The behavior of quarks within protons and neutrons is described by quantum chromodynamics (QCD), which is the theory of the strong nuclear force. The Schrödinger equation, however, is not the appropriate equation to describe the dynamics of quarks within the framework of QCD.
The Schrödinger equation is a non-relativistic equation that describes the quantum behavior of non-relativistic particles. Quarks, on the other hand, are elementary particles that exhibit relativistic behavior and cannot be accurately described solely by the Schrödinger equation.
The dynamics of quarks within QCD are described by a more complex equation called the QCD Lagrangian, which is based on the principles of quantum field theory. Solving this equation analytically is an extremely challenging task due to the strong interactions involved and the mathematical complexity of the theory.
In practice, physicists use numerical methods and approximations, such as lattice QCD, to study the behavior of quarks within protons, neutrons, and other hadrons. These methods involve discretizing spacetime and performing calculations on a lattice to simulate the interactions between quarks and gluons, the carriers of the strong force.
While these approaches have been successful in providing insights into the behavior of quarks within hadrons, a complete analytical solution for the QCD Lagrangian and the behavior of quarks within protons or electrons is not yet available. The study of QCD and the behavior of quarks is an active area of research in theoretical physics.