Stationary States and Separation of Variables

The previous article, Time-Independent Schrödinger Equation, ran the separation ritual and produced the bricks: product solutions $\Psi_n = \psi_n(x)\,e^{-iE_nt/\hbar}$ , one per entry on the energy menu. This article is about what those bricks are and what they buy. Three properties make them the most valuable objects in quantum mechanics. First, they are stationary: every probability you can extract from one — the density, the odds of any measurement outcome — is frozen in time, even though the wavefunction itself turns its phase forever. Second, they are sharp in energy: measure the energy of a system in the state $\Psi_n$ and you get $E_n$ , not on average but every single time, with zero scatter. Third, they are complete: any state whatsoever, however violently non-stationary, is a weighted superposition of them. The third property upgrades a curiosity into a strategy. Expand the initial state over the stationary states, attach to each component its phase clock $e^{-iE_nt/\hbar}$ , and resum whenever you want the future: dynamics becomes phase bookkeeping. All visible motion — a density that sloshes, an atom that radiates — lives in the interference between components whose clocks tick at different rates, and the beat frequencies $(E_m - E_n)/\hbar$ are exactly what spectroscopes have been measuring since before quantum mechanics existed. You arrive with the eigenvalue problem posed and solved in principle; you leave with the reason it was worth posing — and with the words "stationary" and "superposition" doing precise, load-bearing work.

Key takeaways

A stationary state $\Psi_n = \psi_n(x)e^{-iE_nt/\hbar}$ has $|\Psi_n|^2 = |\psi_n|^2$ : the density, and every expectation value of a time-independent observable, never changes.
Stationary states carry definite energy: $\langle H\rangle = E_n$ with zero spread — an energy measurement returns $E_n$ with certainty.
"Stationary" describes the statistics, not the wavefunction: the phase rotates at rate $E_n/\hbar$ always, and even the current can be nonzero (a plane wave is stationary and flowing — an idealized, non-normalizable stationary state; physical states are wave packets built from them).
The general solution of the time-dependent equation is $\Psi(x,t) = \sum_n c_n\psi_n e^{-iE_nt/\hbar}$ — expand once, evolve by clocks.
The weights obey $|c_n|^2$ = probability of finding energy $E_n$ ; they never change under the evolution.
Superpose two energies and the density breathes at the Bohr frequency $\omega_{21} = (E_2 - E_1)/\hbar$ : all visible quantum motion is beats.
Key idea: energy eigenstates freeze all observable statistics; superpositions of them, each with its own phase clock, generate every possible quantum motion.

The core insight

For a time-independent Hamiltonian, separation of variables $\Psi = \psi(x)e^{-iEt/\hbar}$ yields energy eigenstates whose observables are time-independent and which form a complete basis: any state evolves as $\Psi = \sum_n c_n\psi_n e^{-iE_nt/\hbar}$ , with all dynamics carried by the relative phases.

Where this lives today

The most precise humanly made measurements are stationary-state physics. An optical atomic clock traps strontium or ytterbium atoms in a lattice and interrogates one deliberately chosen pair of stationary states whose beat frequency — an optical $\omega_{21}/2\pi$ near $4\times10^{14}\ \mathrm{Hz}$ — is reproducible to nineteen significant figures. The clock does not measure a level; it measures a difference, exactly as equation (2) dictates, and the 2026-era redefinition programme for the SI second is, at bottom, a vote on which pair of stationary states humanity trusts most. Superconducting quantum computers run the same physics at microwave frequencies: a transmon qubit's $|0\rangle$ and $|1\rangle$ are the two lowest stationary states of an engineered anharmonic oscillator, its "qubit frequency" is their Bohr beat (typically $4$ – $6\ \mathrm{GHz}$ ), and the phase bookkeeping of equation (1) — each amplitude turning at its own $E_n/\hbar$ — is what every quantum algorithm choreographs. Decoherence, the field's central enemy, is precisely anything that scrambles those relative phases faster than the gates can use them.

The physical picture

Picture the energy menu of a trapped particle as a shelf of sealed snow globes. Inside globe $n$ sits the shape $|\psi_n|^2$ — and nothing in it ever stirs. Come back in a nanosecond or a year: the same landscape. Yet each globe secretly turns: the wavefunction's complex phase advances at rate $E_n/\hbar$ , faster on the higher shelves. You cannot see the turning by looking at one globe, because the Born rule takes an absolute square and $|e^{-iE_nt/\hbar}| = 1$ — the rotation is real but statistically invisible. Now pour two globes together. The mixture is no longer sealed: the two phases turn at different rates, drift in and out of alignment, and the combined density leans left, then right, then left — breathing at exactly the difference rate $(E_2 - E_1)/\hbar$ . Nothing about either ingredient changed; the motion lives entirely in the relative phase. That is the mechanism of every visible quantum dynamic, from a sloshing wavepacket to a radiating atom. (For the radiating atom, treat this as the first word rather than the last: once the atom couples to the radiation field, the emitted photon perturbs the system and the strict isolated-state picture gives way to quantum electrodynamics — but the frequency it radiates at is exactly the beat this bookkeeping predicts.)

