Operators, observables, and Hermiticity

Intuition [Beginner]

In unit 12.02.01 you learned that a quantum state lives in a Hilbert space — a vector space of kets with an inner product. States are vectors. But quantum mechanics also needs to describe measurements: position, momentum, energy, spin, angular momentum. Each of these measurable quantities corresponds to an operator — a rule that transforms one ket into another ket.

Think of an operator as a machine. You feed it a ket $|\psi ⟩$, and it produces a new ket. The simplest operator is the identity: feed in $|\psi ⟩$, get $|\psi ⟩$ back. More interesting operators stretch, rotate, or flip the ket.

Not every operator corresponds to a physical measurement. The ones that do have a special property called Hermiticity. A Hermitian operator is one that equals its own conjugate transpose — in bra-ket language, $⟨\varphi |(A|\psi ⟩)=⟨(A\varphi )|\psi ⟩$ for all kets $|\varphi ⟩$ and $|\psi ⟩$. The operator and its bra-side partner act identically, just with the bra and ket swapped.

Why does Hermiticity matter? Two reasons, and they are the structural backbone of quantum mechanics.

First, Hermitian operators have real eigenvalues. An eigenvalue is a number $a$ such that $A|\psi ⟩=a|\psi ⟩$ — the operator just scales the ket without changing its direction. When you measure an observable in the lab, the number that appears on the dial is always real. You never measure a complex-valued energy. The Hermiticity condition is what guarantees this: the eigenvalues come out real, matching the real numbers that instruments display.

Second, the eigenvectors of a Hermitian operator form an orthonormal basis. This means every state in the Hilbert space can be written as a combination of the operator's eigenstates. The measurement postulate says: when you measure observable $A$ in state $|\psi ⟩$, the possible outcomes are the eigenvalues $a_1,a_2,\dots$ and the probability of each outcome is the squared overlap of $|\psi ⟩$ with the corresponding eigenstate. This decomposition would not work without the eigenbasis property.

Consider the Stern-Gerlach experiment from unit 12.01.02. The spin operator $S_z$ measures spin along the $z$-axis. It has two eigenvalues: $+\hslash /2$ ("spin up") and $-\hslash /2$ ("spin down"). Its eigenstates are $|\uparrow ⟩$ and $|\downarrow ⟩$. A general spin state $\alpha |\uparrow ⟩+\beta |\downarrow ⟩$ is a superposition of these two eigenstates. The probabilities $|\alpha {|}^2$ and $|\beta {|}^2$ are the Born-rule numbers. All of this rests on $S_z$ being Hermitian.

There is a mathematical operation that combines two operators into a third: the commutator $[A,B]=AB-BA$. If two operators commute ($AB=BA$), they share a common set of eigenstates. This means you can simultaneously know both quantities with certainty. If they do not commute, measuring one disturbs the other — this is the algebraic origin of the uncertainty principle, which we develop in unit 12.04.01.

Not all physically important operators are Hermitian. The time-evolution operator $U(t)=e^{-iHt/\hslash }$ is unitary — it preserves the inner product — rather than Hermitian. Unitary operators describe transformations of the system (rotations, translations, time evolution), while Hermitian operators describe measurements. The two classes are linked: every Hermitian operator generates a unitary one-parameter family via the exponential map, and every continuous one-parameter unitary family has a Hermitian generator. This is Stone's theorem, and it is why the Hamiltonian (a Hermitian operator) drives the Schrödinger equation (a unitary time evolution).

The operator formalism replaces the classical picture in which observables are ordinary functions on phase space. In classical mechanics, energy is a function $H(q,p)$; in quantum mechanics, energy is an operator $\hat{H}$ acting on a Hilbert space. The transition from classical to quantum is, at its algebraic core, the replacement of functions by operators and Poisson brackets by commutators — the theme of unit 09.04.02 carried into the quantum domain.

Operators can be composed, added, and multiplied by scalars, exactly like linear maps on a vector space. They form an algebra — not a commutative one (in general $AB\ne BA$), and this non-commutativity is not a defect but the mathematical signature of quantum interference. The entire structure of quantum mechanics — superposition, measurement, uncertainty, entanglement — is encoded in the algebra of non-commuting operators on Hilbert space.

Visual [Beginner]

Picture the Pauli matrix ${\sigma }_z$ acting on the 2D spin space. The basis kets $|\uparrow ⟩$ and $|\downarrow ⟩$ sit along perpendicular axes. The operator ${\sigma }_z$ leaves $|\uparrow ⟩$ unchanged (eigenvalue $+1$) and flips $|\downarrow ⟩$ to point backward (eigenvalue $-1$). An arbitrary state vector projects onto both axes. The squared lengths of the two projections are the measurement probabilities.

Worked example [Beginner]

The three Pauli matrices are the simplest interesting Hermitian operators. They act on the two-dimensional spin space from unit 12.01.02:

$${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

Take ${\sigma }_z$ first. To verify it is Hermitian, take its conjugate transpose: swap rows with columns and complex-conjugate every entry. Since all entries are real and the off-diagonal entries are zero, the conjugate transpose gives back the same matrix. So ${\sigma }_z$ is Hermitian.

The eigenvalues are read off the diagonal: ${\sigma }_z$ acting on $\left(\begin{array}{c}1\\ 0\end{array}\right)$ gives $+1\cdot \left(\begin{array}{c}1\\ 0\end{array}\right)$, and acting on $\left(\begin{array}{c}0\\ 1\end{array}\right)$ gives $-1\cdot \left(\begin{array}{c}0\\ 1\end{array}\right)$. The eigenvalues are $+1$ and $-1$, both real — as Hermiticity guarantees.

Now take ${\sigma }_x$. Conjugate transpose: the matrix is real and symmetric, so ${\sigma }_x^{†}={\sigma }_x$. Hermitian. The eigenvalues come from the characteristic equation ${\lambda }^2-1=0$, giving $\lambda =\pm 1$ again. The eigenvectors are $\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)$ for eigenvalue $+1$ and $\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -1\end{array}\right)$ for eigenvalue $-1$. These are spin-right and spin-left states — measuring spin along $x$ instead of $z$.

