Assume that the interaction between the neutron and the proton that make up a deuteron can be represented by a square well potential for which \(a=0\) (zero range or \(\delta\) function interaction), and that the only bound energy level of the system has \(l=0\) and \(\epsilon=2.23\) Mev (million electron-volts). Show that no error is involved in the calculation of the photodisintegration cross section if the final-state wave function is taken to be a plane wave. Find the differential and total cross sections for unpolarized photons.

In physics, a 'square well potential' represents a basic model for the interaction between particles. For the deuteron (a bound state of a neutron and a proton), the interaction can be idealized using a square well potential. This simplifies calculations while still capturing the essential physics. The given problem assumes a zero-range interaction or \(\text{delta function}\) interaction, where the potential is represented by a sharp peak, effectively a 'well' of zero width. This approach vastly simplifies our mathematical treatment, focusing all interaction at a single point.

The 'plane wave assumption' states that the wave function of particles after interaction can be approximated as plane waves. This means that, rather than dealing with complex, possibly curved waves, we assume that the outgoing particles move away from the interaction zone in straight, evenly spaced wavefronts. In our deuteron photodisintegration problem, we show that this assumption is valid, due to the high energy of the outgoing particles compared to the binding energy of the deuteron. Thus, taking the final-state wave function as a plane wave introduces no significant error.

The 'differential cross section' quantifies how likely it is that a specific interaction occurs at different angles. For example, in our photodisintegration problem, the differential cross section tells us how likely the emitted particles are to move in a particular direction after the interaction. Mathematically, it is represented as: \[ \frac{d \sigma}{d \Omega} = \frac{16 \pi^{2} \alpha E}{3 k} \left| \left< f \right| \mathbf{d} \left| i \right> \right|^{2} \] where \( \alpha \) is the fine-structure constant, \( E \) is the photon energy, \( k \) is the wavenumber, and \( \mathbf{d} \) represents the dipole operator. This formula helps us understand the scattering pattern of the particles.

The 'total cross section' measures the likelihood that an interaction occurs, regardless of the direction in which the particles scatter. It is an integral of the differential cross section over all scattering angles, effectively summing up the probabilities for all possible outcomes. For our deuteron problem, it is given by: \[\sigma_{total} = \frac{64 \pi^{2} \alpha E}{3 k} \left| \left< f \right| \mathbf{d} \left| i \right> \right|^{2} \] This total cross section allows us to capture the overall effectiveness of the photodisintegration process.

The 'Born approximation' is a method used to simplify the calculation of scattering problems. It assumes that the scattered wave is only slightly altered by the interaction, making it easier to calculate. For high-energy events, like in our deuteron photodisintegration problem, the Born approximation is particularly useful. It justifies our plane wave assumption because the high energy of the final state means the interaction perturbs the scattering wave only minimally. Hence, the approximation is valid and simplifies complex quantum mechanical calculations.