Vector spaces over C and their inner products

The Schrödinger equation is linear: if ψ₁ and ψ₂ solve it, so does c₁ψ₁ + c₂ψ₂ for any complex c₁, c₂. Physics therefore hands you closure under addition and scalar multiplication, and the remaining axioms — associativity, a zero vector, additive inverses, distributivity — are bookkeeping that Cⁿ and function spaces satisfy automatically. The scalars must be complex, not real: a relative phase between components, as in (|0⟩ + e^(iδ)|1⟩)/√2, changes measurable interference terms, so multiplying one component by e^(iδ) genuinely changes the state. Two cautions. The zero vector is a bookkeeping element, not a physical state — it cannot be normalised. And a global phase e^(iα)ψ, applied to the whole vector, changes no inner-product modulus and so no prediction: states are really rays, vectors up to overall complex scale.

The inner product eats two vectors and returns one complex number, but not symmetrically. In the physics convention ⟨φ|ψ⟩ is linear in the ket, ⟨φ|cψ⟩ = c⟨φ|ψ⟩, and conjugate-linear in the bra, ⟨cφ|ψ⟩ = c*⟨φ|ψ⟩. On C² this is a recipe: conjugate the first vector's components, then multiply and add. With u = (2, 1+i) and v = (i, 1): ⟨u|v⟩ = 2*⋅i + (1+i)*⋅1 = 2i + 1 − i = 1 + i. Conjugate symmetry then says ⟨v|u⟩ = (1 + i)* = 1 − i — swapping the slots conjugates the number, and only the modulus survives the swap. Mind the literature: most mathematics texts put the conjugate on the second slot instead. Both conventions are internally consistent; mixing them mid-calculation is the actual error, and it announces itself as a sign flip on every imaginary part.

Set φ = ψ and conjugate symmetry makes ⟨ψ|ψ⟩ real; the positive-definiteness axiom demands more: ⟨ψ|ψ⟩ > 0 for every nonzero ψ. That is what entitles you to define ‖ψ‖ = √⟨ψ|ψ⟩ and call it a length. On C², ⟨ψ|ψ⟩ = Σ|ψᵢ|², a sum of squared moduli, positive term by term. Take |ψ⟩ = (1+i)|0⟩ + 2|1⟩: ⟨ψ|ψ⟩ = |1+i|² + |2|² = 2 + 4 = 6, so ψ̂ = ψ/√6 is the unit representative of the ray, and the numbers |⟨0|ψ̂⟩|² = 2/6 and |⟨1|ψ̂⟩|² = 4/6 already sum to one — Born's rule will simply read them off. The axiom has teeth: drop the conjugate from the recipe and the supposed norm of (1, i) computes to 1 + i² = 0, a nonzero vector of zero length, and the whole probability interpretation dies with the division by ‖ψ‖.

For any two vectors, |⟨φ|ψ⟩|² ≤ ⟨φ|φ⟩⟨ψ|ψ⟩. The proof is one application of positive-definiteness: for φ ≠ 0, the vector ψ − (⟨φ|ψ⟩/⟨φ|φ⟩)φ — ψ with its component along φ removed — has non-negative squared norm, and expanding that squared norm and rearranging gives the inequality. The equality condition rides along free: zero headroom forces the residual vector to vanish, so ψ = cφ. Two consequences arrive immediately. For unit vectors the squared overlap is at most 1, so Born's rule can never manufacture a probability above one. And the triangle inequality ‖φ + ψ‖ ≤ ‖φ‖ + ‖ψ‖ follows, making the norm an honest notion of distance. The payoff deferred to the uncertainty unit: applied to the vectors (A − ⟨A⟩)|ψ⟩ and (B − ⟨B⟩)|ψ⟩ for Hermitian A and B, the same inequality becomes Robertson's bound ΔA ΔB ≥ ½|⟨[A, B]⟩| — the uncertainty principle is Cauchy–Schwarz wearing operators.

Wavefunctions live in L²(R), the functions with ∫|ψ|² dx finite, under ⟨f|g⟩ = ∫ f*(x) g(x) dx. Membership is checked integral by integral: e^(−x²/2) passes; 1/(1+x²) passes; the plane wave e^(ikx) has |ψ|² = 1 everywhere and fails, which is why sharp-momentum states are not vectors of this space and are handled later as δ-normalised distributions. That L² is closed under addition is itself a small theorem — |f+g|² ≤ 2|f|² + 2|g|² does it — and Cauchy–Schwarz guarantees ⟨f|g⟩ converges whenever both members pass the test. Two fine points are assumed rather than proved this week. Functions equal except on a set of measure zero are identified, otherwise positive-definiteness would fail on functions nonzero only at isolated points. And the space is complete — Cauchy sequences converge inside it — which is exactly what the name Hilbert space adds to inner-product space, and what expansion in an infinite basis will silently rely on.

Given a proposed ⟨⋅|⋅⟩, test the axioms in order of how often they fail. Conjugate symmetry first: on C², the unconjugated form Σuᵢvᵢ is symmetric but not conjugate-symmetric, and its self-overlaps are not even real — feed it u = (1, 1+i) and it returns 1 + (1+i)² = 1 + 2i, a complex number no square root can turn into a length. Linearity in the ket second: it usually holds by construction. Positive-definiteness last, because it fails quietly. The weighted product ⟨f|g⟩ = ∫ f* g w(x) dx is a legitimate inner product whenever w(x) > 0 — the Sturm–Liouville problems closing this unit use exactly this — but let w vanish on an interval and any function supported there acquires zero norm: the form degrades to a seminorm, and normalisation divides by zero. The habit to build: name the space, name the inner product, and verify positivity before trusting any norm, probability or orthogonality claim computed from it.