You need to understand the mathematics of Hilbert spaces to understand this issue.

The linear combination may be finite or infinite, there is not restriction.

The observables correspond one-to-one to the self adjoint operators in a separable Hilbert space of infinite dimension,

*H*. The pure states correspond one-to-one to the one dimensional subspaces of

*H*. Every state is a (possibly infinite) convex combination of pure states. (This is a mathematical statement of waveform generation and collapse)

'Collapse of the waveform' is linked to the so called 'measurement problem' or measurement paradox and is very difficult to discuss without some higher maths.

Try reading here

The Measurement Problem
Essentially the measurement problem, waveform collapse and 'paradoxes' like Schroedinger's cat are about the crossover point from deterministic classical mechanics to probabilistic quantum mechanics and how this is handled.