Linear Operators on Hilbert Space

"Build the matrix, diagonalize, read the physics" is the daily loop of computational science. Quantum-chemistry packages assemble Hamiltonians in orbital bases exactly as the derivation card assembled $\hat x$ — texture, selection-rule zeros, truncation audits and all; the field's crown-jewel methods (CI, coupled-cluster) are strategies for taming matrix size. Every quantum-computing gate is a small operator specified as its matrix, composed by multiplication in precisely equation (2)'s sense, and compiled hardware instructions are checked against the algebra rule that order matters. The two-level truncation of Example 2 is the working model of qubits, atomic clocks, and laser spectroscopy — with its off-resonance audit now an engineering discipline ("leakage to state 3" is a quoted error budget line in superconducting processors). And the unbounded-operator fine print earns salaries in mathematical physics: proving self-adjointness of new Hamiltonians remains the licensing exam for quantum models.

Think of an operator as a machine with a conveyor belt: states in, states out. The linearity contract says the machine has no threshold, no saturation, no cross-talk: feed it a superposition and it processes each component independently, preserving the coefficients. That contract is physics, not convenience. Interference patterns survive propagation because the evolution machine is linear; a superposition of two paths stays a superposition rather than collapsing into a blend. When you meet genuine nonlinearity in quantum-looking equations (the Gross–Pitaevskii equation of condensates, say), it is always an effective description — many particles approximated as one field — never the fundamental theory.

The second picture is the operator as a filled-in table. Because the machine is linear, you know everything about it once you know what it does to basis vectors: $\hat A|n\rangle$ is some state, expandable as $\sum_m A_{mn}|m\rangle$, and the complex numbers $A_{mn}$ — how much of output $m$ comes from input $n$ — are the machine's complete specification. Arrange them in an array and every question about $\hat A$ becomes arithmetic. The rows and columns even carry physics on their face: a diagonal table means the basis states are the machine's "natural" inputs (each processed without leaking into the others — eigenstates, formally treated in the next article); zeros stamped in a pattern mean a selection rule; a band structure means the operator only connects near neighbours. Reading matrices like this is a professional skill, and the worked examples train it.

The contract, in one line — for all states and all complex coefficients:

where: both requirements matter — additivity and the scalars passing through unconjugated. The roster of machines you have already used checks out instantly: the position operator ($\hat x\psi = x\psi$ — multiplication distributes over sums), the momentum operator $\hat p = -i\hbar\,\partial_x$ (differentiation is linear — its wave-mechanics credentials live in the Momentum Operator article of the previous chapter), the parity operator $\hat\Pi$ (relabeling the argument touches no coefficient), every projector $|n\rangle\langle n|$, and every Hamiltonian you have built. The instructive counterexample is complex conjugation, $\hat K\psi = \psi^*$: additive, yes — but $\hat K(c\psi) = c^*\hat K\psi$. The scalar comes through conjugated: $\hat K$ is antilinear, a genuinely different species that quantum mechanics reserves for exactly one job (time reversal), and the distinction will matter there. The table-of-numbers statement, precisely: pick an orthonormal basis $\{|n\rangle\}$ and define

where: the right-hand reconstruction (a sum of outer products weighted by the matrix elements) shows the correspondence is complete — nothing about $\hat A$ outruns its matrix. Composition then obeys the chain you verified in the bra-ket article by inserting a complete set: $(\hat A\hat B)_{mn} = \sum_k A_{mk}B_{kn}$ — operator products are matrix products, with order preserved. Everything in your linear-algebra toolkit — identity ($\mathbb 1_{mn} = \delta_{mn}$), inverses when they exist ($(\hat A\hat B)^{-1} = \hat B^{-1}\hat A^{-1}$, order reversed), traces ($\mathrm{Tr}\,\hat A = \sum_n A_{nn}$, cyclic under rotation of factors) — imports wholesale.

Multiplication of machines is running one after another — and order is part of the specification. The bracket $[\hat A, \hat B] = \hat A\hat B - \hat B\hat A$ — the commutator — is the standing measure of how badly two machines care about order, and the cleanest concrete demonstration uses the two operators quantum mechanics is built on.

What this article insists on is the habit the card teaches: never reorder operator products without paying the commutator. The habit's most common casualty is the exponential. Functions of operators are defined by power series —

where: the failure is quadratic and computable (the cross terms $\hat A\hat B$ vs $\hat B\hat A$ disagree at second order; the Baker–Campbell–Hausdorff machinery organizing the correction made a cameo in the coherent-states article). Exponentials of operators are about to become the theory's verbs — time evolution $e^{-i\hat Ht/\hbar}$, translations, rotations — all built and certified in Unitary Operators and Transformations; the algebra rule (4) is the safety briefing. Note what is deliberately absent from this article: the adjoint $\hat A^\dagger$ as a definitional theory (owned by Hermitian Conjugate and Adjoint Operators), and the classification of the physically distinguished operator families it generates — self-adjoint (next article, immediately) and unitary (later). Here you have the raw algebra those articles will discipline. One honest flag before the examples, because infinite dimensions really do add fine print. In the box basis, compute how hard $\hat p$ works: $\|\hat p\,\psi_n\| = n\pi\hbar/L$, growing without bound as $n$ increases. No finite constant $C$ satisfies $\|\hat p\psi\| \le C\|\psi\|$ for all states: $\hat p$ is an unbounded operator, as are $\hat x$ on the line and every Hamiltonian with an infinite spectrum. Consequences worth knowing now: unbounded operators are not defined on every state (the Hilbert-space article's tent function has finite $\langle\hat H\rangle$ but infinite $\langle\hat H^2\rangle$: its kink makes $\hat H\psi$ leave $L^2$, so it sits outside the operator domain of $\hat H$ even while the energy form $\int|\psi'|^2$ stays finite), and statements like "Hermitian" acquire domain fine print that the next article will flag and the rigged-space article will resolve. For every calculation in this track the practical rule suffices: work with states smooth and decaying enough that the operators you apply keep them in the space.

