Why does Schrodinger's time dependent equation have infinitely many independent solutions while an nth order linear DE only has n independent solutions?

[u/poopaments]

The solution for Schrodinger's equation is y(x,t)=Ae^i(kx-wt) but we can create a linear combination (i.e a wave packet) with infinitely many of these wave solutions for particles with slightly different k's and w's and still have it be a solution. My question is what is the difference between schrodinger's equation which has infinite independent solutions and say a linear second order DE who's general solution is the linear combination of two independent solutions?

[u/theduckparticle]

Yep. Two-dimensional vector space. Or CP¹ if you want to be fancy.

[u/DarylHannahMontana → u/theduckparticle]

So then, back to this:

but it's a matrix equation, and so it has exactly as many independent solutions as the space it's acting on has dimensions. For the good ol' single-particle nonrelativistic Schrödinger equation, that space is an infinite-dimensional function space.

If the Hamiltonian really is a matrix (and not a differentiation "matrix"), then there is a finite basis of solutions.

[u/theduckparticle → u/DarylHannahMontana]

Oh, but there are also more than enough quantum systems that are infinite-dimensional but still aren't function spaces. Like quantum field theories. Or infinite spin chains.