When comparing a classical and a quantum string, the resulting wave function is made up of the sum of the individual modes for the former but the product for the latter. That is at least what I gather from a lecture on quantum field theory, cf. particularly the slide at minute 5:07. The narrator says that
[Unlike the classical modes], the quantum mechanical modes are oscillating independently so the quantum state of the entire system is the product of the quantum states of the individual modes.
Is this one of those dreaded postulates? How can we explain this intuitively from first principles?
PS: Incidentally, does the superposition principle (of quantum fame) only apply within any given mode? Why would that be the case given that EM waves can be made up of infinitely many frequencies?