Expectation Values

Quantum mechanics refuses to predict single outcomes; it predicts statistics. The last two articles built the machines — position multiplies, momentum differentiates — and this article builds the universal contract between any such machine and the laboratory: the expectation value,

\langle Q\rangle = \int \Psi^*\,\hat Q\,\Psi\,dx,

the number to which the average of many measurements converges when each measurement is performed once on a freshly prepared copy of the state. Every prediction the theory makes about experiments passes through this sandwich — conjugate wave, operator, wave, integrate — so learning to read it, compute it, and not misread it is a survival skill for the whole track. The misreadings are the real enemy, and they are seductive. $\langle Q\rangle$ is not the result of any single measurement; it may be a value no measurement can ever return. It is not a time average of one system watched patiently — it is an ensemble average over many identical preparations, and the difference is enforced by the theory itself, because the first measurement changes the state. And the sandwich's word order is law, not typography: $\hat Q$ acts on $\Psi$ before the conjugate multiplies, and for operators built from derivatives, rearranging the sandwich changes the number. The payoff for the discipline is that expectation values are where quantum mechanics touches ground. They evolve smoothly even when outcomes jump; they obey conservation laws; and — as Ehrenfest's Theorem will prove next-but-one — they are the quantities through which Newton's mechanics re-emerges from the wave equation. Here you learn to compute them; there you learn how they move.

Key takeaways

The mean of repeated measurements of $Q$ on identically prepared systems: $\langle Q\rangle = \int\Psi^*\hat Q\,\Psi\,dx$ — the operator sandwich.
Order is law: $\hat Q$ acts to the right first. $\int\Psi^*\hat Q\Psi \neq \int(\hat Q\text{-anything else})$ when $\hat Q$ carries derivatives.
Ensemble reading: many copies, one measurement each. NOT a time average, NOT the outcome of one shot, NOT "the value the system has."
For position the sandwich collapses to $\int x|\Psi|^2dx$ ; for momentum it does not collapse at all — that is why the operator grammar exists.
Composite observables ride along: $\langle T\rangle = \int\Psi^*\left(-\frac{\hbar^2}{2m}\partial_x^2\right)\Psi\,dx$ , $\langle H\rangle = \langle T\rangle + \langle V\rangle$ .
In a superposition of energy eigenstates, $\langle Q\rangle$ can oscillate at Bohr frequencies when $Q$ has nonzero matrix elements between those states — expectation values can change even though the energy probabilities remain constant.
Key idea: expectation values are the statistical means predicted by quantum mechanics, computed by sandwiching the observable's operator between the state and its conjugate. In appropriate limits, these expectation values provide the bridge to classical mechanics.

The core insight

The expectation value $\langle Q\rangle = \int\Psi^*\,\hat Q\,\Psi\,dx$ is the statistical mean obtained from many measurements of $Q$ on identically prepared systems — an operator sandwich whose order matters, but which is not itself the outcome of any single measurement.

Where this lives today

Every quantum-computing result you have read is an expectation value. A quantum processor cannot print a wavefunction; it can only be run many times and sampled — precisely the factory-line protocol of this article. The headline quantities of the field are ensemble means: a "fidelity of 99.9%" is $\langle$ projector onto the target state $\rangle$ estimated from repeated shots, and the variational algorithms used for chemistry (VQE and descendants) work by estimating $\langle H\rangle = \sum_k c_k\langle P_k\rangle$ term by term — thousands of shots per Pauli term $P_k$ — then steering parameters downhill in the sampled mean energy. Shot budgeting, the trade between runs and statistical error in $\langle Q\rangle$ , is Bernoulli's theorem as an engineering line item. The swinging-centroid physics of the second worked example is also live technology: attosecond pump–probe experiments now track $\langle x\rangle(t)$ -type observables — time-dependent dipole expectation values — of electrons in atoms and molecules on sub-femtosecond grids, literally watching equation (4)'s cosine in charge-migration data, matrix element, Bohr frequency and all.

The physical picture

Imagine a factory line producing electrons, every one stamped into the same state $\Psi$ — same preparation procedure, same wavefunction, indistinguishable in principle. At the end of the line stands one measuring device. Each electron gets measured once, then discarded. The readings scatter — quantum mechanics guarantees they scatter unless the state is an eigenstate — and pile up into a histogram. The expectation value is the histogram's balance point: the sample mean, in the limit of infinitely many electrons. Every phrase in that paragraph is load-bearing. Same state: the ensemble is of preparations, not of different systems. Measured once, then discarded: the first measurement collapses the state, so measuring the same electron twice samples a different distribution the second time — which is why $\langle Q\rangle$ is not a time average over one patient electron. Balance point: the mean is a property of the whole distribution, not of any single reading — it can sit between allowed outcomes, or at a forbidden one, exactly as Position Operator's mean-at-a-node example showed.

