercise:

• We were given the width of the well, \(a = 2.80 \times 10^{-18} \text{ cm}\), and the energy level \(E = -2.18 \text{ MeV}\).
• By converting the width into meters and using relevant constants such as Planck's constant and the neutron mass, we could substitute these values into our formula.
• Our task was to find \(V_0\), the depth of the well.

The essence of such calculations lies in understanding the relationships and interplay of these constants and values. Converting units, substituting into equations, and solving for unknowns are everyday steps. These calculations not only apply to academic exercises but also have real-world applications in understanding atomic and subatomic particles, which are fundamental to modern physics and technology.
With a solid grasp of these concepts, you can unlock deeper insights into the quantum world and the particles that form the universe.