For ${\sigma }_y$, the conjugate transpose swaps $i$ to $-i$ and $-i$ to $i$, recovering the original matrix. Hermitian again. Eigenvalues $\pm 1$ with eigenvectors $\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)$ and $\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -i\end{array}\right)$.

The spectral decomposition of ${\sigma }_z$ in Dirac notation is ${\sigma }_z=(+1)|\uparrow ⟩⟨\uparrow |+(-1)|\downarrow ⟩⟨\downarrow |$. For a spin state $|\psi ⟩=\frac{1}{\sqrt{3}}|\uparrow ⟩+\sqrt{\frac{2}{3}}|\downarrow ⟩$, the expectation value is $⟨\psi |{\sigma }_z|\psi ⟩=\frac{1}{3}(+1)+\frac{2}{3}(-1)=-\frac{1}{3}$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

An operator on a Hilbert space $\mathcal{H}$ is a linear map $A:\mathcal{H}\to \mathcal{H}$. In Dirac notation, $A$ sends $|\psi ⟩\mapsto A|\psi ⟩$. Linearity means $A(\alpha |\varphi ⟩+\beta |\psi ⟩)=\alpha A|\varphi ⟩+\beta A|\psi ⟩$ for all complex $\alpha ,\beta$ and all kets $|\varphi ⟩,|\psi ⟩$.

The adjoint (conjugate transpose) of $A$, written $A^{†}$, is defined by the relation

$$⟨\varphi |A\psi ⟩=⟨A^{†}\varphi |\psi ⟩\text{for all }|\varphi ⟩,|\psi ⟩\in \mathcal{H}.$$

In matrix language: take the transpose and complex-conjugate every entry. An operator is self-adjoint (or Hermitian, in finite dimensions) if $A=A^{†}$.

Observables as self-adjoint operators. The first postulate of the operator formalism states: every measurable physical quantity (observable) is represented by a self-adjoint operator on $\mathcal{H}$. The reason is the spectral theorem: a self-adjoint operator has real eigenvalues (the measurement outcomes) and an orthonormal eigenbasis (the states of definite value), and decomposes as

$$A=\sum _n{a}_n{P}_n,$$

where $a_n$ are the distinct eigenvalues and $P_n=|n⟩⟨n|$ are the orthogonal projectors onto the corresponding eigenspaces.

The measurement postulate in operator language. If the system is in state $|\psi ⟩$ and observable $A$ is measured, the probability of obtaining outcome $a_n$ is

$$\mathrm{Prob}(a_n)=⟨\psi |P_n|\psi ⟩=|⟨n|\psi ⟩{|}^2,$$

and the post-measurement state is $P_n|\psi ⟩/\Vert P_n|\psi ⟩\Vert$. The expectation value of $A$ in state $|\psi ⟩$ is

$$⟨A{⟩}_{\psi }=⟨\psi |A|\psi ⟩=\sum _n{a}_n|⟨n|\psi ⟩{|}^2,$$

which is the Born-rule weighted average of eigenvalues. The variance is $(\Delta A{)}^2=⟨A^2⟩-⟨A{⟩}^2$, and $\Delta A\ge 0$ for any state.

Commutators and compatibility. The commutator of two operators is $[A,B]:=AB-BA$. Two observables $A,B$ are compatible (simultaneously measurable) iff $[A,B]=0$, in which case they share a complete set of simultaneous eigenvectors. Non-commuting observables obey the Robertson uncertainty relation $\Delta A\Delta B\ge \frac{1}{2}|⟨[A,B]⟩|$ (unit 12.04.01).

The Heisenberg algebra. Position and momentum operators satisfy the canonical commutation relation $[\hat{x},\hat{p}]=i\hslash 1$. This single relation is the algebraic distillation of wave-particle duality: it forces $\hat{x}$ and $\hat{p}$ to be incompatible observables with a lower bound on their joint uncertainty.

Counterexamples to common slips

• Self-adjoint $\ne$ symmetric in infinite dimensions. A symmetric operator satisfies $⟨\varphi |A\psi ⟩=⟨A\varphi |\psi ⟩$ on its domain, but the domain of $A$ and the domain of $A^{†}$ may differ. Self-adjointness requires the domains to coincide. The operator $-id/dx$ on $L^2([0,1])$ with domain $C_0^{\infty }(0,1)$ is symmetric but not self-adjoint — its deficiency indices are $(1,1)$, and it admits a one-parameter family of self-adjoint extensions parametrised by a boundary phase.

• Not all operators are Hermitian. The time-evolution operator $U(t)=e^{-iHt/\hslash }$ is unitary ($U^{†}U=1$), not Hermitian. Projectors $P_n$ are Hermitian. Density operators $\rho$ are Hermitian. The distinction matters: observables are Hermitian, symmetry transformations are unitary (or anti-unitary for time reversal).

• The matrix representation is basis-dependent. The Pauli matrix ${\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ in the $\{|\uparrow ⟩,|\downarrow ⟩\}$ basis. In the eigenbasis of ${\sigma }_x$, the same abstract operator ${\sigma }_z$ has a different matrix with off-diagonal entries. The operator is an abstract map; the matrix is its coordinate representation.

Key theorem with proof [Intermediate+]

Theorem (Spectral theorem for finite-dimensional self-adjoint operators). Let $A$ be a self-adjoint operator on a finite-dimensional Hilbert space $\mathcal{H}$. Then:

(i) Every eigenvalue of $A$ is real.

(ii) Eigenvectors corresponding to distinct eigenvalues are orthogonal.

(iii) $\mathcal{H}$ admits an orthonormal basis consisting of eigenvectors of $A$.

(iv) $A$ admits the spectral decomposition $A={\sum }_n{a}_n|n⟩⟨n|$, where $\{|n⟩\}$ is the orthonormal eigenbasis and $a_n=⟨n|A|n⟩$.