Figure

The ensemble contract: many copies, one measurement each, one balance point

Why does the operator compute this statistical object? For position, no mystery: with $\hat Q=\hat x$ the sandwich $\Psi^*(x)\,\hat x\,\Psi(x)$ gives $x|\Psi(x)|^2$ , whose integral is the probability-density-weighted mean position. The genuinely new content appears for momentum and observables built from it, because the momentum operator acts by differentiation, so the expectation value generally cannot be written using only the position-space probability density $|\Psi(x)|^2$ . The derivation card below shows why the same grammar nonetheless delivers the correct mean for any observable: the sandwich is secretly a sum over the operator's eigenvalues weighted by Born-rule probabilities.

Figure

The sandwich machine: word order is law

The sandwich, earned

The definition first, then its credentials. For any observable $Q$ with operator $\hat Q$ (built from $\hat x$ and $\hat p$ by the substitution rules of the last two articles),

\langle Q\rangle = \int_{-\infty}^{\infty}\Psi^*(x,t)\,\hat Q\,\Psi(x,t)\,dx \qquad (1)

where: the hat acts on everything to its right inside the integrand — $\hat Q\Psi$ is computed first, then conjugate-multiplied and integrated. Two familiar cases calibrate the formula. For $\hat Q = \hat x$ : $\langle x\rangle = \int x|\Psi|^2dx$ , the Born-rule lever-arm average of Position Operator. For $\hat Q = \hat p$ : $\langle p\rangle = -i\hbar\int\Psi^*\partial_x\Psi\,dx$ , the phase-reading integral of Momentum Operator — irreducibly sandwich-shaped, with no density-only form. Composite observables inherit the grammar by substitution. Kinetic energy $T = p^2/2m$ becomes

\langle T\rangle = -\frac{\hbar^2}{2m}\int_{-\infty}^{\infty}\Psi^*\,\frac{\partial^2\Psi}{\partial x^2}\,dx \qquad (2)

where: the second derivative measures the curvature of the wavefunction, and for the kinetic-energy operator $\hat T=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$ this curvature determines the kinetic-energy expectation value. Adding $\langle V\rangle = \int V(x)|\Psi|^2dx$ gives $\langle H\rangle = \langle T\rangle + \langle V\rangle$ — the mean energy, the quantity whose conservation and whose role as "the" energy of a superposition the first worked example makes concrete.

Derivation AWhy the sandwich computes the mean — the eigenvalue bookkeeping

Equation (1) is usually presented as a postulate, but for observables with a known eigenbasis it is a two-line theorem, and seeing the proof once immunizes you against ever misreading the sandwich. The strategy: expand the state in the observable's eigenfunctions, let the operator act diagonally through its eigenvalues, and use orthonormality to collapse the double sum into exactly the probability-weighted average that a "mean" means. We run it for a discrete spectrum (energy is the cleanest example, using the machinery of Stationary States and Separation of Variables); the continuous case swaps sums for integrals.

Step 1. Expand the state. Write $\Psi = \sum_n c_n\psi_n$ with $\hat Q\psi_n = q_n\psi_n$ and $\int\psi_m^*\psi_n dx = \delta_{mn}$ . Why: completeness (the doctrine behind it lives with the Formalism chapter's Complete Orthonormal Bases) guarantees the expansion exists; the $c_n$ are the state's coordinates along the observable's own axes.

Step 2. Let the operator act. $\hat Q\Psi = \sum_n c_n q_n\psi_n$ — each eigenstate component is rescaled by its eigenvalue. Why: in its eigenbasis, the operator acts diagonally, so its action is reduced to multiplication by the corresponding eigenvalue.

Step 3. Sandwich and collapse. $\int\Psi^*\hat Q\Psi\,dx = \sum_m\sum_n c_m^*c_n q_n\int\psi_m^*\psi_n dx = \sum_n |c_n|^2 q_n$ . Why: orthonormality kills every cross term $m \neq n$ ; what survives is a sum of eigenvalues weighted by $|c_n|^2$ .

