superposition principle

The superposition principle says that if two wavefunctions each describe a valid quantum state, then so does any weighted sum of them. A particle can be in a state that is part 'here' and part 'there', or part one energy and part another, all at once. This is not a statement that we are merely ignorant of a single hidden value; the combined state has genuinely different physical behaviour from either piece alone.

Superposition follows directly from the mathematics of the wavefunction and from the linearity of the Schrödinger equation, which guarantees that combinations of solutions are again solutions. The weights in the sum are complex amplitudes, so it is their phases, not just their sizes, that matter. Change the relative phase between two superposed parts and you change where interference fringes appear, even though the probabilities of the individual parts are untouched.

The honest subtlety is what happens on measurement. While left alone, a superposition is real and its parts interfere. But when you measure, you find one definite outcome, with probabilities set by the Born rule — you never observe the eerie 'both at once' directly. Whether the other branches vanish, hide, or persist unseen is exactly the interpretive question quantum mechanics leaves open; the principle itself only tells you how the states combine before you look.