This decomposition is unique when all eigenvalues are non-degenerate; for degenerate eigenvalues the basis within each eigenspace is unique up to unitary rotation within that subspace.

Proof.

Part (i): eigenvalues are real. Let $A|v⟩=a|v⟩$ with $|v⟩\ne 0$. Then $⟨v|A|v⟩=a⟨v|v⟩$. Self-adjointness gives $⟨v|A|v⟩=⟨Av|v⟩=\overline{a}⟨v|v⟩$. Since $⟨v|v⟩>0$, we conclude $a=\overline{a}$, so $a\in \mathbb{R}$.

Part (ii): orthogonality of distinct eigenspaces. Let $A|v⟩=a|v⟩$ and $A|w⟩=b|w⟩$ with $a\ne b$. Compute $⟨w|A|v⟩$ two ways: directly, $⟨w|A|v⟩=a⟨w|v⟩$; via self-adjointness, $⟨w|A|v⟩=⟨Aw|v⟩=b⟨w|v⟩$ (using $b\in \mathbb{R}$ from part (i)). Subtracting: $(a-b)⟨w|v⟩=0$. Since $a\ne b$, we get $⟨w|v⟩=0$.

Part (iii): existence of an orthonormal eigenbasis. By induction on $n=\dim \mathcal{H}$.

Base case $n=1$: the single vector is an eigenvector of every operator.

Inductive step: the characteristic polynomial of $A$ has at least one root $a_1\in \mathbb{C}$ (fundamental theorem of algebra), and by part (i) $a_1\in \mathbb{R}$. Let $|1⟩$ be a corresponding unit eigenvector. Define $V_1^{\perp }=\{|\psi ⟩\in \mathcal{H}:⟨1|\psi ⟩=0\}$, a subspace of dimension $n-1$.

Self-adjointness of $A$ preserves $V_1^{\perp }$: if $⟨1|\psi ⟩=0$, then $⟨1|A\psi ⟩=⟨A1|\psi ⟩=a_1⟨1|\psi ⟩=0$. So $A$ restricts to a self-adjoint operator on $V_1^{\perp }$. By the inductive hypothesis, $V_1^{\perp }$ has an orthonormal eigenbasis $\{|2⟩,\dots ,|n⟩\}$ for $A$. Together with $|1⟩$, these form an orthonormal eigenbasis for $\mathcal{H}$.

Part (iv): spectral decomposition. Define the operator $B:={\sum }_{k=1}^n{a}_k|k⟩⟨k|$. For each basis vector $|j⟩$:

$$B|j⟩=\sum _{k=1}^n{a}_k|k⟩⟨k|j⟩=\sum _{k=1}^n{a}_k|k⟩{\delta }_{kj}=a_j|j⟩=A|j⟩.$$

Two linear operators that agree on a basis are equal, so $A=B$. $\square$

Corollary (functional calculus). For any function $f:\mathbb{R}\to \mathbb{C}$, the operator $f(A)$ is defined by $f(A)={\sum }_{n}f(a_n)|n⟩⟨n|$. In particular, $e^{iAt}$ is unitary for real $t$, and $A^k={\sum }_n{a}_n^k|n⟩⟨n|$.

Corollary (projection-valued measure). The spectral projectors $P_n=|n⟩⟨n|$ satisfy $P_n{P}_m={\delta }_{nm}{P}_n$, ${\sum }_n{P}_n=1$, and $A={\sum }_n{a}_n{P}_n$. This is the finite-dimensional instance of the projection-valued-measure decomposition that extends to unbounded operators in the Master tier.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib formalises bounded linear operators (LinearMap, ContinuousLinearMap), the adjoint (LinearMap.adjoint), and self-adjoint operators (LinearMap.IsSelfAdjoint). The finite-dimensional spectral theorem for self-adjoint operators is available via Matrix.IsHermitian.eigenvalues and the associated decomposition machinery.

What Mathlib does not contain is a physics-layer formalisation of the quantum measurement postulate. Specifically, there is no definition of "quantum observable" as a bundled structure combining a self-adjoint operator, its spectral projectors, and the Born-rule probability distribution. There is no Dirac-notation layer mapping kets to vectors and bras to linear functionals with the physicist's linearity convention (linear in the second argument). There is no formalisation of the expectation value, variance, or Robertson uncertainty relation in the quantum-mechanical context. Building these would require a physics layer on top of the existing spectral-theory infrastructure — a straightforward but substantial formalisation project.

Unbounded self-adjoint operators and the spectral theorem [Master]

Physical observables like position $\hat{x}$ and momentum $\hat{p}=-i\hslash d/dx$ are unbounded operators: they cannot be defined on all of $L^2(\mathbb{R})$, and the finite-dimensional spectral theorem does not apply. The correct framework is the spectral theorem for (possibly unbounded) self-adjoint operators on separable Hilbert spaces, due to von Neumann (1932) and refined by Stone, Riesz, and Nagy.

Domains and self-adjointness

An unbounded operator $A$ on $\mathcal{H}$ is specified by a domain $\mathcal{D}(A)\subseteq \mathcal{H}$ (a dense subspace) and a linear map $A:\mathcal{D}(A)\to \mathcal{H}$. The adjoint $A^{†}$ has domain $\mathcal{D}(A^{†})=\{|\varphi ⟩\in \mathcal{H}:\exists |\chi ⟩\text{ with }⟨\varphi |A\psi ⟩=⟨\chi |\psi ⟩\forall |\psi ⟩\in \mathcal{D}(A)\}$, and $A^{†}|\varphi ⟩:=|\chi ⟩$.

$A$ is self-adjoint if $\mathcal{D}(A)=\mathcal{D}(A^{†})$ and $A|\varphi ⟩=A^{†}|\varphi ⟩$ for all $|\varphi ⟩\in \mathcal{D}(A)$. This is strictly stronger than symmetry ($⟨\varphi |A\psi ⟩=⟨A\varphi |\psi ⟩$ for all $|\varphi ⟩,|\psi ⟩\in \mathcal{D}(A)$). A symmetric operator that is not self-adjoint can have complex eigenvalues, non-orthogonal eigenfunctions, or no spectral decomposition at all.