Step 4. Recognize the mean. By the Born rule, $|c_n|^2$ is the probability of obtaining the eigenvalue $q_n$ when the eigenvalue is nondegenerate; for a degenerate eigenvalue, the probabilities of all states in the corresponding eigenspace must be added. Thus $\sum_n |c_n|^2q_n$ is the mean of the measurement-outcome distribution:

\langle Q\rangle = \sum_n P(q_n)\,q_n \qquad (3)

Why: the sandwich and the histogram's balance point are the same number---proven, not decreed. For a Hermitian $\hat Q$ , $\langle Q\rangle$ is real because its eigenvalues are real; in a finite or bounded spectrum, it lies between the smallest and largest eigenvalues, and no single outcome needs to equal it.

Go deeper · Derivation B · 2 minThe ensemble doctrine — why one system is not enough

Derivation BThe ensemble doctrine — why one system is not enough

"Average of many measurements" hides a fork: many measurements on one system, or one measurement each on many systems? Classically the two often agree; quantum mechanically the first is not even well-defined without new physics, and this card pins down why. The argument needs only the collapse rule from Born Statistical Interpretation and produces the operational definition every laboratory actually uses — plus the reason "the particle has the value $\langle Q\rangle$ " is not just sloppy but wrong.

Step 1. Try the single-system protocol. Measure $Q$ on one system in state $\Psi$ : outcome $q_k$ , with the state now collapsed to $\psi_k$ . Measure again: the second reading samples the new state's distribution — for an ideal measurement, it returns $q_k$ again with certainty. Why: repetition on one system after collapse measures the eigenstate you created, not the state you prepared; the running average converges to $q_k$ , not to $\langle Q\rangle$ .

Step 2. State the correct protocol. Prepare $N$ systems identically in $\Psi$ ; measure each once; average the outcomes. As $N \to \infty$ the sample mean converges to $\sum P(q_n)q_n = \langle Q\rangle$ by the law of large numbers. Why: each trial samples the same distribution — the one $\Psi$ encodes — because no trial contaminates another. "Identically prepared ensemble" is not pedantry; it is the only protocol the theory licenses.

Step 3. Kill the property reading. Suppose $\langle Q\rangle$ were "the value the system has." For the ISW superposition of the first worked example, $\langle H\rangle = 0.94\ \mathrm{eV}$ — but every energy measurement returns $0.376$ or $1.504\ \mathrm{eV}$ , never $0.94$ . A "possessed value" that no measurement can ever elicit is empty. Why: the mean is a property of the distribution, i.e. of the preparation, not of any individual system. Quantities that fluctuate around the mean get their measure — the standard deviation — in Standard Deviation and Quantum Uncertainty.

Step 4. Note the honest exception. If $\Psi$ is an eigenstate of the observable $\hat Q$ , the measurement distribution is a single spike: every measurement of $Q$ returns the same eigenvalue $q$ . In that specific sense, saying that the system has the definite value $q$ is justified — the certainty case discussed in Operators and Observables. Why: eigenstates are exactly the states for which the classical language of possessed values survives; everywhere else, speak of distributions.

When expectation values move

Sandwich a time-dependent state and the number inherits time dependence. The cleanest laboratory is the two-state superposition of Stationary States and Separation of Variables: energy statistics frozen, yet $\langle x\rangle$ oscillates. Take $\Psi = \frac{1}{\sqrt2}(\psi_1 e^{-iE_1t/\hbar} + \psi_2 e^{-iE_2t/\hbar})$ with real $\psi_{1,2}$ ; the sandwich for $\hat x$ picks up the beating cross term:

\langle x\rangle(t) = \frac{\langle x\rangle_1 + \langle x\rangle_2}{2} + X_{12}\cos(\omega_{21}t), \qquad X_{12} = \int\psi_1\,x\,\psi_2\,dx \qquad (4)

where: $\langle x\rangle_n$ are the static means in each eigenstate; $X_{12}=\int \psi_1^*(x)\,x\,\psi_2(x)\,dx$ is the cross matrix element, which sets the swing's amplitude; and $\omega_{21}=(E_2-E_1)/\hbar$ is the Bohr frequency, as usual. The oscillation occurs when $X_{12}\neq0$ . The centroid of the probability mist genuinely swings back and forth, forever, at the beat frequency — this is the quantitative content of the "sloshing" pictures two articles back, and the second worked example puts numbers on it. Matrix elements like $X_{12}$ are the currency of spectroscopy: they decide which transitions radiate and how brightly, a story the Time-Dependent Phenomena chapter tells in full. How expectation values move in general — the equations of motion for $\langle x\rangle$ and $\langle p\rangle$ themselves — is precisely Ehrenfest's Theorem, two articles ahead; equation (4) is your preview of what it must explain.

