Quantum states, Dirac notation and mixtures

The first postulate assigns an isolated system a unit vector |ψ⟩ in a complex separable Hilbert space H — two dimensions for a spin-½, the square-integrable functions L² for a particle on a line. But the assignment is many-to-one: |ψ⟩ and e^(iα)|ψ⟩ satisfy |⟨φ|e^(iα)ψ⟩|² = |⟨φ|ψ⟩|² for every |φ⟩, so no experiment distinguishes them. The physical state is therefore the ray — the one-dimensional subspace {c|ψ⟩ : c ∈ ℂ} — with the unit ket a convenient representative. Separability earns its place in the sentence: it guarantees a countable orthonormal basis, which is what lets a state be a column of amplitudes and an operator a matrix rather than something worse.

The Riesz theorem pairs every ket |ψ⟩ with exactly one bra ⟨ψ| — every continuous linear functional on H is of that form — and the pairing ⟨φ|ψ⟩ is the inner product: conjugate-symmetric, ⟨φ|ψ⟩ = ⟨ψ|φ⟩*, and therefore conjugate-linear in the bra slot in the physics convention. Everything measurable is built from it: ⟨ψ|ψ⟩ = 1 is the normalisation Born's rule needs, |⟨φ|ψ⟩|² is the probability that a system prepared in |ψ⟩ passes a test for |φ⟩, and ⟨φ|A|ψ⟩ is a matrix element once an operator sits between. Try it on |ψ⟩ = (3|0⟩ + 4i|1⟩)/5: ⟨ψ|ψ⟩ = (9 + 16)/25 = 1 — but only because the bra conjugates 4i to −4i, making the |1⟩ contribution (−4i)(4i) = +16. Drop the conjugate and you get (9 − 16)/25, a negative probability; the star is not decoration.

Any orthonormal basis (|n⟩) resolves the identity: Σ|n⟩⟨n| = 1̂. Insert it before a ket and the expansion falls out, |ψ⟩ = Σ⟨n|ψ⟩|n⟩, each coefficient a projection cₙ = ⟨n|ψ⟩, with Parseval guaranteeing Σ|cₙ|² = 1 — probability survives a change of representation because the insertion is the identity, not an approximation. Concretely: the z-basis ket |0⟩ meets the x basis |±⟩ = (|0⟩ ± |1⟩)/√2 through ⟨+|0⟩ = ⟨−|0⟩ = 1/√2, so |0⟩ = (|+⟩ + |−⟩)/√2 and a σₓ measurement returns ±1 at ½ each. The same insertion between an operator and a ket, ⟨m|A|ψ⟩ = Σₙ ⟨m|A|n⟩⟨n|ψ⟩, is nothing but matrix-times-column — Dirac notation makes the linear algebra self-assembling.

Multiply an entire ket by e^(iα) and every probability keeps its value: the phase is global and unphysical. Move the same factor inside a superposition and it becomes a relative phase, which measurement can see. Take |ψ_φ⟩ = (|0⟩ + e^(iφ)|1⟩)/√2. In the z basis both amplitudes have modulus 1/√2 whatever φ is: P(0) = P(1) = ½ and φ is invisible. Project on |+⟩ instead: ⟨+|ψ_φ⟩ = (1 + e^(iφ))/2, so P(+x) = ½(1 + cos φ) — a full interference fringe. At φ = 0 the state is |+⟩ itself and P = 1; at φ = π it is |−⟩ and P = 0; at φ = π/2 it is the σ_y eigenstate and P = ½. The working rule: a phase you can factor out of every term is gauge; one you cannot is data.

A ket encodes maximal knowledge. A machine that emits |ψₖ⟩ with probability pₖ has handed you less, and no single ket in H reproduces its statistics. The right object is ρ = Σ pₖ|ψₖ⟩⟨ψₖ| — Hermitian, positive, unit trace — with every prediction read off as ⟨A⟩ = Tr(ρA) and P = Tr(ρP̂) for the outcome's projector. A pure state is the special case ρ = |ψ⟩⟨ψ|, a rank-one projector with ρ² = ρ; anything else is mixed, and the basis-free test is purity: Tr ρ² = 1 iff pure, falling to 1/d at the maximally mixed 1̂/d. One warning the notation hides: the decomposition into pₖ and |ψₖ⟩ is not unique. Mixing |0⟩ and |+⟩ at ½ each gives ρ = [[¾, ¼],[¼, ¼]], and mixing that ρ's own eigenstates at 0.854 and 0.146 gives exactly the same matrix — the preparations differ, the physics does not.

Write both candidates in the (|0⟩, |1⟩) basis. The superposition (|0⟩+|1⟩)/√2 has ρ = ½[[1,1],[1,1]]; the even mixture has ρ = ½[[1,0],[0,1]]. Their diagonals — the populations — agree, so every z measurement agrees. The whole difference sits in the off-diagonal coherence ⟨0|ρ|1⟩: ½ against 0. The Pauli decomposition ρ = ½(1̂ + a⋅σ) makes this geometric: a = (⟨σₓ⟩, ⟨σ_y⟩, ⟨σ_z⟩) is a point in the unit ball, pure states on the surface, the even mixture at dead centre, and the coherences are the equatorial components aₓ = 2 Re ρ₀₁, a_y = −2 Im ρ₀₁. A σₓ measurement returns +1 with probability ½(1 + aₓ): 1 for the superposition, ½ for the mixture. Interference is nothing exotic — it is a coherence being read out in a rotated basis, and decoherence is that matrix element draining away.