The classic example: the momentum operator $\hat{p}=-id/dx$ on $L^2([0,1])$. With domain $C_0^{\infty }(0,1)$ (smooth functions vanishing at endpoints), $\hat{p}$ is symmetric but not self-adjoint — the domain of ${\hat{p}}^{†}$ includes functions with nonzero boundary values. The deficiency indices are $(1,1)$, and a one-parameter family of self-adjoint extensions exists, parametrised by a phase $e^{i\theta }$ relating the boundary values $\psi (1)=e^{i\theta }\psi (0)$. Each extension is a different physical system (a particle on a ring with different flux threading).

The spectral theorem (unbounded case)

Theorem (von Neumann). Let $A$ be a self-adjoint operator on a separable Hilbert space $\mathcal{H}$. Then there exists a unique projection-valued measure $E_A$ on the Borel $\sigma$-algebra of $\mathbb{R}$ such that

$$A={\int }_{\mathbb{R}}\lambda d{E}_A(\lambda ),$$

with domain $\mathcal{D}(A)=\{|\psi ⟩\in \mathcal{H}:{\int }_{\mathbb{R}}{\lambda }^{2}d⟨\psi |E_A(\lambda )|\psi ⟩<\infty \}$.

The measure ${\mu }_{\psi }(\Omega ):=⟨\psi |E_A(\Omega )|\psi ⟩$ is a probability measure when $\Vert |\psi ⟩\Vert =1$ — the Born measure for observable $A$ in state $|\psi ⟩$. The probability that a measurement of $A$ yields a value in the Borel set $\Omega$ is ${\mu }_{\psi }(\Omega )$.

The spectral theorem replaces the discrete sum $A={\sum }_n{a}_n|n⟩⟨n|$ with the projection-valued integral. For operators with pure point spectrum, the integral reduces to a sum. For operators with continuous spectrum (position, momentum, free-particle Hamiltonian), the integral is genuinely continuous and there are no normalisable eigenstates in $\mathcal{H}$ — only approximate eigenstates in the rigged Hilbert space of unit 12.02.01.

Stone's theorem

Theorem (Stone, 1932). There is a bijection between self-adjoint operators $H$ on $\mathcal{H}$ and strongly continuous one-parameter unitary groups $\{U(t){\}}_{t\in \mathbb{R}}$ on $\mathcal{H}$, given by $U(t)=e^{-itH}$ (defined via the functional calculus). Conversely, $H|\psi ⟩=i{\lim }_{ϵ\to 0}(U(ϵ)-1)|\psi ⟩/ϵ$ on the domain of differentiability.

Stone's theorem is the mathematical foundation of the Schrödinger equation. The Hamiltonian $H$ generates the time-evolution group $U(t)=e^{-iHt/\hslash }$, and $H$'s self-adjointness guarantees that $U(t)$ is unitary (probabilities preserved). If $H$ were merely symmetric but not self-adjoint, $U(t)$ would not be unitary and probability would not be conserved.

The commutator and the Stone-von Neumann theorem

For unbounded operators, the commutator $[\hat{x},\hat{p}]=i\hslash$ holds on a dense domain (e.g., Schwartz space $\mathcal{S}(\mathbb{R})$). The Stone-von Neumann theorem states that any irreducible representation of the canonical commutation relations on a separable Hilbert space is unitarily equivalent to the standard Schrödinger representation on $L^2(\mathbb{R})$. This justifies the claim that position and momentum are uniquely defined (up to unitary equivalence) by the commutation relation. The theorem fails for infinitely many degrees of freedom (quantum field theory), where inequivalent representations of the CCR proliferate.

Self-adjoint extensions and deficiency indices

Given a symmetric operator $A$, the deficiency subspaces are ${\mathcal{N}}_{\pm }=\ker (A^{†}\mp i)$. Their dimensions $n_{\pm }=\dim {\mathcal{N}}_{\pm }$ are the deficiency indices. By a theorem of von Neumann:

• $A$ has self-adjoint extensions iff $n_{+}=n_{-}$.
• When $n_{+}=n_{-}=n$, the self-adjoint extensions are parametrised by unitary maps $U:{\mathcal{N}}_{+}\to {\mathcal{N}}_{-}$, i.e., by $U(n)$.

For the momentum operator on $[0,1]$ with domain $C_0^{\infty }(0,1)$: $n_{+}=n_{-}=1$, and the self-adjoint extensions are parametrised by $U(1)\cong S^1$ — the boundary phase $e^{i\theta }$. For the hydrogen-atom Hamiltonian: the Coulomb Hamiltonian is essentially self-adjoint on $C_0^{\infty }({\mathbb{R}}^3\setminus \{0\})$ (deficiency indices $(0,0)$), so no extension ambiguity arises.

Advanced results [Master]

Theorem (Spectral mapping). Let $A$ be a self-adjoint operator on $\mathcal{H}$ with spectrum $\sigma (A)$, and let $f:\mathbb{R}\to \mathbb{C}$ be continuous. Then $\sigma (f(A))=\overline{f(\sigma (A))}$, the closure of $\{f(\lambda ):\lambda \in \sigma (A)\}$.

This is the operator-valued analogue of the fact that $f$ maps eigenvalues to eigenvalues. Its physical consequence: if $A$ is an observable with possible values in $\sigma (A)$, then $f(A)$ is an observable with possible values in $\overline{f(\sigma (A))}$. The kinetic-energy operator ${\hat{p}}^2/(2m)$ has spectrum $\{p^2/(2m):p\in \sigma (\hat{p})\}$. The spectral mapping theorem is the functional-calculus backbone that makes the operator-to-observable correspondence consistent under arbitrary coordinate transformations.

