We chose our basis because it is convenient. It's called a Fock space[1]. You can choose any basis you want, the equations of motion derived from the Lagrangian

We chose our basis because it is convenient. It's called a Fock space[1]. You can choose any basis you want, the equations of motion derived from the Lagrangian will tell you how any wavefunction evolves, independently from your choice of basis. The number of waveforms isn't finite by the way. There's a fundamental waveform for every momentum (which can be any real number) and for every number of particles (which must be a positive integer). [1]: http://en.wikipedia.org/wiki/Fock_space http://en.wikipedia.org/wiki/Fock_space