Operators and Observables

Position multiplies. Momentum differentiates. Energy does both at once. Three articles in, a pattern is begging to be named: every measurable quantity of quantum mechanics arrives as an operator acting on the wavefunction. This article names the pattern and states its rules — which operators get to represent physical observables, and what that privilege guarantees. The membership test is a single symmetry of the sandwich integral. An operator $\hat Q$ is Hermitian when it can act on either slice of bread indifferently: $\int f^*\hat Qg\,dx = \int(\hat Qf)^*g\,dx$ for all admissible states. That one line has teeth. It forces every expectation value to be a real number — no laboratory dial ever reads $2+3i$ — and it will ultimately force real eigenvalues, orthogonal eigenfunctions, and complete eigenbases, the full spectral scaffolding on which measurements stand. The credential is not automatic: the innocent-looking derivative $d/dx$ fails the test, and this article makes it fail numerically, so you can see with your own eyes what an anti-Hermitian "observable" would predict — imaginary averages, meaningless dials. The second half of the contract runs the other way. If the state happens to be an eigenfunction of $\hat Q$ , then measurement of $Q$ is not statistical at all: every shot returns the eigenvalue, with certainty. Determinate states exist, they are exactly the eigenstates, and a $2\times2$ worked model lets you diagonalize an observable end to end. Between "Hermitian ⇒ real statistics" and "eigenstate ⇒ certain outcome," this article is where quantum mechanics signs its measurement contract — the deeper clauses (why eigenfunctions are orthogonal and complete, what happens in the measurement) are signed in the Formalism chapter.

Key takeaways

Observables ↔ Hermitian operators: $\int f^*\hat Qg\,dx = \int(\hat Qf)^*g\,dx$ for all admissible $f, g$ — the operator moves across the sandwich unchanged.
Hermiticity ⇔ every expectation value is real, in every state — the theorem runs both directions.
Membership audit: $\hat x$ (real multiplier) and $\hat p = -i\hbar\partial_x$ (the $-i$ earns it) pass; bare $d/dx$ is anti-Hermitian and predicts imaginary averages.
Hermiticity survives assembly: real combinations of Hermitian operators, and $\hat H = \hat p^2/2m + V(x)$ in particular, are Hermitian.
Eigenstate ⇒ certainty: if $\hat Q\Psi = q\Psi$ , every measurement of $Q$ returns exactly $q$ — determinate states are eigenstates.
Finite-dimensional observables are Hermitian matrices; diagonalizing one is solving its measurement theory completely.
Key idea: physical observables are represented by Hermitian operators — the symmetry that guarantees real measured values, and whose eigenstates are the states of certainty.

The core insight

Physical observables correspond to Hermitian (self-adjoint) operators: their expectation values are real in every state, their eigenvalues are the possible measurement outcomes, and a system in an eigenstate returns that eigenvalue with certainty.

Where this lives today

Ask a quantum-hardware group what they actually measure and the answer is a list of Hermitian matrices. Qubit readout is the $\pm1$ -outcome observable $\hat Z$ ; two-qubit gate benchmarking estimates expectation values of Pauli strings — Hermitian by construction — and quantum state tomography reconstructs an unknown state precisely by measuring a complete set of Hermitian observables and inverting the statistics, the "full catalogue of means" idea made procedure. When hardware papers report stabilizer expectation values, every number quoted is $\langle\text{Hermitian operator}\rangle$ estimated from shots; cross-entropy benchmarks sit one step further out — statistical scores built from the whole sampled bitstring distribution rather than the average of a single Hermitian observable, though every bitstring still arrives through projective $\hat Z$ readout. The anti-Hermitian side of the audit has its own modern career. Engineered non-Hermitian systems — photonic waveguides with balanced gain and loss, "PT-symmetric" lattices — deliberately violate the contract to produce one-way light transport and exotic degeneracies called exceptional points, now used in ultra-sensitive sensors. The field's entire vocabulary is defined against this article's theorem: "non-Hermitian physics" is interesting precisely because Hermiticity is what ordinary observables must obey.

The physical picture

Think of an observable's operator as a dial-reading machine wired into the sandwich integral. The machine's output — an expectation value — must land on the real axis every time, for every conceivable state, because laboratory dials have no imaginary tick marks. What property of the machine guarantees that? Watch the sandwich under complex conjugation: conjugating $\int\Psi^*\hat Q\Psi\,dx$ swaps which copy of $\Psi$ the operator touches. If the operator doesn't care which side it acts on — if it slides across the sandwich unchanged — then the integral equals its own conjugate, and a number equal to its own conjugate is real. Hermiticity is exactly that indifference, and reality of every reading is its immediate wage.