Theorem (Polar decomposition). Every closed densely-defined operator $A$ on $\mathcal{H}$ admits a decomposition $A=U|A|$, where $|A|:=\sqrt{A^{†}A}$ is a positive self-adjoint operator with $\mathcal{D}(|A|)=\mathcal{D}(A)$, and $U$ is a partial isometry from $\ker (A{)}^{\perp }$ onto $\mathrm{ran}(A)$ with $\ker (U)=\ker (A)$.

The polar decomposition is the operator analogue of writing a complex number as $z=e^{i\theta }|z|$. In finite dimensions, $U$ is unitary when $A$ is invertible. The modulus $|A|$ is always self-adjoint and positive; its spectral decomposition gives the singular values of $A$, and these are the operator-theoretic generalisation of the singular-value decomposition from linear algebra.

Theorem (Kato-Rellich). Let $H_0$ be a self-adjoint operator on $\mathcal{D}(H_0)\subseteq \mathcal{H}$. Let $V$ be a symmetric operator on $\mathcal{D}(H_0)$ that is $H_0$-bounded with relative bound $a<1$: there exist $a<1$ and $b\ge 0$ such that $\Vert V\psi \Vert \le a\Vert H_0\psi \Vert +b\Vert \psi \Vert$ for all $|\psi ⟩\in \mathcal{D}(H_0)$. Then $H=H_0+V$ is self-adjoint on $\mathcal{D}(H_0)$.

Kato's theorem (Kato 1951, J. Math. Soc. Japan 3) is the workhorse of quantum-mechanical perturbation theory. The Coulomb Hamiltonian $H={\hat{p}}^2/(2m)-e^2/r$ is self-adjoint because the kinetic energy is self-adjoint and the Coulomb potential is relatively bounded with respect to it, with relative bound 0. The same argument extends to Hamiltonians with magnetic fields and spin-orbit couplings, guaranteeing that these physically central operators have real spectra and well-defined time evolution.

Theorem (Complete sets of commuting observables). A collection $\{A_1,\dots ,A_k\}$ of pairwise-commuting self-adjoint operators on $\mathcal{H}$ forms a complete set of commuting observables (CSCO) if every simultaneous eigenspace is one-dimensional. Equivalently, the joint spectrum $\sigma (A_1,\dots ,A_k)$ labels a unique (up to phase) simultaneous eigenstate $|a_1,\dots ,a_k⟩$.

The CSCO concept, introduced by Dirac (1930), resolves the ambiguity that arises when an observable has degenerate eigenvalues. The hydrogen-atom Hamiltonian alone has degenerate energy levels; adding ${\hat{L}}^2$ and ${\hat{L}}_z$ forms a CSCO $\{H,{\hat{L}}^2,{\hat{L}}_z\}$ whose joint eigenstates $|n,l,m⟩$ are uniquely labelled by three quantum numbers. The mathematical content is that the CSCO generates a maximal abelian subalgebra of the operator algebra, and its joint spectral measure provides a resolution of the identity.

Theorem (Hellinger-Toeplitz). If $A$ is a symmetric operator defined on all of $\mathcal{H}$ (i.e., $\mathcal{D}(A)=\mathcal{H}$ and $⟨\varphi |A\psi ⟩=⟨A\varphi |\psi ⟩$ for all $|\varphi ⟩,|\psi ⟩$), then $A$ is bounded.

Hellinger-Toeplitz (1927) has an immediate corollary of physical importance: no unbounded observable — position, momentum, or Hamiltonian — can be defined on all of $\mathcal{H}$. This is the functional-analytic reason why domain questions are unavoidable in quantum mechanics. The theorem follows from the uniform boundedness principle: for each fixed $|\varphi ⟩$, the linear functional $|\psi ⟩\mapsto ⟨\varphi |A\psi ⟩=⟨A\varphi |\psi ⟩$ is pointwise bounded, hence norm-bounded, which forces $A$ to be bounded.

Theorem (Rayleigh-Ritz variational principle). Let $H$ be a self-adjoint operator bounded below with discrete spectrum $E_0\le E_1\le \cdots$. Then $E_0={\inf }_{\Vert |\psi ⟩\Vert =1}⟨\psi |H|\psi ⟩$, and more generally $E_n=\inf \{⟨\psi |H|\psi ⟩:\Vert |\psi ⟩\Vert =1,|\psi ⟩\perp {\mathcal{V}}_{n-1}\}$ where ${\mathcal{V}}_{n-1}$ is the span of the first $n$ eigenstates.

The variational principle is the theoretical foundation of computational quantum chemistry: the ground-state energy of any system can be approximated from above by evaluating $⟨\psi |H|\psi ⟩$ on any trial state $|\psi ⟩$. The infimum is attained exactly when $|\psi ⟩$ is an eigenstate of $H$. The principle generalises to the min-max theorem for excited states: $E_n={\inf }_{\dim W=n+1}{\sup }_{|\psi ⟩\in W,\Vert |\psi ⟩\Vert =1}⟨\psi |H|\psi ⟩$, which provides the rigorous basis for variational methods in atomic and molecular physics.

Theorem (Schrödinger uncertainty relation). For any two Hermitian operators $A,B$ and any state $|\psi ⟩$ with $\Vert |\psi ⟩\Vert =1$:

$$\Delta A\cdot \Delta B\ge \sqrt{\frac{1}{4}|⟨[A,B]{⟩}_{\psi }{|}^2+\frac{1}{4}|⟨\{A-⟨A⟩,B-⟨B⟩\}{⟩}_{\psi }{|}^2},$$

where $\{X,Y\}=XY+YX$ is the anticommutator.

Schrödinger (1930, Sitzungsber. Preuss. Akad. Wiss.) derived this refinement of the Robertson relation by retaining the anticommutator term discarded in Robertson's proof. The additional term is non-negative and vanishes only when the centred observables $A-⟨A⟩$ and $B-⟨B⟩$ anticommute in expectation. The Schrödinger relation is saturated when $(A-⟨A⟩)|\psi ⟩=i\lambda (B-⟨B⟩)|\psi ⟩$ for some real $\lambda$ — the condition defining minimum-uncertainty (coherent) states.

