Observables, Hermitian Operators, and the Spectral Theorem

A state is a vector; a measurable quantity is an operator that acts on vectors. The previous lesson built the space of states. This lesson gives observables their algebraic identity. The central claim of quantum mechanics is narrow and consequential: every observable is represented by a Hermitian operator, and the outcomes of measuring it are its eigenvalues. Everything about spectra, orthogonality of outcomes, and compatibility of measurements follows from that one requirement.

Linear operators

An operator $\hat{A}$ maps vectors to vectors, $\hat{A} ∣ α ⟩ = ∣ α^{'} ⟩$ , and is linear when $\hat{A} (c ∣ α ⟩ + d ∣ β ⟩) = c \hat{A} ∣ α ⟩ + d \hat{A} ∣ β ⟩$ . Linearity is what preserves superpositions, so every operator in quantum mechanics is linear (with the single exception of the antilinear time-reversal operator, treated separately). Resolving the identity on both sides fixes an operator by its matrix elements in a basis,

\hat{A} = 1 \hat{A} 1 = i, j \sum ∣ e_{i} ⟩ ⟨ e_{i} ∣ \hat{A} ∣ e_{j} ⟩ ⟨ e_{j} ∣ = i, j \sum A_{ij} ∣ e_{i} ⟩ ⟨ e_{j} ∣, A_{ij} = ⟨ e_{i} ∣ \hat{A} ∣ e_{j} ⟩ .

Operators compose by matrix multiplication, $(\hat{A} \hat{B})_{ik} = \sum_{j} A_{ij} B_{j k}$ , and in general do not commute: $\hat{A} \hat{B} \neq = \hat{B} \hat{A}$ . The obstruction is measured by the commutator $[\hat{A}, \hat{B}] = \hat{A} \hat{B} - \hat{B} \hat{A}$ , which controls compatibility and uncertainty in the lessons that follow.

The adjoint

Every operator has a partner defined through the inner product. The adjoint $\hat{A}^{†}$ is the operator for which

⟨ α ∣ \hat{A} β ⟩ = ⟨ \hat{A}^{†} α ∣ β ⟩ for all ∣ α ⟩, ∣ β ⟩ .

Moving an operator across the inner-product bar turns it into its adjoint. In components the adjoint is the conjugate transpose, $(A^{†})_{ij} = A_{j i}^{*}$ , because the antilinear first slot conjugates the matrix element and the swap of slots transposes the indices. Three properties follow directly from the definition:

Involution. $(\hat{A}^{†})^{†} = \hat{A}$ .
Reversal. $(\hat{A} \hat{B})^{†} = \hat{B}^{†} \hat{A}^{†}$ , the order reversing exactly as for matrix transpose.
Antilinearity in scalars. $(c \hat{A})^{†} = c^{*} \hat{A}^{†}$ .

The bra corresponding to $\hat{A} ∣ α ⟩$ is $⟨ α ∣ \hat{A}^{†}$ ; this is the rule for taking the Hermitian conjugate of any Dirac expression — reverse the order of every factor, swap kets with bras, and conjugate the scalars.

The trace

A basis-independent number attached to any operator is its trace, the sum of diagonal matrix elements,

Tr \hat{A} = i \sum ⟨ e_{i} ∣ \hat{A} ∣ e_{i} ⟩ .

The value does not depend on the basis: inserting the identity shows $Tr (\hat{U}^{†} \hat{A} \hat{U}) = Tr \hat{A}$ for any unitary $\hat{U}$ , so the trace is invariant under change of basis. It is linear and cyclic, $Tr (\hat{A} \hat{B}) = Tr (\hat{B} \hat{A})$ , from which $Tr [\hat{A}, \hat{B}] = 0$ for any operators on a finite-dimensional space — a fact that already forbids the canonical commutator $[\overset{x}{^}, \overset{p}{^}] = i ℏ$ from being realized by finite matrices, since $Tr (i ℏ 1) = i ℏ dim H \neq = 0$ . The trace of a projector counts the dimension of its range, $Tr \hat{P}_{n} = g_{n}$ , and it is the tool that computes expectation values of mixed states through the density operator, $⟨ A ⟩ = Tr (\overset{ρ}{^} \hat{A})$ .

Hermitian operators

An operator equal to its own adjoint is the object that represents a physical quantity.

In infinite dimensions Hermiticity requires care with the domain of the operator and its boundary behavior; the physically correct notion is self-adjointness, which fixes the boundary conditions that quantize a spectrum. For the finite and well-behaved cases here the two coincide, and the distinction is flagged where it matters. Three consequences make Hermitian operators the right representatives of observables.

Real eigenvalues are non-negotiable for an observable: a measurement returns a real number. This is the first reason observables must be Hermitian.

Distinct measurement outcomes correspond to orthogonal states. This is what lets a measurement discriminate outcomes cleanly: the states associated with different readings do not overlap.