Figure

Hermitian machines read real; the bare derivative reads imaginary

The second picture concerns outcomes rather than averages. A general state fed to a measurement produces a scattered histogram. But feed the machine one of its own eigenfunctions and the scatter vanishes: the histogram is a single spike at the eigenvalue. Eigenstates are the machine's fixed points of certainty — the only states about which the observable has nothing statistical to say. That is why "solve the eigenvalue problem" and "understand the measurement" are the same task, and why diagonalizing a matrix, in the worked example below, amounts to a complete measurement theory for a two-outcome observable.

Figure

General state: scattered outcomes · Eigenstate: a single certain spike

The Hermiticity contract

State the definition with care, because every later theorem hangs on it. An operator $\hat Q$ on the space of admissible wavefunctions is Hermitian if

\int_{-\infty}^{\infty} f^*\,(\hat Q\,g)\,dx = \int_{-\infty}^{\infty} (\hat Q\,f)^*\,g\,dx \quad \text{for all admissible } f, g \qquad (1)

where: $f, g$ — any two normalizable states (with the boundary behaviour cataloged by Admissibility Conditions on Wavefunctions); the star on the right conjugates the whole of $\hat Qf$ . Read it as promised: the operator may be applied to either function without changing the number. (Mathematicians distinguish "Hermitian" from the stricter "self-adjoint" — a domain subtlety that matters for pathological potentials and is deferred, with the rest of the fine print, to the Formalism chapter's Hermitian Operators and Observables; at this register the words are interchangeable.)

Derivation AReal averages — the two-way theorem that defines an observable

The postulate "observables are Hermitian operators" earns its keep here: we prove that Hermiticity is equivalent to every expectation value being real. The forward direction is three lines of conjugation bookkeeping. The reverse direction — every-average-real forces Hermiticity — is the striking one: reality of dials is not merely a nice consequence but a complete characterization, so nature's requirement (real readings) picks out exactly the Hermitian class, no more and no less. We use the polarization trick, quantum mechanics' favourite way to upgrade diagonal information to off-diagonal.

Step 1. Forward: conjugate the sandwich. $\langle Q\rangle^* = \left(\int\Psi^*\hat Q\Psi\,dx\right)^* = \int(\hat Q\Psi)^*\Psi\,dx$ . Why: conjugating an integral conjugates the integrand; the star lands on the factor the operator acted on.

Step 2. Apply Hermiticity with $f = g = \Psi$ . The right side of (1) says $\int(\hat Q\Psi)^*\Psi\,dx = \int\Psi^*\hat Q\Psi\,dx = \langle Q\rangle$ . Hence $\langle Q\rangle^* = \langle Q\rangle$ : real, in every state. Why: indifference-to-side is exactly the licence to un-star the operator; a number equal to its own conjugate has no imaginary part.

Step 3. Reverse: assume all averages are real. For every admissible $\Psi$ , $\int\Psi^*\hat Q\Psi\,dx \in \mathbb{R}$ . Feed the assumption the combined states $\Psi = f + g$ and $\Psi = f + ig$ and expand. The diagonal terms are real by assumption; collecting what remains gives, from the first, $\mathrm{Im}\!\left[\int f^*\hat Qg\,dx + \int g^*\hat Qf\,dx\right] = 0$ and, from the second, $\mathrm{Re}\!\left[\int f^*\hat Qg\,dx - \int g^*\hat Qf\,dx\right] = 0$ . Why: this is polarization — probing the operator with superpositions extracts the cross terms that single states cannot see.

Step 4. Combine. The two conditions together say $\int g^*\hat Qf\,dx = \left(\int f^*\hat Qg\,dx\right)^*= \int (\hat Qg)^* f\,dx$ for all $f, g$ — which is (1) with the roles relabelled. Hermiticity is forced. Why: real diagonals in every state leave the operator no room to cheat off-diagonally. The equivalence is complete: Hermitian ⇔ all averages real.