Theorem (Weyl's criterion for the essential spectrum). A number $\lambda \in \mathbb{R}$ belongs to the essential spectrum ${\sigma }_{\mathrm{ess}}(A)$ of a self-adjoint operator $A$ if and only if there exists a sequence $\{|{\psi }_n⟩\}$ of unit vectors in $\mathcal{D}(A)$ with no convergent subsequence, such that $\Vert (A-\lambda 1)|{\psi }_n⟩\Vert \to 0$ as $n\to \infty$.

Weyl's criterion distinguishes isolated eigenvalues of finite multiplicity (the discrete spectrum) from the rest (the essential spectrum, comprising continuous spectrum, accumulation points, and eigenvalues of infinite multiplicity). For the hydrogen atom, the negative energy levels form the discrete spectrum and the non-negative energies form the continuous spectrum. The scattering states live in the essential spectrum. Weyl's criterion underlies Weyl's theorem on the invariance of the essential spectrum under relatively compact perturbations, which is why adding a compact perturbation to a Hamiltonian does not change its continuous spectrum.

Synthesis. The advanced results above are the foundational reason that the operator-observable correspondence is mathematically robust: Kato-Rellich guarantees self-adjointness survives physical perturbations, the spectral mapping theorem ensures that functions of observables are themselves well-defined observables, and the polar decomposition provides the canonical factorisation of every operator into a modulus (self-adjoint) and a phase (partial isometry). Putting these together with Stone's theorem and the spectral theorem from the previous section, the central insight emerges: the entire measurement postulate rests on identifying observables with self-adjoint operators, and this is exactly the structure that the projection-valued measure formalises. The pattern generalises to the algebraic approach, where the bridge is between the concrete Hilbert-space operator formalism developed here and the abstract C*-algebraic framework, which appears again in the quantum-field-theoretic setting of 12.13.01 where inequivalent representations of the canonical commutation relations necessitate an algebraic framework beyond a single Hilbert space.

Full proof set [Master]

Proposition (Robertson uncertainty relation). Let $A,B$ be Hermitian operators on $\mathcal{H}$ and $|\psi ⟩\in \mathcal{H}$ with $\Vert |\psi ⟩\Vert =1$. Define centred operators $\tilde{A}=A-⟨A{⟩}_{\psi }1$ and $\tilde{B}=B-⟨B{⟩}_{\psi }1$. Then $(\Delta A{)}^2(\Delta B{)}^2\ge \frac{1}{4}|⟨\psi |[A,B]|\psi ⟩{|}^2$.

Proof. Define $|\alpha ⟩=\tilde{A}|\psi ⟩$ and $|\beta ⟩=\tilde{B}|\psi ⟩$. Then $(\Delta A{)}^2=⟨\alpha |\alpha ⟩$ and $(\Delta B{)}^2=⟨\beta |\beta ⟩$. By the Cauchy-Schwarz inequality:

$$⟨\alpha |\alpha ⟩⟨\beta |\beta ⟩\ge |⟨\alpha |\beta ⟩{|}^2.$$

Now $⟨\alpha |\beta ⟩=⟨\psi |\tilde{A}\tilde{B}|\psi ⟩$. Decompose $\tilde{A}\tilde{B}=\frac{1}{2}[\tilde{A},\tilde{B}]+\frac{1}{2}\{\tilde{A},\tilde{B}\}$. Since $[\tilde{A},\tilde{B}]=[A,B]$ (the identity terms cancel) and $\{\tilde{A},\tilde{B}\}$ is Hermitian:

$$⟨\alpha |\beta ⟩=\frac{1}{2}⟨[A,B]{⟩}_{\psi }+\frac{1}{2}⟨\{\tilde{A},\tilde{B}\}{⟩}_{\psi }.$$

The commutator $[A,B]$ of Hermitian operators is anti-Hermitian, so $⟨[A,B]{⟩}_{\psi }$ is purely imaginary. The anticommutator $\{\tilde{A},\tilde{B}\}$ is Hermitian, so $⟨\{\tilde{A},\tilde{B}\}{⟩}_{\psi }$ is real. Since a purely imaginary number and a real number have no cross term in the squared modulus:

$$|⟨\alpha |\beta ⟩{|}^2=\frac{1}{4}|⟨[A,B]{⟩}_{\psi }+⟨\{\tilde{A},\tilde{B}\}{⟩}_{\psi }{|}^2\ge \frac{1}{4}|⟨[A,B]{⟩}_{\psi }{|}^2.$$

Combining with Cauchy-Schwarz yields $(\Delta A{)}^2(\Delta B{)}^2\ge \frac{1}{4}|⟨[A,B]{⟩}_{\psi }{|}^2$. $\square$

Proposition (Polar decomposition in finite dimensions). Every invertible matrix $A\in \mathrm{GL}(n,\mathbb{C})$ admits a unique decomposition $A=UP$ where $U$ is unitary and $P$ is positive-definite Hermitian. In particular, $P=\sqrt{A^{†}A}$ and $U=A{P}^{-1}$.

Proof. Since $A$ is invertible, $A^{†}A$ is positive-definite Hermitian: for any $|v⟩\ne 0$, $⟨v|A^{†}A|v⟩=\Vert A|v⟩{\Vert }^2>0$. By the spectral theorem, $A^{†}A={\sum }_k{\lambda }_k|k⟩⟨k|$ with all ${\lambda }_k>0$. Define $P=\sqrt{A^{†}A}={\sum }_k\sqrt{{\lambda }_k}|k⟩⟨k|$, which is positive-definite and invertible.