A Hermitian operator has real eigenvalues (marked on the real line) and mutually orthogonal eigenvectors; distinct outcomes correspond to perpendicular state directions.

The spectral theorem

Real eigenvalues and orthogonality are half the story. The completing fact is that the eigenvectors span the whole space, so any state expands in them.

The decomposition says a Hermitian operator is diagonal in its own eigenbasis, with the eigenvalues down the diagonal. Any function of the operator is then computed eigenvalue by eigenvalue: $f (\hat{A}) = \sum_{n} f (a_{n}) \hat{P}_{n}$ , since the projectors are orthogonal and idempotent. The exponential $e^{- i \hat{H} t /ℏ}$ that governs time evolution comes from applying this construction to the Hamiltonian.

The construction $f (\hat{A}) = \sum_{n} f (a_{n}) \hat{P}_{n}$ defines the functional calculus: any function of a real variable becomes a function of a Hermitian operator by acting on eigenvalues. It is well defined because the projectors are mutually orthogonal, so cross terms vanish and $\hat{A}^{k} = \sum_{n} a_{n}^{k} \hat{P}_{n}$ reproduces the power series of $f$ term by term.

For Hermitian operators on infinite-dimensional Hilbert space the theorem still holds with two amendments: the spectrum may include a continuous part (position and momentum have no normalizable eigenvectors), and the sum becomes a sum plus an integral over the continuous spectrum. The projector formalism carries over with projectors onto eigenspaces replaced by a projection-valued measure; the working version used throughout is that a Hermitian operator's eigenstates form a complete basis, discrete or continuous, in which any state expands.

Spectral decomposition: a Hermitian operator resolves into a weighted sum of orthogonal projectors, one per eigenvalue, each scaling its eigenspace by the corresponding real eigenvalue.

Degeneracy

When one eigenvalue is shared by several independent eigenvectors it is degenerate, and its eigenspace has dimension greater than one. The spectral theorem still applies: within a degenerate eigenspace any orthonormal set is an equally valid eigenbasis, because every vector in the eigenspace has the same eigenvalue. The projector $\hat{P}_{n}$ then projects onto the full eigenspace,

\hat{P}_{n} = k = 1 \sum g_{n} ∣ a_{n}, k ⟩ ⟨ a_{n}, k ∣,

where $g_{n}$ is the degeneracy and $k$ labels an orthonormal basis of the eigenspace. Degeneracy is not a nuisance to be removed but a signature of symmetry: a symmetry that commutes with $\hat{A}$ maps one eigenvector to another of the same eigenvalue, forcing the degeneracy. That connection is developed in the lesson on symmetries and conservation laws. Inside a degenerate subspace a second observable is often used to relabel the states, which is the subject of compatible observables below.

Unitary operators and change of basis

Alongside Hermitian operators sit the operators that preserve the inner product.

Unitaries are the rotations of Hilbert space. They carry orthonormal bases to orthonormal bases, so a change from one orthonormal basis ${∣ e_{i} ⟩}$ to another ${∣ f_{i} ⟩}$ is implemented by the unitary $\hat{U} = \sum_{i} ∣ f_{i} ⟩ ⟨ e_{i} ∣$ . Components and matrix elements transform by conjugation,

c_{i}^{'} = j \sum U_{ij} c_{j}, A^{'} = \hat{U} A \hat{U}^{†},

which leaves eigenvalues, traces, and determinants invariant. A Hermitian operator is diagonalized precisely by the unitary whose columns are its eigenvectors: $\hat{U}^{†} \hat{A} \hat{U}$ is the diagonal matrix of eigenvalues. Physical symmetries and time evolution are unitary, because both must preserve total probability; the connection between Hermitian generators and unitary transformations is $\hat{U} = e^{- i \hat{G} θ}$ with $\hat{G}$ Hermitian, developed in the time-evolution lesson.

A unitary carries one orthonormal frame rigidly onto another, rotating every vector while preserving all lengths and angles; diagonalizing a Hermitian operator is choosing the frame in which it is diagonal.

Compatible observables and the commutator

Two observables can be measured together without disturbance exactly when they share a complete eigenbasis, and that condition is algebraic.

Commuting observables are compatible: a state can have definite values of both at once, and measuring one does not scramble the other. When $[\hat{A}, \hat{B}] \neq = 0$ the observables are incompatible and no basis diagonalizes both, so sharp values cannot coexist — the seed of the generalized uncertainty principle. A maximal set of mutually commuting observables whose joint eigenvalues label each basis vector uniquely is a complete set of commuting observables (CSCO); the standard ${\hat{H}, \hat{L}^{2}, \hat{L}_{z}}$ for the hydrogen atom is the archetype.

Commuting observables share one eigenbasis, so a single frame diagonalizes both; incompatible observables have eigenbases rotated relative to each other, and no frame diagonalizes both at once.

The eigenbases of Hermitian operators are the frames in which measurement outcomes are read. The next lesson turns that geometry into physical predictions through the postulates and the Born rule.