With the contract signed, audit the membership of the operators you own. $\hat x$ : multiplication by a real number slides through the conjugate untouched — Hermitian for free, as Position Operator noted. $\hat p = -i\hbar\,\partial_x$ : Hermitian by the integration-by-parts derivation of Momentum Operator — the boundary term dies by normalizability and the $(-i)^* = +i$ flip pays for the parts-sign; that card is cited here, not re-run. The instructive member is the one that fails:

\int f^*\,\frac{dg}{dx}\,dx = \big[f^*g\big]_{-\infty}^{\infty} - \int\left(\frac{df}{dx}\right)^{\!*}g\,dx = -\int\left(\frac{df}{dx}\right)^{\!*}g\,dx \qquad (2)

where: the boundary term vanishes as always, leaving a minus sign: $d/dx$ crosses the sandwich only by flipping sign — anti-Hermitian. Its averages are consequently pure imaginary in every state (run Step 1–2 above with the extra sign and $\langle Q\rangle^* = -\langle Q\rangle$ falls out). The first worked example measures this failure numerically. And the repair is visible in equation (2) itself: multiply by any pure-imaginary constant and the constant's conjugation flip cancels the parts flip — which is precisely why momentum wears its $-i\hbar$ . Assembly rules complete the audit kit. If $\hat A, \hat B$ are Hermitian, so is $\hat A + \hat B$ (integrals add) and so is $c\hat A$ for real $c$ ; products need care — $\hat A\hat B$ is Hermitian only when the two commute — but any power $\hat A^n$ is safe. Hence the Hamiltonian passes by construction:

\hat H = \frac{\hat p^2}{2m} + V(x) \qquad (3)

where: $\hat p^2$ — a power of a Hermitian operator; $V(x)$ — multiplication by a real function, Hermitian exactly as $\hat x$ was. The reality of $V$ is once again load-bearing — the same clause that conserved probability in Conservation of Normalization is here certifying the energy as an honest observable; complex "optical" potentials forfeit both certificates at once, and that is no coincidence but one fact in two ledgers.

Go deeper · Derivation B · 1 minEigenstate in, certainty out

Derivation BEigenstate in, certainty out

The measurement contract's second clause: if the state is an eigenfunction of the observable, the statistics degenerate to certainty. We prove it with the moment machinery of Expectation Values — mean equals eigenvalue, mean square equals eigenvalue squared, so the distribution has zero width and only one possible outcome. The logic is the same two-line pattern used for sharp energy in Stationary States and Separation of Variables, now stated for arbitrary observables, which is exactly how a good pattern earns the name "theorem." Mind the direction: this card proves eigenstate ⇒ certainty; the converse (zero spread ⇒ eigenstate) needs the variance machinery and is owned by Standard Deviation and Quantum Uncertainty.

Step 1. Set up. Let $\hat Q\Psi = q\Psi$ with $\Psi$ normalized, $q$ real (guaranteed: Hermitian operators have real eigenvalues — take the expectation of $\hat Q$ in $\Psi$ and apply the reality theorem). Why: the eigenvalue equation is the hypothesis "the state is a fixed point of the machine."

Step 2. First moment. $\langle Q\rangle = \int\Psi^*\hat Q\Psi\,dx = q\int|\Psi|^2dx = q$ . Why: the operator surrenders its eigenvalue and normalization finishes the job.

Step 3. Second moment. $\hat Q^2\Psi = \hat Q(q\Psi) = q^2\Psi$ , so $\langle Q^2\rangle = q^2$ . Why: a fixed point of the machine is a fixed point of the machine run twice.

Step 4. Zero spread, certain outcome. The variance $\langle Q^2\rangle - \langle Q\rangle^2 = q^2 - q^2 = 0$ ; a probability distribution with zero variance is a single spike, so every measurement returns $q$ . Why: this is the precise sense in which eigenstates are "states of definite $Q$ " — and the only states for which the classical phrase "the system has the value $q$ " survives the ensemble doctrine of Expectation Values.