Set $U=A{P}^{-1}$. Then $U^{†}U=P^{-1}{A}^{†}A{P}^{-1}=P^{-1}{P}^2{P}^{-1}=1$, so $U$ is unitary. For uniqueness, if $A=U_1{P}_1=U_2{P}_2$, then $A^{†}A=P_1^2=P_2^2$, and since the square root of a positive-definite operator is unique, $P_1=P_2$. Then $U_1=A{P}_1^{-1}=A{P}_2^{-1}=U_2$. $\square$

Proposition (Self-adjointness is preserved under unitary conjugation). If $A$ is self-adjoint on $\mathcal{D}(A)$ and $U$ is unitary, then $\tilde{A}=UA{U}^{†}$ is self-adjoint on $\mathcal{D}(\tilde{A})=U\mathcal{D}(A)$.

Proof. For $|\varphi ⟩,|\psi ⟩\in U\mathcal{D}(A)$, write $|\varphi ⟩=U|{\varphi }_0⟩$ and $|\psi ⟩=U|{\psi }_0⟩$ with $|{\varphi }_0⟩,|{\psi }_0⟩\in \mathcal{D}(A)$. Then:

$$⟨\varphi |\tilde{A}\psi ⟩=⟨U{\varphi }_0|UA{U}^{†}U{\psi }_0⟩=⟨{\varphi }_0|A{\psi }_0⟩=⟨A{\varphi }_0|{\psi }_0⟩=⟨UA{U}^{†}U{\varphi }_0|U{\psi }_0⟩=⟨\tilde{A}\varphi |\psi ⟩.$$

So $\tilde{A}$ is symmetric. The domain of ${\tilde{A}}^{†}$ coincides with $U\mathcal{D}(A)$ because $U$ is a bijection of $\mathcal{H}$ mapping $\mathcal{D}(A)$ onto $U\mathcal{D}(A)$. Hence $\tilde{A}$ is self-adjoint. $\square$

Proposition (Functional calculus preserves self-adjointness). Let $A$ be self-adjoint and $f:\mathbb{R}\to \mathbb{R}$ be a real-valued Borel function. Then $f(A)$ is self-adjoint.

Proof. By the spectral theorem, $f(A)={\int }_{\mathbb{R}}f(\lambda )d{E}_A(\lambda )$. For any $|\varphi ⟩,|\psi ⟩$ in the domain of $f(A)$:

$$⟨\varphi |f(A)\psi ⟩={\int }_{\mathbb{R}}f(\lambda )d⟨\varphi |E_A(\lambda )|\psi ⟩={\int }_{\mathbb{R}}\overline{f(\lambda )}d\overline{⟨\psi |E_A(\lambda )|\varphi ⟩}=⟨f(A)\varphi |\psi ⟩,$$

where the last equality uses $f(\lambda )\in \mathbb{R}$ and the self-adjointness of the projection-valued measure ($E_A(\Omega )=E_A(\Omega {)}^{†}$). The domain $\mathcal{D}(f(A))=\{|\psi ⟩:\int |f(\lambda ){|}^{2}d{\mu }_{\psi }<\infty \}$ is symmetric under the interchange of $|\varphi ⟩$ and $|\psi ⟩$ because $|f{|}^2$ is real-valued, so $\mathcal{D}(f(A{)}^{†})=\mathcal{D}(f(A))$. $\square$

Operator algebras and the algebraic approach [Master]

The Hilbert-space formalism of this unit — self-adjoint operators as observables, unit vectors as pure states — suffices for quantum mechanics with a fixed finite number of degrees of freedom. For systems with infinitely many degrees of freedom (quantum fields, thermodynamic limits), a more abstract framework is needed. The algebraic approach replaces the concrete Hilbert space with an abstract C*-algebra of observables.

C*-algebras of observables

A C-algebra* $\mathfrak{A}$ is a complex Banach algebra with an involution $\ast :\mathfrak{A}\to \mathfrak{A}$ satisfying the C*-identity $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$ for all $a\in \mathfrak{A}$. A state on $\mathfrak{A}$ is a positive linear functional $\omega :\mathfrak{A}\to \mathbb{C}$ with $\omega (1)=1$. The observables are the self-adjoint elements $a=a^{\ast }$, and the expectation of observable $a$ in state $\omega$ is $\omega (a)\in \mathbb{R}$.

By the Gelfand-Naimark theorem (Gelfand and Naimark 1943, Mat. Sb. 12), every abstract C*-algebra is isometrically $\ast$-isomorphic to a norm-closed $\ast$-subalgebra of bounded operators on some Hilbert space. By the Gelfand-Naimark-Segal (GNS) construction (Segal 1947, Bull. AMS 53), every state $\omega$ on $\mathfrak{A}$ determines a representation ${\pi }_{\omega }:\mathfrak{A}\to \mathcal{B}({\mathcal{H}}_{\omega })$ and a cyclic vector $|{\Omega }_{\omega }⟩\in {\mathcal{H}}_{\omega }$ such that $\omega (a)=⟨{\Omega }_{\omega }|{\pi }_{\omega }(a)|{\Omega }_{\omega }⟩$. The Hilbert space is recovered from the algebra and the state.

Von Neumann algebras and the double commutant theorem

A von Neumann algebra $\mathfrak{M}$ acting on $\mathcal{H}$ is a $\ast$-subalgebra of $\mathcal{B}(\mathcal{H})$ containing $1$ that is closed in the weak-operator topology. The double commutant theorem (von Neumann 1930, Ann. Math. 31) states that $\mathfrak{M}$ is a von Neumann algebra if and only if ${\mathfrak{M}}^{\prime \prime }=\mathfrak{M}$, where ${\mathfrak{M}}^{\prime }=\{T\in \mathcal{B}(\mathcal{H}):TS=ST\text{ for all }S\in \mathfrak{M}\}$ is the commutant.

Von Neumann algebras capture the observables local to a subsystem. For a composite system ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$, the algebra of $A$-local observables is $\mathcal{B}({\mathcal{H}}_A)\otimes 1_B$, a von Neumann algebra whose commutant is $1_A\otimes \mathcal{B}({\mathcal{H}}_B)$ — the $B$-local observables. Entanglement is characterised algebraically: a pure state $|\psi ⟩$ on ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$ is entangled if and only if the reduced density operator ${\rho }_A={\mathrm{tr}}_B|\psi ⟩⟨\psi |$ has rank greater than 1.

