In relativistic quantum mechanics, we can solve the Dirac's equation with an added condition that the momentum of the particle is $0$. However, such independence isn't provided by the Schrodinger's equation(I think so). Why is it so? Is there any physical reason to why this is possible in one case and not in the other?
Also, another doubt, the Dirac Matrices are used in the form of a 4-vector while contracting with other stuff in the Dirac equation and elsewhere but the so called "components" of the "Dirac matrix 4 vector" are rather matrices and not numbers as one would expect from the components of a 4 vector. Is there something mathematically or physically deeper about it or is it just efficient use of notation?