Limiting cases

Case	Result	Meaning
$\hat Q$ = real multiplier ( $\hat x$ , $V(x)$ )	Hermitian for free	conjugates slide past real functions
$\hat Q = c\,d/dx$	Hermitian iff $c$ pure imaginary	the parts-flip must be paid by the conjugation-flip
$\hat A\hat B$ , $[\hat A,\hat B] \neq 0$	not Hermitian; $\tfrac12(\hat A\hat B + \hat B\hat A)$ is	symmetrize products before calling them observables
eigenstate of $\hat Q$	zero spread, outcome certain	the spike histogram; converse proven via $\sigma$ later
finite basis	observables = Hermitian matrices, $Q_{mn} = Q_{nm}^*$	diagonalization = complete measurement theory

Common trap“Any operator you can write down is an observable”

The grammar tempts you: if $\hat x$ and $\hat p$ are operators, surely $d/dx$ , or $\hat x\hat p$ , or $i\hat H$ are observables too? All three fail the contract. The bare derivative is anti-Hermitian — its "averages" are pure imaginary, as this article's first worked example computes to the digit. The product $\hat x\hat p$ fails because its factors do not commute: conjugating the sandwich returns $\hat p\hat x$ , which differs from $\hat x\hat p$ by exactly $i\hbar$ (the commutator of Momentum Operator), so its expectation ( $\langle \hat{x}\hat{p} \rangle=\left\langle \frac{\hat{x}\hat{p}+\hat{p}\hat{x}}{2} \right\rangle + \frac{i\hbar}{2}$ ) carries a spurious constant imaginary part $i\hbar/2$ ; the honest observable is the symmetrized $\tfrac12(\hat x\hat p + \hat p\hat x)$ . And $i\hat H$ is anti-Hermitian by construction — useful as a generator (it drives time evolution), meaningless as a dial. The moral: Hermiticity is a checked credential, not a default. Audit every operator you promote to "measurable" — the check takes one integration by parts, and skipping it produces predictions no experiment can even express.

Worked examples

1 · Measuring the failure of d/dx (confidence builder)

Take the normalized ISW (infinite square well) superposition $\Psi = \frac{1}{\sqrt2}(\psi_1 - i\psi_2)$ of the well's two lowest stationary states on $[0, L]$ (the $-i$ is essential — it puts a genuine phase twist into the state). Compute $\langle d/dx\rangle = \int\Psi^*\Psi'\,dx$ and $\langle\hat p\rangle$ , and compare their characters.

Step 1. Expand the integrand: $\Psi^*\Psi' = \tfrac12(\psi_1 + i\psi_2)(\psi_1' - i\psi_2') = \tfrac12(\psi_1\psi_1' + \psi_2\psi_2') + \tfrac{i}{2}(\psi_2\psi_1' - \psi_1\psi_2')$ .

Step 2. The real part integrates to zero: $\psi_n\psi_n' = \tfrac12(\psi_n^2)'$ , and $\psi_n^2$ vanishes at both walls. The imaginary part survives: with $\psi_n = \sqrt{2/L}\sin(n\pi x/L)$ , direct integration gives $\int_0^L(\psi_2\psi_1' - \psi_1\psi_2')dx = \dfrac{16}{3L}$ .

Step 3. So $\langle d/dx\rangle = \dfrac{i}{2}\cdot\dfrac{16}{3L} = \boxed{i\,\dfrac{8}{3L}}$ — pure imaginary, exactly as anti-Hermiticity demands. For $L = 1\ \mathrm{nm}$ : $i\times2.67\times10^{9}\ \mathrm{m^{-1}}$ . No instrument can read it as it is imaginary.

Step 4. Now the repaired operator: $\langle\hat p\rangle = -i\hbar\langle d/dx\rangle = -i\hbar\cdot i\dfrac{8}{3L} = \boxed{\dfrac{8\hbar}{3L}}$ — real and positive: $2.8\times10^{-25}\ \mathrm{kg\,m/s}$ for $L = 1\ \mathrm{nm}$ . The $-i$ rotated the reading onto the real axis, and the state's phase twist (the relative $i$ between components) shows up as genuine net momentum. One factor of $-i\hbar$ is the entire difference between nonsense and physics.

2 · Diagonalizing a two-outcome observable (exam level)

A two-state system (basis states $|1\rangle, |2\rangle$ — think of the two-well model of Time-Independent Schrödinger Equation) carries the observable $\hat S = \dfrac{\hbar}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ . (a) Verify $\hat S$ is Hermitian. (b) Find the possible outcomes and the states of certainty. (c) For the state $\mathbf{c} = (1, 0)$ , give the measurement statistics.

Step 1. Hermiticity for matrices reads $S_{mn} = S_{nm}^*$ : here $S_{12} = S_{21} = \hbar/2$ , both real — pass. (The matrix rule is the sandwich rule (1) restricted to a finite basis.)

