Probabilities must always total one
Start from a demand so obvious it is easy to overlook: whenever you measure something, *some* result must happen. The electron's spin will come out up or down; the odds of all the possible answers, added together, must equal exactly one — a 100% chance that *something* occurs. We baked this in earlier by insisting every state arrow have length one, the normalization condition. The question this guide answers is: as time passes and the state changes, what keeps that total pinned at one? It cannot be left to luck.
Quantum states are not frozen; they evolve. An undisturbed system glides smoothly from one state to another as the seconds tick by — this is time evolution, governed by the Schrödinger equation. So as the state arrow swings through Hilbert space, we need a guarantee that it never quietly grows or shrinks, because a change in its length would mean probabilities that no longer sum to one — a 90% universe, or a 110% one, both nonsensical.
The rule: evolution is a rotation, never a stretch
Here is the elegant answer. Quantum time evolution is required to be length-preserving: it may turn the state arrow to point in a new direction, but it may never lengthen or shorten it. Picture a rigid arrow pinned at the origin, free to swivel anywhere on the unit sphere but forbidden from stretching. That "rotation without stretching" property has a name — the evolution is described by a unitary operator, and the principle that all evolution behaves this way is called unitarity.
Because a unitary evolution preserves not just lengths but all overlaps ⟨φ|ψ⟩, it preserves angles between states too. Two states that were perfectly distinguishable (overlap zero) stay perfectly distinguishable as time goes on; two that were partly alike keep their exact degree of resemblance. The whole geometry of the state space turns as one rigid piece. Nothing is created, nothing destroyed, nothing blurred — the configuration is merely rotated.
Why we can trust this
Unitarity is not an extra assumption bolted on for tidiness — it follows from the Schrödinger equation itself. That equation has a particular shape (built around a special object called the Hamiltonian) which mathematically forces the resulting evolution to be a length-preserving rotation. So once you accept the basic equation of motion, conservation of total probability comes along for free, automatically, at every instant. You do not have to keep checking the books; the dynamics balances them for you.
- Total probability must equal one — "something happens" is certain. That fixes the state arrow's length at one.
- States evolve in time, so the arrow swings — it must swing without changing length, or the total would drift off one.
- Length-preserving swinging is a rotation: a unitary operator. The principle that evolution is always unitary is unitarity.
- The Schrödinger equation guarantees it, so probability is conserved automatically — no manual balancing needed.
The one apparent exception: measurement
An honest guide must flag one place where this smooth picture seems to break. When you actually *measure* a system, the state appears to jump abruptly — a superposition suddenly snaps to a single definite outcome. That jump is *not* a gentle unitary rotation; it is sharp and probabilistic, and reconciling it with smooth evolution is the famous, still-debated measurement problem of quantum mechanics. We are not going to solve it here. The fair summary is: between measurements, evolution is perfectly unitary and probability is exactly conserved; the apparent jump happens only at the moment of measurement, and what really goes on there is one of physics' deepest open questions.
And with that, the language is in your hands. You can write a state as a ket, probe it with a bra, find its overlaps, place it in Hilbert space, unfold it across a basis, and watch it rotate unitarily through time. These five ideas — state-as-arrow, bra-ket, Hilbert space, basis, unitarity — are the load-bearing beams of the whole subject. Everything more advanced is built on top of them, and you now know what holds it all up.