Dirac's bra-ket notation: |ψ⟩ is a state vector, ⟨φ| is its dual. ⟨φ|ψ⟩ is an inner product (probability amplitude).
Quick check
In Dirac notation, what does the inner product ⟨φ|ψ⟩ give you?
The vocabulary
- Ket |ψ⟩ — a column vector describing a quantum state.
- Bra ⟨φ| — the matching row vector (conjugate transpose).
- Bracket ⟨φ|ψ⟩ — a number; the overlap / probability amplitude.
- |φ⟩⟨ψ| — an operator (outer product), e.g. a projector onto a state.
Born rule
Probabilities of all outcomes sum to 1 because the state is normalised: ⟨ψ|ψ⟩ = 1.
It's just linear algebra with elegant brackets. A qubit is |ψ⟩ = α|0⟩ + β|1⟩ with |α|² + |β|² = 1 — you could even plot α as a point on the complex plane.