Step 2. Outcomes = eigenvalues: $\det(\hat S - s\mathbb{1}) = s^2 - \hbar^2/4 = 0$ , so $s_\pm = \boxed{\pm\hbar/2}$ — real, as Hermiticity guarantees. Any single measurement returns one of these two numbers, never anything else.

Step 3. States of certainty = eigenvectors: $\mathbf{c}_\pm = \tfrac{1}{\sqrt2}(1, \pm1)$ . Prepared in $\mathbf{c}_+$ , every measurement of $S$ returns $+\hbar/2$ ; the histogram is a spike. Note the eigenvectors are orthogonal — a Hermitian bonus formalized in the Formalism chapter.

Step 4. Statistics for $\mathbf{c} = (1,0)$ : expand in the certainty basis, $(1,0) = \tfrac{1}{\sqrt2}(\mathbf{c}_+ + \mathbf{c}_-)$ , so $P(+\hbar/2) = P(-\hbar/2) = \boxed{\tfrac12}$ , giving $\langle S\rangle = 0$ and maximal scatter. One diagonalization delivered everything: outcomes, certainty states, and full statistics for arbitrary states. (This matrix is no toy — it reappears in the Angular Momentum chapter as the spin operator $\hat S_x$ , where these eigenvectors become the states a Stern–Gerlach magnet transmits.)

Origins · 1 minCharles Hermite and how this was discovered

Charles Hermite

1822–1901 · Paris

The adjective on every observable in quantum mechanics honours a man who nearly failed his way out of mathematics. Hermite bombed the École Polytechnique entrance examinations twice (his geometry was superb; his rote algebra was not), was admitted at last near the bottom of the list, and was then eased out within a year over a congenitally deformed right foot the school deemed disqualifying. He became, regardless, the central French mathematician of his century. The relevant work is his 1855 memoir on what are now called Hermitian forms and matrices: arrays obeying $A_{mn} = A_{nm}^*$ , for which he proved the theorem this article runs on — their eigenvalues are all real, however complex the entries. It was pure algebra, motivated by number theory, seventy years before anyone needed a guarantee that measured energies come out real; when matrix mechanics arrived in 1925 and observables became matrices, his theorem was the reason the theory's predictions were readable on dials. Hermite's other bequest to this track arrives in the oscillator chapter — the polynomial family bearing his name, a story with its own article — but it is the 1855 reality theorem that put his name inside the word "observable."

History

1855

— Hermite proves that matrices with A(mn) = A(nm)* have real eigenvalues — algebra waiting for physics.

1904

— Hilbert's spectral theory extends the reality theorem to operators on infinite-dimensional spaces.

1925

— Matrix mechanics casts observables as Hermitian matrices; real spectra become physical predictions.

1926

— Schrödinger's wave mechanics needs the same credential for differential operators; integration by parts supplies it.

1932

— Von Neumann's self-adjointness analysis completes the operator-observable dictionary for the continuous case.

Going further

Texts. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §3.2–3.3 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 1 & 4 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §1.3–1.4 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. II · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 2 & 6. Key results collected here for reference: the reality⇔Hermiticity equivalence (including the polarization step); the anti-Hermiticity of $d/dx$ via integration by parts, with $\langle d/dx\rangle = 8i/3L$ and $\langle\hat p\rangle = 8\hbar/3L = 2.8\times10^{-25}\ \mathrm{kg\,m/s}$ at $L = 1\ \mathrm{nm}$ ; the eigenstate-certainty argument $\langle Q\rangle = q$ , $\langle Q^2\rangle = q^2$ ; and the $2\times2$ diagonalization giving eigenvalues $\pm\hbar/2$ , eigenvectors $(1, \pm1)/\sqrt2$ , and 50/50 statistics for the state $(1,0)$ . The orthogonality and completeness theorems for Hermitian eigenfunctions, the degenerate case, and the self-adjointness fine print are treated in the Formalism chapter (Hermitian Operators and Observables, Eigenvalues and Eigenstates, Complete Orthonormal Bases); the $\sigma = 0 \Rightarrow$ eigenstate converse is covered in Standard Deviation and Quantum Uncertainty.

Prerequisites. The Wavefunction · Born Statistical Interpretation · Probability Density and Normalization · de Broglie Matter Waves · Position Operator · Momentum Operator · Expectation Values. Leads to. Standard Deviation and Quantum Uncertainty · Ehrenfest's Theorem · the Formalism chapter's operator theory.

