The state vector and its position representation

Start in two dimensions, where nothing is hidden. A ket |v⟩ in C² is one arrow; pick an orthonormal basis (|e₁⟩, |e₂⟩) and it acquires components vᵢ = ⟨eᵢ|v⟩ — two complex numbers that change the moment you rotate the basis, while the arrow does not. Position representation is the same move with a continuous label: the basis vectors are |x⟩, one for each real number, the components are ψ(x) = ⟨x|ψ⟩, and the sum over an index becomes an integral over x. So "the wavefunction" is a coordinate list — an uncountable column vector — and writing |ψ⟩ instead of ψ(x) is not pedantry: it is the reminder that the same state will later be reread as momentum components φ(p) = ⟨p|ψ⟩ without anything physical changing. The notation splits the labour cleanly: kets carry the physics, brackets extract numbers from it.

The space is L²(ℝ): complex-valued functions of x with ∫|ψ|²dx finite. Run candidates through the test. A Gaussian e^(−x²/2) passes. 1/(1+x²) passes: its squared modulus falls as x⁻⁴. The plane wave e^(ikx) fails — |ψ|² = 1 everywhere integrates to infinity — so a state of perfectly sharp momentum is not in the space, however often the phrase gets used. Boundedness is not the point either: ψ = |x|^(−1/4) on [−1, 1] blows up at the origin yet ∫|x|^(−1/2)dx = 4 is finite, so it qualifies, while the gentler-looking |x|^(−1/2) fails on the same interval because ∫|x|⁻¹dx diverges logarithmically. And "Hilbert" adds one clause beyond the inner product: the space is complete, so Cauchy sequences of functions converge to members. That clause is what will later let an infinite eigenfunction expansion actually equal the state it represents.

The inner product ⟨φ|ψ⟩ = ∫φ*(x)ψ(x)dx — conjugate on the first slot, in the physics convention — is the only measuring device the space owns. It gives lengths: ‖ψ‖² = ⟨ψ|ψ⟩, real and non-negative, scaled to 1 by normalisation. It gives geometry: swapping the slots conjugates the number, ⟨φ|ψ⟩ = ⟨ψ|φ⟩*, and Cauchy–Schwarz bounds |⟨φ|ψ⟩| by ‖φ‖‖ψ‖, so for unit vectors the squared overlap sits in [0, 1] — exactly the interval a probability needs, which is why Born's rule (next topic) can be stated at all. And it gives distinguishability a shape: two unit-norm Gaussians of width σ whose centres sit a distance d apart overlap by e^{(−d²/4σ²)}. At d = 2σ that is e⁻¹ ≈ 0.37; at d = 6σ it is e⁻⁹ ≈ 1.2 × 10⁻⁴, and the states are orthogonal for every practical purpose. Overlap, not distance in x, is how the space measures how different two states are.

A basis labelled by a continuum cannot be normalised to 1. Demand that the expansion ψ(x) = ⟨x|ψ⟩ and the completeness relation I = ∫|x⟩⟨x|dx hold together, and orthonormality is forced into the distributional form ⟨x|x′⟩ = δ(x − x′): insert the identity into ⟨x′|ψ⟩ and ψ(x′) = ∫⟨x′|x⟩ψ(x)dx must hold for every ψ, which only the δ can do. Insert it into ⟨φ|ψ⟩ instead and the abstract bracket becomes ∫φ*(x)ψ(x)dx, which is the whole point of the basis. The price appears at x′ = x: ⟨x|x⟩ = δ(0) is infinite, so |x⟩ fails the very membership test it administers — it is an improper vector, a distribution that only means something under an integral. Physically this is no defect. No instrument resolves position exactly; a real position measurement of width ε leaves the particle in a normalisable packet of that width, and |x⟩ is the idealisation those packets approach in the bookkeeping but never in the space.

Multiply the state by e^(iα): every component becomes e^(iα)ψ(x), every inner product picks up the same phase, and every measurable — each |⟨φ|ψ⟩|², each |ψ(x)|² — is untouched. So the physical state is not a unit vector but a ray, the set (e^(iα)|ψ⟩), and "normalise ψ" fixes only the modulus of the constant. The immunity stops inside a superposition. Take |ψ⟩ = (|a⟩ + e^(iθ)|b⟩)/√2 with ⟨a|b⟩ = 0, and test it against |c⟩ = (|a⟩ + |b⟩)/√2, for which ⟨c|a⟩ = ⟨c|b⟩ = 1/√2: the amplitude is (1 + e^(iθ))/2, so the probability is cos²(θ/2) — 1 at θ = 0, 1/2 at θ = π/2, 0 at θ = π. That θ lives between two components of one vector, and no global rotation can remove it. One rule carries the whole distinction: a phase you can factor out of the entire ket is gauge; a phase you cannot is physics.

Discretise the line: N points xⱼ spaced Δx apart, and ψ collapses to the column vector ψⱼ = ψ(xⱼ) in C^N. The inner product becomes a conjugate dot product carrying the measure, ⟨φ|ψ⟩ ≈ Δx Σ φⱼ* ψⱼ — in NumPy, dx * np.vdot(φ, ψ), and vdot rather than dot because vdot conjugates its first argument exactly as the bra does. Try it: a unit-norm Gaussian of σ = 1 sampled at the N = 256 points xⱼ = −8 + jΔx with Δx = 0.0625 returns dx * np.vdot(ψ, ψ) = 1 to machine precision, because the tails the box cuts off are already below 10⁻¹⁴. Two habits transfer from the mathematics: check the norm before trusting any other number, and remember what truncation assumes — amplitude outside the box, or structure finer than Δx, is simply not representable, and any probability that belongs there is silently lost. Every numerical method later in the course — eigh, the FFT, split-step propagation — is built on this identification.