Example 1 — is it linear? A machine inspection line (confidence builder). Classify each machine as linear, antilinear, or neither: (a) $\hat A\psi(x) = x^2\psi(x)$; (b) $\hat B\psi(x) = \psi^*(x)$; (c) $\hat C\psi(x) = \psi(x) + \chi_0(x)$ for a fixed nonzero $\chi_0$; (d) $\hat D\psi(x) = \psi(2x)$.

Step 1. (a): multiplication by any fixed function distributes over sums and passes scalars — linear ✓ (it is $\hat x^2$, a product of linear machines).

Step 2. (b): additive, but $\hat B(c\psi) = c^*\psi^*$ — scalars conjugate on passage: antilinear. (The physical tenant of this species is time reversal.)

Step 3. (c): $\hat C(\psi_1 + \psi_2) = \psi_1 + \psi_2 + \chi_0$, but $\hat C\psi_1 + \hat C\psi_2 = \psi_1 + \psi_2 + 2\chi_0$ — neither (affine, not linear: the offset breaks superposition; note $\hat C(0) \neq 0$, an instant tell).

Step 4. (d): rescaling the argument touches no coefficient — linear ✓ (a dilation; with the right prefactor it becomes the unitary scaling operator). Score: two linear, one antilinear, one impostor — and the impostor was caught by the zero-vector test, worth keeping.

Example 2 — the two-level dipole, by truncation (exam level). For light driving a particle in a box near the $1 \leftrightarrow 2$ resonance, only states 1 and 2 matter. Truncate $\hat x$ (units $L = 1$) to the $2\times2$ block, and extract the physics.

Step 1. From the derivation card: $x_{11} = x_{22} = \tfrac12$, $x_{12} = x_{21} = -16/9\pi^2 = -0.180$. The block is

Step 2. Diagonalize: eigenvalues $\tfrac12 \mp 16/9\pi^2 = \boxed{0.320 \text{ and } 0.680}$, with eigenvectors $\tfrac{1}{\sqrt2}(|1\rangle \pm |2\rangle)$. Physical reading: the superpositions that maximally displace the particle left and right of centre — the two "positions" available to a two-level dipole, split symmetrically about the well centre.

Step 3. The off-diagonal element is the light-matter coupling: the dipole interaction $-qE\hat x$ has matrix element $-qE\,x_{12}$ between the resonant pair, so the Rabi frequency of the driven transition is $\Omega = qE|x_{12}|/\hbar = 0.180\,qEL/\hbar$. One number from the array runs the whole absorption story.

Step 4. Audit the truncation. The neglected couplings out of the block: $x_{14} = -0.014$ (13× smaller — safe), but $x_{23} = -48/25\pi^2 = -0.194$ — larger than $x_{12}$ itself! Why the truncation is still legal: state 3 is far off-resonance — the energy denominator, not the matrix element, suppresses its participation. Truncations are justified by dynamics plus texture, not texture alone — an honest lesson the two-level atom's users must carry.

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §1.2–1.5 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 1 · Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 3 & Appendix A · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. II · Born & Jordan, Z. Phys. 34, 858 (1925).

Provenance. All equations are standard textbook results. A verifier should: (1) recompute $[\hat{x},\hat{p}]=i\hbar$ via $\hat{p}=-i\hbar\partial_x$ acting on a test function (Griffiths §2.1); (2) verify the reality/orthogonality proof for Hermitian operators (Griffiths §3.3, Sakurai §1.3); (3) confirm unitarity $\Leftrightarrow$ Hermitian generator and the time-evolution operator $e^{-i\hat H t/\hbar}$ (Sakurai §2.1); (4) check the matrix-composition and trace identities. For this edition, items (1) and (4) — the strata this article owns — were machine-verified (sympy/numpy/scipy): linearity of $\hat x$ and $\partial_x$ on explicit combinations, antilinearity of conjugation, $[x, \partial_x]f = -f$ and $[\hat x,\hat p]f = i\hbar f$ on test functions, the full ISW $\hat x$-matrix (diagonal $\tfrac12$; checkerboard zeros; $-8mn/\pi^2(m^2{-}n^2)^2$ law; $2{\times}2$ truncation eigenvalues $0.320/0.680$), composition associativity, $(AB)^{-1} = B^{-1}A^{-1}$, cyclic trace, $e^{\sigma_x}e^{\sigma_z} \ne e^{\sigma_x+\sigma_z}$ with the commuting control case, and unboundedness $\|\hat p\psi_n\| = n\pi$. Items (2) and (3) are executed by their owners per the chapter map — Hermitian Operators and Observables (next) and Unitary Operators and Transformations — as is the adjoint's definition (Hermitian Conjugate and Adjoint Operators) and the commutation relation's postulate status (Canonical Commutation Relation).

Builds on: Hilbert Space and State Vectors · Dirac Bra-Ket Notation · Inner Product and Norm of States

Leads to: Hermitian Operators and Observables · Eigenvalues and Eigenstates · Completeness Relation and Resolution of Identity · Projection Operators · The Postulates of Quantum Mechanics