The Hamiltonian as a self-adjoint operator

Canonical quantisation is a substitution with rules. Keep the classical Hamiltonian H(x, p) = p²/2m + V(x), then promote: x̂ acts as multiplication by x, p̂ as −iℏ d/dx, both on the position-basis wavefunction ψ(x) in L²(ℝ). Kinetic energy becomes −(ℏ²/2m) d²/dx² — in three dimensions the Laplacian −(ℏ²/2m)∇² — and the potential becomes multiplication by V(x). The pair is calibrated by the commutator: [x̂, p̂] = iℏ mirrors the Poisson bracket {x, p} = 1 with the factor iℏ, which is the entire content of the word 'quantisation'. Numbers anchor it: for an electron ℏ²/2mₑ = 3.81 eV Ų, so a wavefunction forced to bend on the scale of an ångström carries electron-volt kinetic energies — atomic physics fixed by one coefficient.

The substitution is only well defined because p²/2m + V(x) never multiplies x by p. The classical observable xp has at least three quantum images — x̂p̂, p̂x̂ and (x̂p̂ + p̂x̂)/2 — differing by multiples of iℏ, since x̂p̂ − p̂x̂ = iℏ. The first two are not even Hermitian: (x̂p̂)† = p̂x̂ = x̂p̂ − iℏ. Hermiticity therefore forces the symmetrised product, and Weyl ordering extends the rule to any polynomial in x and p. Nor is the ambiguity exotic: a charged particle in a magnetic field carries (p̂ − qA(x̂))²/2m, whose expansion produces the cross term p̂⋅A + A⋅p̂, collapsing to 2A⋅p̂ only in Coulomb gauge ∇⋅A = 0, where the factors commute. Every ordering choice shifts the operator at order ℏ — invisible classically, measurable quantum mechanically.

In ℂⁿ every matrix acts on every vector. In L² that fails: Ĥ is unbounded and cannot act on the whole space. The function ψ = 1/(1 + |x|) is square-integrable, yet x̂ψ is not, because ∫ x²/(1 + |x|)² dx diverges; the kink e^(−|x|) is normalisable, but its second derivative contains −2δ(x), which the kinetic term cannot digest. So defining Ĥ means naming its domain D(Ĥ): the dense set of states smooth enough, decaying fast enough, and — on an interval — obeying stated boundary conditions. Change the domain and you change the operator, spectrum included: −(ℏ²/2m) d²/dx² on [0, L] with Dirichlet walls has levels 1, 4, 9, … in units of π²ℏ²/2mL², while the periodic domain ψ(0) = ψ(L), ψ′(0) = ψ′(L) delivers 0, 4, 16, … in the same unit, each nonzero level twice. Same expression, different physics.

Test the kinetic term against two states φ and ψ on [0, L]. Two integrations by parts give ⟨φ|T̂ψ⟩ − ⟨T̂φ|ψ⟩ = (ℏ²/2m)[φ̄′ψ − φ̄ψ′]₀ᴸ: the interior cancels identically, and the entire question of hermiticity condenses onto four boundary numbers. The figure below computes this live: φ = sin(2πx/L) respects the Dirichlet walls, while ψ is given a leak ψ(0) = β, and the defect comes out as −4πβ in units of ℏ²/2mL — growing linearly with the leak and utterly indifferent to the interior amplitude. Killing the bracket for every pair of states in the domain is a condition on the domain, and its solutions form a four-parameter family: Dirichlet ψ = 0, Neumann ψ′ = 0, periodic identification, and the Robin interpolations ψ′ = κψ with real κ — every one a legitimate, distinct, self-adjoint Hamiltonian.

Symmetry — ⟨φ|Ĥψ⟩ = ⟨Ĥφ|ψ⟩ on D(Ĥ) — can hold while the adjoint Ĥ†, defined on every φ for which some η satisfies ⟨φ|Ĥψ⟩ = ⟨η|ψ⟩ for all ψ in D(Ĥ), lives on a strictly larger domain. The clean counterexample is momentum on the half-line: p̂ = −iℏ d/dx on x ≥ 0 with ψ(0) = 0. The boundary term iℏφ̄(0)ψ(0) dies through ψ(0) = 0 alone, so the adjoint imposes no condition on φ — and it owns the normalisable eigenfunction e^(−x/a) with the imaginary eigenvalue iℏ/a. Von Neumann's deficiency indices count the L² solutions of Ĥ†χ = ±iλχ with λ > 0: equal indices mean self-adjoint extensions exist and form a family; here they are (1, 0), so no boundary condition whatever can repair p̂. The physics is honest — momentum generates translations, and translating a half-line state pushes it off its space. A confined particle keeps a kinetic energy, since −d²/dx² there has indices (1, 1), but it has no momentum observable.

Self-adjointness is the hypothesis of the two theorems this unit runs on. The spectral theorem gives Ĥ a real spectrum and a complete spectral resolution: an orthonormal eigenbasis of levels Eₙ when the spectrum is discrete, as in a box, joined by a continuous part when it is not. Stone's theorem exponentiates Ĥ into the unitary U(t) = e^{(−iĤt/ℏ)}, and unitarity is probability conservation; a merely symmetric operator generates no such group. Numerically the issue seems to evaporate: discretise on N points and Ĥ becomes a finite real symmetric matrix, and in finite dimensions symmetric and self-adjoint coincide. But the domain has not vanished — it has moved into the stencil. Ending the three-point Laplacian at the first and last rows silently sets ψ = 0 one step beyond each wall, so scipy.linalg.eigh_tridiagonal is solving the Dirichlet extension; modelling periodic or Robin physics means editing the corner entries of the matrix, which is the boundary condition wearing numerical clothes.