In the derivation of two-neutrino oscillation probabilities, the assumption that the mass states have the same momentum is made for simplicity and to facilitate the mathematical calculations. This assumption is known as the "ultra-relativistic approximation" and is valid when the neutrinos involved in the oscillation process have very high energies compared to their masses.
When neutrinos are highly relativistic, their energy is dominated by their momentum, and their mass can be neglected in comparison. In this regime, the energy-momentum relation for a relativistic particle, such as a neutrino, can be approximated as:
E ≈ pc
Where E is the energy, p is the momentum, and c is the speed of light. Since the neutrinos involved in oscillation experiments are typically highly energetic, this approximation is reasonable.
By assuming that the mass states have the same momentum, the equations describing neutrino oscillations can be simplified. The assumption allows us to express the neutrino flavor states as superpositions of the mass states, and the time evolution of the neutrino flavor states can be derived using quantum mechanics. This simplification is often referred to as the "flavor basis" or "mass eigenstate basis" approximation.
It's important to note that the assumption of equal momentum is an approximation that may introduce some small errors in certain scenarios, especially for lower-energy neutrinos where the mass becomes more significant. In general, for precise calculations or in situations where the energy is not much larger than the mass, a more accurate treatment involving the full energy-momentum relation is necessary. Nonetheless, the equal momentum assumption is widely used in introductory discussions and derivations of two-neutrino oscillation probabilities due to its simplicity and ease of understanding.