Assume that the neutron-proton differential scattering cross section is symmetric about \(90^{\circ}\) in the center-of-mass system, and neglect the difference between triplet and singlet interactions. Show that this is equivalent to the requirement that $$ \sum_{l} \sum_{l^{\prime}}(2 l+1)\left(2 l^{\prime}+1\right) \sin \delta_{l} \sin \delta_{l^{\prime}} \cos \left(\delta_{l}-\delta_{l^{\prime}}\right) P_{l}(\cos \theta) P_{l^{\prime}}(\cos \theta)=0 $$ for all \(\theta\), where all \(l\) are even and all \(l^{\prime}\) are odd. Show that a sufficient condition that this relation is satisfied is that all \(\delta_{l}\) are either \(0(\bmod \pi)\) or \(\eta(\bmod \pi)\) and that all \(\delta_{l^{\prime}}\) are either \(0(\bmod \pi)\) or \(\eta+\frac{1}{2} \pi(\bmod \pi)\), where \(\eta\) is arbitrary. \(^{1}\) Show also that, if only one of the \(\delta_{l^{\prime}}\) is different from \(0(\bmod \pi)\), a necessary and sufficient condition that the above relation is satisfied is that all \(\delta_{l}\) are either \(0(\bmod \pi)\) or \(\delta_{l^{\prime}}+\frac{1}{2} \pi(\bmod \pi)\).

Legendre polynomials, denoted as \(P_l(\cos \theta)\), are solutions to Legendre's differential equation, which arises in physics problems with spherical symmetry, like neutron-proton scattering.
These polynomials have orthogonality properties that make them particularly useful in expanding functions of angles.
The orthogonality can be expressed as:
\[ \int_{-1}^{1} P_l(x) P_{l'}(x) \, dx = \frac{2}{2l+1} \delta_{ll'} \]
where \(\delta_{ll'}\) is the Kronecker delta, indicating that the integral is zero unless \(l = l'\).
This property helps simplify complex sums involving Legendre polynomials, which is crucial in solving scattering problems.
In neutron-proton scattering, these polynomials help to describe the angular distribution of scattered particles.
When examining the given expression:
\[ \sum_{l} \sum_{l^{\prime}}(2 l+1)\left(2 l^{\prime}+1\right) \sin \delta_{l} \sin \delta_{l^{\prime}} \cos \left(\delta_{l}-\delta_{l^{\prime}}\right) P_{l}(\cos \theta) P_{l^{\prime}}(\cos \theta)=0 \]
The orthogonality of \(P_l(\cos \theta)\) aids in ensuring the sum equals zero for all \(\theta\), provided specific conditions on phase shifts are met.
This function clearly plays a pivotal role in simplifying complex quantum mechanical problems.

Quantum angular momentum is a fundamental concept in quantum mechanics.
It's described by quantum numbers \(l\), which are associated with the eigenvalues of the angular momentum operators.
In scattering processes, the angular momentum quantum numbers \(l\) and \(l'\) characterize the rotational properties of the particles involved.
Each quantum number corresponds to the total angular momentum in units of \(\hbar\) (reduced Planck’s constant).
For neutron-proton scattering, the angular momentum quantum numbers specify the partial wave contributions to the scattering cross-section.
In the summation:
\[ \sum_{l} \sum_{l^{\prime}}(2 l+1)\left(2 l^{\prime}+1\right) \sin \delta_{l} \sin \delta_{l^{\prime}} \cos \left(\delta_{l}-\delta_{l^{\prime}}\right) P_{l}(\cos \theta) P_{l^{\prime}}(\cos \theta)=0 \]
The coefficients \((2l+1)\) and \((2l'+1)\) reflect the degeneracy of the quantum states with angular momentum \(l\) and \(l'\).
Understanding these contributions is critical for analyzing and predicting the scattering behavior.
The phase shifts \(\delta_l\) and \(\delta_{l'}\) represent the interaction potential's effect on the partial waves.
They play a key role in determining how the quantum states interfere and lead to the observed cross-section.

The scattering cross-section quantifies the likelihood of scattering events as a function of angle, energy, and other parameters.
For neutron-proton scattering, the differential cross-section measures how particles scatter at different angles relative to the incoming direction.
The provided expression:
\[ \sum_{l} \sum_{l^{\prime}}(2 l+1)\left(2 l^{\prime}+1\right) \sin \delta_{l} \sin \delta_{l^{\prime}} \cos \left(\delta_{l}-\delta_{l^{\prime}}\right) P_{l}(\cos \theta) P_{l^{\prime}}(\cos \theta)=0 \]
Shows how specific angular momentum states (characterized by \(l\) and \(l'\)) contribute to the cross-section.
If the contributions from even and odd angular momenta interfere destructively (cancel each other out), the resulting cross-section remains symmetric about \(90^{\circ}\).
The symmetry is crucial for simplifying the analysis and ensuring consistency with physical observations.
Practical calculations often involve approximations, where only a few terms are significant, depending on the energy and interaction potential.
Understanding how the phase shifts \(\delta_l\) depend on the potential and energy provides insight into the interaction dynamics, making the cross-section a vital tool for experimental and theoretical analyses.