Self-adjoint operators and their spectra

In a finite orthonormal basis the adjoint is the conjugate transpose, (A†)ᵢⱼ = (Aⱼᵢ)*, and ⟨φ|A|ψ⟩* = ⟨ψ|A†|φ⟩ says the same thing without naming a basis. Nothing more is needed there. In L²(ℝ) it is not that easy, because the operators physics cares about are unbounded: there is no constant c with ‖pψ‖ ≤ c‖ψ‖, and applying −iħ d/dx to a merely square-integrable ψ can throw you straight out of the space. So an operator is a pair — a rule together with a dense domain D(A) — and the adjoint has to be constructed rather than written down. A vector φ belongs to D(A†) exactly when some χ satisfies ⟨φ|Aψ⟩ = ⟨χ|ψ⟩ for every ψ ∈ D(A), and only then is A†φ defined, as χ. Read that definition twice: the smaller you make D(A), the fewer conditions φ must satisfy, so the larger D(A†) becomes. Every difficulty in this topic is a consequence of that inverse relation.

Symmetric asks only that ⟨φ|Aψ⟩ = ⟨Aφ|ψ⟩ whenever both vectors sit in D(A) — equivalently D(A) ⊆ D(A†), with A† agreeing with A there. Self-adjoint asks for the two domains to be equal. Most textbooks say "Hermitian" and mean the first. The example that separates them is p = −iħ d/dx on L²(0, ∞) with the domain ψ(0) = 0. It is symmetric. It owns no eigenvector: pψ = λψ forces ψ ∝ e^(iλx/ħ), whose modulus is constant for real λ and therefore not square-integrable on a half-line. Its adjoint meanwhile carries no boundary condition at all, so for every complex z with Im z > 0 the function e^(izx/ħ) decays, lies in L²(0, ∞) with squared norm ħ/(2 Im z), and satisfies p†ψ = zψ. Each such ψ is orthogonal to the whole range of p − z*, so the spectrum of this perfectly symmetric operator is the closed lower half plane, the mirror image of the adjoint's eigenvalues. Real eigenvalues and a real spectrum are not the same claim, and only self-adjointness delivers the second.

The working test is one line. For p = −iħ d/dx on an interval [0, L], integration by parts gives ⟨φ|pψ⟩ − ⟨pφ|ψ⟩ = −iħ [φ*(x)ψ(x)]₀^L. The factor of i is what makes this possible at all: bare d/dx is antisymmetric, and multiplying by −i flips the sign back. Self-adjointness is now the demand that the bracket vanish for every pair drawn from one and the same domain. Dirichlet, ψ(0) = ψ(L) = 0, kills it — but only by constraining ψ, leaving φ entirely free, so D(p†) is strictly larger and the operator is symmetric only. The repair is a condition that ties the two ends together: ψ(L) = e^(iθ)ψ(0). Then φ*(L)ψ(L) = e^(−iθ)φ*(0) × e^(iθ)ψ(0) = φ*(0)ψ(0), the bracket cancels identically, and D(p†) = D(p). Its eigenfunctions e^(ipx/ħ)/√L are honest members of L²(0, L), with pₙ = (2πn + θ)ħ/L spaced h/L apart — 3.31 × 10⁻²⁵ kg m s⁻¹ for L = 2.00 nm.

Let A|a⟩ = a|a⟩ with |a⟩ ∈ D(A) and A symmetric. Then a⟨a|a⟩ = ⟨a|Aa⟩ = ⟨Aa|a⟩ = a*⟨a|a⟩, and since ⟨a|a⟩ > 0 the eigenvalue satisfies a = a*. The identical computation on two eigenvectors gives (a − b)⟨b|a⟩ = 0, so distinct eigenvalues carry orthogonal eigenvectors. Degeneracy weakens that: a repeated eigenvalue fixes a subspace, orthogonality is guaranteed between eigenspaces but not automatically inside one, and you pick a basis there yourself, usually by diagonalising a second commuting observable. And every line of the argument is conditional on the eigenvector existing in the space. The position operator on L²(ℝ) is self-adjoint on D(x) = (ψ : xψ ∈ L²), yet (x − a)ψ(x) = 0 forces ψ = 0 almost everywhere: not one eigenvector, anywhere in the spectrum.

For a self-adjoint A the theorem supplies a projection-valued measure E, with A = ∫ λ dE(λ), I = ∫ dE(λ), and P(outcome lies in Δ) = ⟨ψ|E(Δ)|ψ⟩. Where the spectrum is discrete, dE is a sum of projectors and you recover the familiar orthonormal basis with Σₙ |n⟩⟨n| = I. Where it is continuous there is no basis of the space at all: x and p on L²(ℝ) both have spectrum ℝ, and their so-called eigenvectors δ(x − a) and e^(ipx/ħ)/√(2πħ) are δ-normalised objects living in a rigged space, not in L². The completeness statements ∫|x⟩⟨x| dx = I and ∫|p⟩⟨p| dp = I still hold as operator identities, and they carry dimensions, since ⟨x|x′⟩ = δ(x − x′) has units of inverse length. So "Hermitian implies an eigenbasis" is a finite-dimensional theorem. Its infinite-dimensional replacement is the spectral measure, and what you compute from it is the probability of an interval.

Sample ψ on N points spaced a = L/N and build p as −iħ times the central-difference matrix. What you write in the first and last rows is a choice of self-adjoint extension, made by your array indexing. Wrap the ends with a phase, ψ_N = e^(iθ)ψ₀, and the matrix is a circulant twisted by that phase, with eigenvectors e^{(i(2πn+θ)j/N)} and eigenvalues ħ sin((2πn + θ)/N)/a, real by construction and converging to (2πn + θ)ħ/L for |n| ≪ N: that is the θ member of the family above. The central difference also carries spurious doubler modes near n = N/2, which lay a second ladder beside the true one; an FFT derivative with the same twist gives the θ ladder alone. Drop the corner entries instead — the naive Dirichlet grid — and the matrix stays exactly antisymmetric, so −iħD is still Hermitian and every eigenvalue is still real, although the operator it imitates owns no eigenvector and has the whole complex plane as its spectrum. Finite dimensions cannot reproduce that, so the grid returns a real ladder that never settles: for L = 2.00 nm the eigenvalues nearest zero approach 0, ±1.657, ±3.313 (×10⁻²⁵ kg m s⁻¹) for odd N but ±0.828, ±2.485 for even N. A Hermitian matrix is not evidence that the operator you meant is self-adjoint.