If I understand it a free particle is not really quantised; it can have pretty much any wavelength. But when it is confined by boundary conditions (e.g. within an atom) we then get interference and reinforcement in the wave equations.

Boundary-induced quantisation does explain the limited set of acceptable/stable wavelengths around e.g. an atom.

However, what it doesn't explain, is why the amplitudes of such wavefunctions should be what they are. For example, why should the integral of the squared norm of the wavefunction for an electron be 1, if it is not a particle?