Send electrons one at a time through two slits and count where they land. With one slit open you get one smooth hump; with the other, another. With both open you do not get the sum of the two humps: you get stripes, with places no electron ever reaches even though each slit alone sent electrons there. Quantum mechanics explains this with one rule: each way something can happen has a complex amplitude, and when the ways cannot be told apart the amplitudes are added before squaring.

Probabilities are like heights of sand piles: two piles always make a bigger pile. Amplitudes are like water waves: two crests make a bigger crest, but a crest meeting a trough leaves flat water. Nature adds the waves first and only then asks how much is there.

Where the electrons land, across the screen. The dashed curve is what adding the two one-slit probabilities predicts. The measured pattern swings between four times one slit's contribution and zero: $|\psi_{1}+\psi_{2}|^{2}$, not $|\psi_{1}|^{2}+|\psi_{2}|^{2}$.

To every way an event can happen, quantum mechanics assigns a complex number, its amplitude, and the probability of the event is the modulus squared of the total amplitude. When an event can happen in two ways that nothing in the universe distinguishes, the total amplitude is the sum of the two.

• A probability is the modulus squared of an amplitude: $P=|\psi|^{2}$. Amplitudes can be negative or complex; probabilities cannot.
• Indistinguishable alternatives add amplitudes. The last term above, the interference term, is what makes dark fringes: at a point where $\psi_{2}=-\psi_{1}$ the probability is zero.

When an event can happen in two ways that are not distinguished, add the two complex amplitudes and take the modulus squared of the sum.

When two amplitudes are equal, the total is twice either one, and the probability is four times either alone: twice what adding probabilities would give.

Where the two amplitudes are equal in size and opposite in sign, they cancel, and the probability is zero — even though each slit alone would send electrons there.

Adding a second path adds an amplitude, and an amplitude can point against the first. More ways of getting somewhere can mean less chance of getting there.

If anything records which slit the electron used — a detector, a photon scattered off it, a change in the slit — the two alternatives are distinguishable, and their probabilities add.

Great work! You now know how to:

• turn an amplitude into a probability by taking its modulus squared
• add amplitudes for undistinguished alternatives and probabilities for distinguished ones
• find bright and dark fringes from the relative phase
• multiply amplitudes along the stages of one route