Is Hermiticity an axiom or a theorem — could nature have used non-Hermitian observables?

Within standard quantum mechanics it is an axiom, but the reverse theorem of this article shows how little freedom nature had: demand only that every ensemble average land on a real dial, and Hermiticity is forced — there is no larger class of operators with universally real statistics. The modern caveat is interesting: PT-symmetric non-Hermitian Hamiltonians can have entirely real spectra, and for a while looked like a genuine generalization; the resolution is that such systems are equivalent to Hermitian ones under a redefined inner product, or describe open systems (gain/loss) where probability genuinely isn't conserved — the same trade-off as the optical potentials of Conservation of Normalization. For isolated systems with the standard probability rules, Hermitian is the whole menu.

Why do observables have to be linear operators in the first place?

Because superposition is physical. If $\Psi_1$ and $\Psi_2$ are possible states, so is $\alpha\Psi_1 + \beta\Psi_2$ , and the statistical machinery must process the superposition consistently with how it processes the parts — which is exactly linearity of the sandwich's inner workings. There are also hard theorems backing the choice: attempts to build nonlinear observables (or nonlinear evolution) generically allow superluminal signalling or break the probability interpretation. The linearity is not modesty; it is one of the most experimentally corroborated structural facts in physics — every interference experiment tests it.

The reality theorem needed 'for all admissible f, g'. Do the boundary conditions ever actually bite?

They bite exactly when the physics has edges. On the full line, normalizability kills the boundary terms and $\hat p$ is safely Hermitian. On a half-line (radial problems, $r \geq 0$ ), the boundary term at the origin survives unless the wavefunctions vanish there — which is why the radial momentum operator needs careful dressing in the Hydrogen chapter. On a ring (periodic boundary), $\hat p$ is Hermitian but its spectrum becomes discrete. And for a particle in a box, $\hat p$ alone is Hermitian but not self-adjoint in the full technical sense — the resolution (domains, deficiency indices) is the Formalism chapter's fine print. Practical rule at this register: check the boundary term explicitly whenever the domain is not the whole line.

What is the observable for 'which state is it in' — is Ψ itself measurable?

$\Psi$ is not an observable — no single Hermitian operator returns "the wavefunction" — and this is a feature, not a gap. What exists: for any state $\phi$ , the projector onto $\phi$ is Hermitian with eigenvalues 1 and 0, and measuring it asks "is the system in state $\phi$ — yes or no?", with probability $|\langle\phi|\Psi\rangle|^2$ of yes. Full knowledge of $\Psi$ requires measuring many such projectors (or other observables) across an ensemble — quantum state tomography — because a single copy of an unknown state cannot be read out completely (the no-cloning theorem, at the track's later register, makes this precise). The projector machinery is built in the Formalism chapter's Projection Operators.

Do the eigenvalues of an observable always exhaust the possible measurement outcomes?

Yes — that is the spectral half of the measurement postulate: outcomes are eigenvalues, discrete or continuous, and nothing else ever appears on the dial. The subtle direction is coverage: for the postulate to make sense, the eigenfunctions must be complete, so that every state has a well-defined outcome distribution over the spectrum. Completeness is a theorem for Hermitian (self-adjoint) operators — the spectral theorem — with the continuous-spectrum version requiring the delta-normalized machinery previewed in Position Operator. The full statement, degenerate cases included, is the Formalism chapter's centrepiece. At this register: trust the spectrum to be the complete outcome menu, and check Hermiticity before trusting the spectrum.

Why does the symmetrized product ½(x̂p̂ + p̂x̂) deserve the name 'the observable xp'?

Because it is the unique Hermitian operator that reduces to the classical product $xp$ in the classical limit and is linear in both factors. The recipe — average the orderings — is the simplest instance of a genuine ambiguity: classical expressions do not dictate operator ordering, and different orderings differ by commutator terms of order $\hbar$ . For $xp$ the symmetrization settles it, and the resulting observable (the generator of dilations, as it happens) plays a role in advanced scattering theory. For worse cases ( $x^2p^2$ and beyond) several inequivalent Hermitian orderings exist, and which one nature uses is settled by experiment or by deeper principles (canonical quantization from the classical Poisson structure — the story told at the track's Dynamics register). Ordering ambiguity is the price of promoting commuting classical variables to non-commuting operators.