Superposition and measurement

Everything in the first six lessons — photons, matter waves, the uncertainty principle — points toward the same conclusion: a quantum system doesn't have a single definite story before you look at it. The mathematical object that describes it, the quantum state (often written |ψ⟩, a 'ket'), is allowed to be a superposition of multiple possible outcomes at once, weighted by complex-number amplitudes. In the double-slit video from earlier in this unit, a single photon or electron isn't secretly going through one slit or the other — its state is a superposition of 'went through slit A' and 'went through slit B,' and the interference pattern you saw building up, one dot at a time, is the observable signature of that superposition genuinely existing, not just our ignorance of which slit it 'really' used.

How do amplitudes turn into the probabilities you actually observe? That's the Born rule, proposed by Max Born in 1926: the probability of getting a particular outcome equals the squared magnitude of that outcome's amplitude in the superposition. Double an amplitude and you quadruple its probability; two amplitudes with equal size are equally likely outcomes. This is the qualitative version of a rule you'll use quantitatively throughout the rest of this course — once Unit 2 introduces the Schrödinger equation and the wavefunction ψ(x), the Born rule becomes |ψ(x)|², a genuine probability density for where you'd find the particle if you measured its position.

Measurement itself is where quantum mechanics parts ways most sharply with classical intuition. Before you measure, a system can be in a genuine superposition — not merely an unknown but pre-existing single value, the way a shuffled, face-down card has a definite rank you just haven't seen yet. Upon measurement, the system yields one definite outcome, with relative frequency set by the Born rule, and every subsequent measurement of the same quantity agrees with that outcome — the superposition has 'collapsed.' Precisely what happens during that collapse, and whether 'collapse' is even the right way to describe it, is not settled physics so much as it is a live question about how to interpret a spectacularly well-tested formalism — the subject Unit 8 returns to once you have the Schrödinger equation, angular momentum, and the hydrogen atom in hand to make the question concrete.