Limiting cases

Case	Result	Meaning
$\Psi$ an eigenstate of $\hat Q$	$\langle Q\rangle = q$ , every shot returns $q$	the only case where "has the value" is honest
stationary state, $\hat Q$ time-independent	$\langle Q\rangle$ constant	frozen statistics, per Stationary States and Separation of Variables
two-state superposition	$\langle x\rangle$ swings at $\omega_{21}$ , amplitude $X_{12}$	beats made observable; the seed of spectroscopy
$\hat Q = \hat x$	sandwich collapses to $\int x\lvert\Psi\rvert^2dx$	the density-only special case — deceptively simple
classical limit (narrow packet)	$\langle Q\rangle \to$ classical value $Q(x_c, p_c)$	means track classical mechanics — Ehrenfest's Theorem

Common trap“⟨Q⟩ is what you'll get if you measure Q”

The single most durable misreading in the subject, and it fails on every word. What you'll get: a single measurement returns an eigenvalue of $\hat Q$ , and $\langle Q\rangle$ is generally not one — the superposition below has $\langle H\rangle = 0.94\ \mathrm{eV}$ while every energy measurement returns $0.376$ or $1.504\ \mathrm{eV}$ , without exception. You: the contract is about an ensemble; one shot tells you almost nothing about $\langle Q\rangle$ , and repeated shots on the same system measure the collapsed state, not the original. If you measure: between preparations, the theory declines to say the system "has" any $Q$ at all. The safe sentence — say it until it is reflex — is: " $\langle Q\rangle$ is the mean of the outcome distribution over many identical preparations." Its fluctuating partner, the width of that distribution, is the next-but-two article's whole subject.

Worked examples

1 · Mean energy of a superposition — computed three ways (confidence builder)

An electron in a $L = 1.0\ \mathrm{nm}$ infinite square well (levels $E_n = n^2E_1$ , $E_1 = 0.376\ \mathrm{eV}$ , per Infinite Square Well) is in the equal superposition $\Psi = \frac{1}{\sqrt2}(\psi_1 + \psi_2)$ . Find $\langle H\rangle$ .

Step 1. Via the eigenvalue bookkeeping (3): $P(E_1) = P(E_2) = \tfrac12$ , so $\langle H\rangle = \tfrac12 E_1 + \tfrac12 E_2 = \tfrac12(0.376) + \tfrac12(1.504) = \boxed{0.940\ \mathrm{eV}}$ .

Step 2. Via the sandwich (1): $\int\Psi^*\hat H\Psi\,dx = \tfrac12\left(E_1\!\int\!\psi_1^2 + E_2\!\int\!\psi_2^2\right) + \text{cross terms} \propto \int\psi_1\psi_2\,dx = 0$ — orthogonality deletes the cross terms and the diagonal terms reproduce Step 1. Same number, now from the integral definition.

Step 3. Sanity by bounds: $E_1 < 0.940 < E_2$ ✓ — a mean must sit between the extreme outcomes. And note what it is not: no single energy measurement on this state ever returns $0.940\ \mathrm{eV}$ ; the outcome histogram has two spikes and empty space between.

Step 4. Constancy: the weights $\tfrac12, \tfrac12$ are frozen (per Stationary States and Separation of Variables), so $\langle H\rangle$ is conserved even while the state sloshes — mean energy is a constant of the motion for any time-independent $\hat H$ .

2 · The swinging centroid, with numbers (exam level)

Same state, same well. The cross matrix element for the box is $X_{12} = \int_0^L\psi_1 x\,\psi_2\,dx = -\frac{16L}{9\pi^2}$ (a result derived in Infinite Square Well; recompute it). Find the amplitude and period of the oscillation of $\langle x\rangle$ , and its extreme values.

Step 1. Recompute the matrix element: $X_{12}=\frac{2}{L}\int_0^L x\sin\frac{\pi x}{L}\sin\frac{2\pi x}{L}\,dx=-\frac{16L}{9\pi^2}=-0.180L$ . The sign depends on the relative phase convention chosen for the two eigenstates; the magnitude $|X_{12}|=0.180L$ determines the oscillation amplitude.