Superselection rules

A superselection rule is a restriction on the superposition principle: certain superpositions are physically forbidden because no observable connects the two sectors. Formally, the algebra of observables $\mathfrak{A}$ is a proper subalgebra of $\mathcal{B}(\mathcal{H})$, and its centre $Z(\mathfrak{A})=\mathfrak{A}\cap {\mathfrak{A}}^{\prime }$ is non-scalar. The Hilbert space decomposes as $\mathcal{H}={⨁}_j{\mathcal{H}}_j$ where each ${\mathcal{H}}_j$ is a superselection sector and $Z(\mathfrak{A})$ acts as a scalar ${\lambda }_{j}1$ on ${\mathcal{H}}_j$. The charge superselection rule (electric charge, baryon number) and the Bose-Fermi superselection rule are the standard examples.

The algebraic approach is essential for quantum field theory, where different representations of the same abstract C*-algebra describe physically distinct phases (the vacuum of a free theory versus a broken-symmetry phase). This structure appears again in 12.13.01 where the Fock-space formalism requires algebraic methods to handle inequivalent representations.

Connections [Master]

Unit 12.02.01 (Hilbert spaces, kets, bras) is the direct prerequisite: the operator formalism acts on the Hilbert-space structure defined there. The resolution of identity and the projector concept introduced in 12.02.01 are the building blocks of the spectral decomposition proved here.

Unit 12.03.01 (Schrödinger equation) takes the Hamiltonian operator $\hat{H}$ — a Hermitian operator on $\mathcal{H}$ — and makes it the generator of time evolution via $i\hslash d|\psi ⟩/dt=\hat{H}|\psi ⟩$. The Hermiticity of $\hat{H}$ proved here is what makes the time-evolution unitary and probability-conserving.

Unit 12.04.01 (uncertainty principle) derives the Robertson relation $\Delta A\Delta B\ge \frac{1}{2}|⟨[A,B]⟩|$ from the Cauchy-Schwarz inequality applied to the commutator. The commutator algebra developed here — especially the anti-Hermiticity of $[A,B]$ for Hermitian $A,B$ — is the algebraic input to that derivation.

Unit 12.05.01 (angular momentum) studies the operators ${\hat{L}}_x,{\hat{L}}_y,{\hat{L}}_z$, which form an $\mathfrak{su}(2)$ Lie algebra under the commutator. The Pauli matrices computed here are the spin-1/2 representation of this algebra.

Unit 14.04.01 (hydrogen atom QM in chemistry) applies the operator formalism to the Coulomb Hamiltonian $H={\hat{p}}^2/(2m)-e^2/r$, a self-adjoint operator on $L^2({\mathbb{R}}^3)$ whose discrete spectrum gives the energy levels. The spectral theorem proved here guarantees that these energy levels are real and the eigenstates orthogonal.

Unit 20.07.01 (measurement problem in philosophy of physics) centres on the projection postulate — the map $|\psi ⟩\to P_n|\psi ⟩/\Vert P_n|\psi ⟩\Vert$ — and its tension with unitary Schrödinger evolution. The spectral decomposition and the projection-valued measure defined here are the mathematical substrate of that debate.

Historical and philosophical context [Master]

The operator formalism was introduced by Dirac in The Principles of Quantum Mechanics (1930, 1st ed.). Dirac's insight was to treat dynamical variables — position, momentum, energy, angular momentum — as abstract operators satisfying algebraic relations (the commutation relations) rather than as numerical functions. The $q$-number algebra (Dirac's term for non-commuting quantities) replaced the $c$-number algebra of classical observables. Dirac's notation — kets, bras, and the bra-ket pairing — made the algebraic structure computationally transparent, though the formal justification of continuous-spectrum manipulations was left to later mathematicians.

von Neumann (1932, Mathematische Grundlagen der Quantenmechanik) placed the theory on rigorous mathematical foundations. He proved the spectral theorem for unbounded self-adjoint operators, established the equivalence between the matrix mechanics of Heisenberg, Born, and Jordan and the wave mechanics of Schrödinger (via unitary equivalence of representations), and introduced the density-operator formalism for mixed states. The spectral theorem is the single most important mathematical result in quantum mechanics: it guarantees that every physical observable has a well-defined set of measurement outcomes (the spectrum) and a well-defined probability distribution for those outcomes (the projection-valued measure).

Stone (1932, On One-Parameter Unitary Groups in Hilbert Space, Ann. Math. 33, 643) proved the correspondence between self-adjoint operators and strongly continuous unitary groups. This result connects the Hermitian operators (observables) to the unitary operators (symmetries and time evolution) via the exponential map — the mathematical statement that "energy generates time translation."

The distinction between symmetric and self-adjoint operators was not fully appreciated until the work of Friedrichs (1935, the Friedrichs extension) and Krein (1940s, the theory of self-adjoint extensions via deficiency indices). The physics literature often uses "Hermitian" and "self-adjoint" interchangeably, which is correct in finite dimensions but fails for unbounded operators where domain issues arise. This distinction has physical consequences: the boundary conditions on a particle-in-a-box problem are precisely the choice of self-adjoint extension, and different extensions correspond to different physical systems.

Bibliography [Master]

• Dirac, P. A. M. The Principles of Quantum Mechanics, 4th ed. Oxford University Press, 1958. Ch. II-III.

• von Neumann, J. Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1955 (trans. from 1932 German edition). Ch. II-III.

• Stone, M. H. "On One-Parameter Unitary Groups in Hilbert Space." Annals of Mathematics 33, 643-649, 1932.

• Sakurai, J. J. and Napolitano, J. Modern Quantum Mechanics, 2nd ed. Cambridge University Press, 2017. Ch. 1.2-1.4.

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