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?

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    $\begingroup$ "we can solve the Dirac's equation with an added condition that the momentum of the particle is 0" - If we can solve the Dirac equation when the momentum is zero in some frame, then we can also solve it when the momentum is any constant value (because it's the same situation viewed from another reference frame). This amounts to saying that we have a closed-form solution of the Dirac equation for a free particle. Basically the same is true of the Schrodinger equation. $\endgroup$ – probably_someone Dec 10 '18 at 15:41
  • $\begingroup$ In general, though, you probably need to define what you mean by "solve." By "solve" do you mean "write down an analytical, closed-form solution"? If so, then it's true that not many situations exist where we can currently do that, but that's also true for the Schrodinger equation. Or by "solve" do you mean "arrive at a possibly-numerically-found solution to whatever tolerance we require"? Because we can definitely do that for many more situations, for both the Dirac and Schrodinger equations. $\endgroup$ – probably_someone Dec 10 '18 at 15:44
  • $\begingroup$ So, that means, in principle, we can solve the schrodinger's equation and then find the solution in another inertial frame where the momentum of the particle is $0$ by simply doing a transformation to another inertial frame using Galilean transformation? $\endgroup$ – Naman Agarwal Dec 10 '18 at 15:44
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    $\begingroup$ Yes, you can use Galilean transformations to transform between frames in non-relativistic quantum mechanics (see physics.stackexchange.com/questions/56024/…). $\endgroup$ – probably_someone Dec 10 '18 at 15:47
  • $\begingroup$ Care needs to be exercised in systems that aren't a free particle, though, because the operators that form the potential may not be Galilean-invariant. $\endgroup$ – probably_someone Dec 10 '18 at 15:49

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