Step 2. Assemble equation (4): each eigenstate has $\langle x\rangle_n = L/2$ by symmetry, so $\langle x\rangle(t) = \frac{L}{2} - 0.180L\cos(\omega_{21}t)$ — the centroid swings about the centre with amplitude $\boxed{0.180L = 0.180\ \mathrm{nm}}$ .

Step 3. Period: $\omega_{21} = 3E_1/\hbar = 1.71\times10^{15}\ \mathrm{rad/s}$ , so $T = 2\pi/\omega_{21} = \boxed{3.67\ \mathrm{fs}}$ — the beat clock of Stationary States and Separation of Variables, now driving a measurable displacement.

Step 4. Extremes: $\langle x\rangle$ ranges over $L/2 \mp 0.180L$ , i.e. from $0.320L$ to $0.680L$ — the centroid never approaches the walls, even though individual position measurements land everywhere in the box. A femtosecond-resolved measurement of the ensemble mean position could trace this cosine directly; an oscillating charge distribution at $1.71\times10^{15}\,\mathrm{rad/s}$ is also the kind of motion that can couple to electromagnetic radiation, providing a classical picture related to the $2\to1$ transition.

Origins · 1 minJacob Bernoulli and how this was discovered

Jacob Bernoulli

1655–1705 · Basel

The word "expectation" entered science through gambling, and its rigorous backbone — the reason a sample mean converges to $\langle Q\rangle$ at all — is Jacob Bernoulli's. His Ars Conjectandi, twenty years in the writing and published only in 1713, eight years after his death, proved the first law of large numbers: the fraction of successes in independent trials converges (in probability, with quantified error bounds) to the underlying probability. Bernoulli called it his "golden theorem" and understood exactly what it bought — a licence to infer the unmeasurable odds from measurable frequencies. Quantum mechanics runs on that licence daily: the theory outputs distribution parameters like $\langle Q\rangle$ and $\sigma_Q$ ; the laboratory records finite histograms; Bernoulli's theorem is the bridge that lets ten thousand shots on a qubit certify $\langle Z\rangle$ to three decimals. He never saw an atom and would have needed no physics lecture to audit our protocols: identical preparations, independent trials, sample means — the Ars Conjectandi checklist, verbatim. The Bernoulli name became a dynasty (his brother Johann, nephew Daniel — a different row in this wiki's glossary), but the law of large numbers that justifies estimating expectation values from repeated measurements is one of Jacob's lasting contributions.

History

1713

— Jacob Bernoulli's Ars Conjectandi proves the law of large numbers: frequencies converge to probabilities.

1926

— Born's statistical interpretation makes |c|² the probability of an outcome; quantum predictions become distributional.

1926

— Schrödinger computes wave-mechanical averages; the operator sandwich takes its modern form.

1927

— Ehrenfest shows the sandwiched means obey Newton-like equations — expectation values become the classical bridge.

1932

— Von Neumann's axiomatization writes ⟨Q⟩ = ⟨Ψ|Q|Ψ⟩ as the theory's fundamental statistical statement.

Going further

Texts. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §1.5 & §3.4 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 4 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §1.4 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. III · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 6. Key results to keep in view: the eigenbasis collapse $\int\Psi^*\hat Q\Psi\,dx = \sum|c_n|^2q_n$ ; the sandwich forms for $\langle x\rangle$ , $\langle p\rangle$ , and $\langle T\rangle$ ; and the anchor numbers for the equal ISW mix at $L = 1\ \mathrm{nm}$ — $\langle H\rangle = 0.940\ \mathrm{eV}$ with outcomes only $0.376/1.504\ \mathrm{eV}$ , cross element $X_{12} = -16L/9\pi^2 = -0.180L$ , swing amplitude $0.180\ \mathrm{nm}$ , period $3.67\ \mathrm{fs}$ , and centroid range $0.320L$ – $0.680L$ . The variance and spread machinery is developed in Standard Deviation and Quantum Uncertainty, and the Dirac-notation formulation in the Formalism chapter's Expectation Values and Variance.

Prerequisites. The Wavefunction · Born Statistical Interpretation · Probability Density and Normalization · Position Operator · Momentum Operator. Leads to. Operators and Observables · Ehrenfest's Theorem · Standard Deviation and Quantum Uncertainty · the Formalism chapter's Expectation Values and Variance.

Why is the sandwich Ψ·Q̂Ψ and not, say, (Q̂Ψ)·Ψ or Ψ·Q̂Ψ*?

