The Spectral Theorem and Unbounded Operators

The postulates say observables are "self-adjoint operators" and measurement outcomes are their "eigenvalues." For finite systems (spin) this is elementary linear algebra. But the central observables of wave mechanics — $\hat{x}$, $\hat{p}$, $\hat{H}$ — are unbounded, defined only on dense subspaces, and $\hat{x},\hat{p}$ have no eigenvectors in $\mathcal{H}$ at all. Making the postulates rigorous requires the spectral theorem for unbounded self-adjoint operators and the rigged-Hilbert-space framework that houses the "eigenkets" $|x⟩,|p⟩$. This page collects the functional analysis that the physics pages use formally.

1. Why Naive Eigenstates Fail

The position "eigenstate" $|x_0⟩$ would satisfy $\hat{x}|x_0⟩=x_0|x_0⟩$, but $⟨x|x_0⟩=\delta (x-x_0)$ is a distribution, not a square- integrable function — it is not in $\mathcal{H}=L^2$. Likewise the momentum eigenstate $e^{ipx/\hslash }$ is not normalizable. The spectrum of $\hat{x}$ (all of $\mathbb{R}$) is continuous: there are no genuine eigenvectors, yet the operator is a perfectly good observable. The resolution is to enlarge the setting.

2. Symmetric vs. Self-Adjoint

For an unbounded operator $\hat{A}$ with dense domain $D(\hat{A})$, one must distinguish:

• Symmetric (Hermitian): $⟨\varphi |\hat{A}\psi ⟩=⟨\hat{A}\varphi |\psi ⟩$ for all $\varphi ,\psi \in D(\hat{A})$.
• Self-adjoint: symmetric and $D({\hat{A}}^{†})=D(\hat{A})$ — the operator and its adjoint have the same domain.

Only self-adjoint operators generate unitary evolution (Stone's theorem, §5) and have a real spectral decomposition; merely symmetric operators may not. The gap is measured by the deficiency indices $n_{\pm }=\dim \ker ({\hat{A}}^{†}\mp i)$: the operator is self-adjoint iff $n_{+}=n_{-}=0$, and admits self-adjoint extensions (a family of them) iff $n_{+}=n_{-}\ne 0$. Physically, the deficiency indices count the boundary conditions needed to make an observable well-defined.

Example — momentum on a half-line/interval. $\hat{p}=-i\hslash d/dx$ on $[0,\infty )$ has $n_{+}\ne n_{-}$ and is not self-adjoint — there is no momentum observable for a particle on a half-line. On $[0,L]$ it has a one-parameter family of self-adjoint extensions, the twisted periodic boundary conditions $\psi (L)=e^{i\theta }\psi (0)$. Choosing boundary conditions is choosing the physics.

3. The Spectral Theorem

Every self-adjoint operator $\hat{A}$ admits a projection-valued measure $\{{\hat{E}}_{\lambda }\}$ (a resolution of the identity) such that

$$\hat{A}={\int }_{\sigma (\hat{A})}\lambda d{\hat{E}}_{\lambda },\hat{1}=\int d{\hat{E}}_{\lambda }.$$

The spectrum $\sigma (\hat{A})$ splits into a discrete (point) part — genuine eigenvalues with $L^2$ eigenvectors, as for bound states — and a continuous part — as for $\hat{x}$, $\hat{p}$, and scattering states. This is the precise version of the physicists' "$\hat{A}={\sum }_n{a}_n{\hat{P}}_n$" from preliminaries: the sum becomes an integral over the continuous spectrum. Functions of the operator are defined by $f(\hat{A})=\int f(\lambda )d{\hat{E}}_{\lambda }$, giving rigorous meaning to $e^{-i\hat{H}t/\hslash }$ and the Born rule probability $⟨\psi |{\hat{E}}_{\Delta }|\psi ⟩$ of finding a value in a set $\Delta$.

4. Rigged Hilbert Space (Gel'fand Triple)

To keep Dirac's convenient eigenket notation, embed $\mathcal{H}$ in a Gel'fand triple

$$\Phi \subset \mathcal{H}\subset {\Phi }^{\times },$$

where $\Phi$ is a dense space of "nice" (rapidly decreasing, smooth) test functions and ${\Phi }^{\times }$ its dual of distributions. The generalized eigenkets $|x⟩,|p⟩$ live in ${\Phi }^{\times }$: they act as functionals on $\Phi$, $⟨x|\psi ⟩=\psi (x)$, and satisfy the completeness $\hat{1}=\int |x⟩⟨x|dx$ used throughout wave mechanics. This is the rigorous home of the formal manipulations physicists perform with continuous bases — the rigged Hilbert space (Gel'fand–Maurin) formulation of QM.

5. Stone's Theorem: Dynamics Needs Self-Adjointness

Stone's theorem establishes the one-to-one correspondence

$$\text{self-adjoint }\hat{H}⟺\text{strongly continuous one-parameter unitary group }\hat{U}(t)=e^{-i\hat{H}t/\hslash }.$$

This is why Postulate 5 demands the Hamiltonian be self-adjoint (not merely symmetric): only then does unitary, probability-conserving evolution exist for all time. A Hamiltonian that is symmetric but not self-adjoint (e.g. an ill-posed boundary problem, or a potential that lets a particle reach infinity in finite time) fails to generate deterministic evolution until a self-adjoint extension is chosen. Self-adjointness is thus not a technicality but the precise statement of quantum determinism.

See also

• Wave Mechanics — where $|x⟩,|p⟩$ and the continuous bases are used.
• Preliminaries — the discrete/finite-dimensional spectral theorem.
• Postulates — observables and unitary evolution made rigorous here.
• math/analysis — functional analysis and spectral theory.