For a Hermitian operator — the only kind that represents an observable — the first two are provably equal: $\int(\hat Q\Psi)^*\Psi = \int\Psi^*\hat Q\Psi$ is precisely the Hermiticity condition (built for $\hat p$ in Momentum Operator, formalized next article). So the convention costs nothing physical; it is fixed so that mixed matrix elements $\int\psi_m^*\hat Q\psi_n$ have a consistent reading (operator acts on the right-hand state). The third form, $\Psi\hat Q\Psi^*$ , differs for complex states and derivative operators — it computes $\langle Q\rangle^*$ , which for Hermitian $\hat Q$ is again the same real number, but writing it invites order errors on paper. Adopt the standard order and every textbook formula matches.

Is there any single-shot meaning to ⟨Q⟩ at all?

A weak one, worth stating precisely: $\langle Q\rangle$ is your best bet — the value minimizing the expected squared error of a one-number prediction of the next outcome. That is a statement about your forecasting, not about the system. There are also genuinely single-shot-flavoured protocols in modern labs (weak measurements, where a feeble coupling shifts a pointer by something proportional to an expectation-like quantity without collapsing the state), but they buy their gentleness with enormous per-shot noise — ensembles return through the back door. For this course: distributions first, single shots never.

Do expectation values obey the classical formulas — is ⟨p²⟩ = ⟨p⟩², or ⟨T⟩ = ⟨p⟩²/2m?

No, and the gap is physics, not error. Means of nonlinear functions differ from functions of means: $\langle p^2\rangle \geq \langle p\rangle^2$ always, with the difference being the variance — the whole subject of Standard Deviation and Quantum Uncertainty. Concretely, the box ground state has $\langle p\rangle = 0$ but $\langle T\rangle = E_1 \neq 0$ : the kinetic energy lives entirely in the spread. Classical-looking relations hold between means only when the relevant operator is linear in the fluctuating quantity — hence $\langle H\rangle = \langle T\rangle + \langle V\rangle$ is exact, while $\langle T\rangle = \langle p\rangle^2/2m$ is generically false. When narrow packets make spreads negligible, the classical formulas return approximately — the regime Ehrenfest's Theorem delimits.

What exactly counts as 'identically prepared'? No two lab runs are truly identical.

Identical in the quantum-mechanical sense: same preparation procedure, hence same wavefunction — not same microscopic history of the apparatus. The theory's claim, massively confirmed, is that the wavefunction is a complete description of the preparation: two runs of the same protocol are as identical as nature permits, and residual outcome scatter is not a hidden difference between runs but irreducible quantum statistics. Practically, labs verify identical preparation statistically — state tomography on samples — and treat drifts (laser power, temperature) as classical noise to be characterized separately. The clean separation of quantum scatter from preparation drift is a working skill of experimental physics.

The cross element X₁₂ decided the oscillation amplitude. What decides whether it's zero?

Symmetry, mostly — and this is your first selection rule. In the box, $\psi_1$ and $\psi_2$ have opposite parity about the centre, and $x$ (measured from the centre) is odd, so the integrand $\psi_1 x\psi_2$ is even and the integral survives. Pair two states of the same parity and $\int\psi_m x\psi_n dx = 0$ : no centroid oscillation, and — since the same matrix element controls dipole radiation — no allowed radiative transition between them. That parity bookkeeping generalizes into the full selection-rule apparatus of the Time-Dependent Phenomena chapter; the habit of checking symmetry before integrating will save you hours across the track.

Can two different states share all the same expectation values?

For a single observable, easily — states with identical $\langle Q\rangle$ but different distributions abound (same mean, different spread). But specify the expectation values of enough observables and the state is pinned uniquely: knowing $\langle Q\rangle$ for every Hermitian $\hat Q$ determines $\Psi$ up to global phase (density-matrix language makes this exact, at the track's later register). Practical state tomography does exactly this with a finite, cleverly chosen set. The moral runs both directions: one mean tells you little; the full catalogue of means is the state.

Does ⟨Q⟩ depend on when I measure, if the state is evolving?

Yes — $\langle Q\rangle(t)$ is computed by sandwiching the state at the measurement time, and equation (4) showed it can oscillate forever. The ensemble protocol then reads: prepare $N$ copies, evolve each for the same duration $t$ , measure each once — a different $t$ is a different experiment with a different mean. Stationary states are the exception (all their means are frozen, per Stationary States and Separation of Variables), and conserved quantities like $\langle H\rangle$ are exceptions for every state. The general law of motion for $\langle Q\rangle(t)$ — which observables move, and how fast — is exactly what Ehrenfest's Theorem delivers next-but-one.