Figure

Stationary states are sealed; superpositions breathe

The pouring metaphor also fixes the right mental model for the expansion coefficients. When a state is a blend $c_1\psi_1 + c_2\psi_2 + \cdots$ , the numbers $|c_n|^2$ are the recipe — how much of each globe went into the pour — and, by the Born statistics of Born Statistical Interpretation, they are simultaneously the probabilities that an energy measurement returns $E_n$ . The recipe never changes while the system evolves: the clocks turn, the weights hold. Energy statistics are eternal; position statistics dance.

Figure

The master strategy: expand, attach clocks, resum

The three properties, earned

Each headline property is a two-line calculation away, and the calculations teach the grammar of everything to come. Throughout, $\Psi_n(x,t) = \psi_n(x)\,e^{-iE_nt/\hbar}$ with $\hat H\psi_n = E_n\psi_n$ , exactly as delivered by Time-Independent Schrödinger Equation — the separation itself is recalled, not re-run.

Derivation AFrozen statistics — nothing observable moves in a stationary state

The claim to earn: in the state $\Psi_n$ , the probability density and the expectation value of any observable built without explicit time dependence are constants. The strategy is disarmingly simple — write the quantity, watch the phase factor meet its own conjugate, and see unit modulus erase the time. The one subtlety worth guarding: the argument needs the same energy in bra and ket, which is precisely what fails in a superposition, and that failure is where all real motion will come from.

Step 1. The density. $|\Psi_n(x,t)|^2 = \psi_n^*(x)\,e^{+iE_nt/\hbar}\;\psi_n(x)\,e^{-iE_nt/\hbar} = |\psi_n(x)|^2$ . Why: the clock and its conjugate wind oppositely and cancel exactly — $|e^{-iE_nt/\hbar}|^2 = 1$ at every instant. The Born-rule map of the particle's whereabouts is a still photograph.

Step 2. Expectation values. For an operator $\hat Q$ containing $x$ and derivatives but no explicit $t$ (the machinery of Expectation Values, used here in preview): $\langle Q\rangle = \int \Psi_n^*\,\hat Q\,\Psi_n\,dx = \int \psi_n^*\,\hat Q\,\psi_n\,dx$ — the clocks pass straight through $\hat Q$ and annihilate as before. Why: $\hat Q$ touches only the spatial factor, so the phases still meet face to face; every moment, every spread, every probability computed from $\Psi_n$ is time-independent.

Step 3. What is NOT claimed. The wavefunction itself moves — its phase turns at $E_n/\hbar$ without rest — and even the probability current can be nonzero: a plane wave is a stationary state with steady flow, like a river in steady state — idealized, to be sure: a plane wave is a generalized eigenstate, not a normalizable physical state, but every realizable current-carrying packet inherits the behavior from it. Why: "stationary" is a statement about statistics being time-independent, not about stillness; the current object itself is built next, in Probability Current and the Continuity Equation.

Go deeper · Derivation B · 1 minSharp energy — the eigenstate passes every energy audit with zero scatter

Derivation BSharp energy — the eigenstate passes every energy audit with zero scatter

"Definite energy" is a statistical claim, and it deserves a statistical proof: we show the mean energy is $E_n$ and the mean square is $E_n^2$ , so the spread — mean square minus squared mean — is exactly zero. No distribution with zero spread has anything but a single value in it: every energy measurement on $\Psi_n$ returns $E_n$ . The expectation-value grammar is borrowed in preview from Expectation Values, and the spread concept gets its full treatment in Standard Deviation and Quantum Uncertainty; what matters here is the eigenvalue equation doing the work twice.

Step 1. Mean. $\langle H\rangle = \int \psi_n^*\,\hat H\psi_n\,dx = \int\psi_n^*\,(E_n\psi_n)\,dx = E_n\int|\psi_n|^2dx = E_n$ . Why: the eigenvalue equation converts the operator into a number, and normalization does the rest.

Step 2. Mean square. $\hat H^2\psi_n = \hat H(E_n\psi_n) = E_n^2\psi_n$ , so $\langle H^2\rangle = E_n^2$ . Why: an eigenfunction of $\hat H$ is automatically an eigenfunction of $\hat H^2$ — apply the machine twice, collect the number twice.

Step 3. Spread. $\sigma_H^2 = \langle H^2\rangle - \langle H\rangle^2 = E_n^2 - E_n^2 = 0$ . Why: zero variance means the energy distribution is a single spike at $E_n$ — certainty, not merely a good average. This is the precise sense in which the menu entries of the previous article are the system's "allowed energies."

Step 4. The converse orientation. In a superposition $c_1\psi_1 + c_2\psi_2$ with $E_1 \neq E_2$ , Step 2 fails — $\hat H$ returns different numbers on the two parts — and the energy distribution has genuine width: outcomes $E_1$ or $E_2$ , probabilities $|c_1|^2, |c_2|^2$ . Why: sharpness is special to eigenstates; the general eigenvalue-certainty doctrine is developed for all observables in Operators and Observables.

Building every motion: the superposition engine

Linearity now converts the bricks into the building. Because the time-dependent equation is linear (Time-Dependent Schrödinger Equation made this structural), any linear combination of solutions is a solution, so the most general state built from the menu is

\Psi(x,t) = \sum_n c_n\,\psi_n(x)\,e^{-iE_nt/\hbar} \qquad (1)

where: $c_n$ — complex constants fixed by the initial shape $\Psi(x,0) = \sum_n c_n\psi_n(x)$ ; the sum becomes an integral over the continuum for unbound problems (per Bound versus Scattering States). Two facts make (1) the master formula of the subject. Completeness: the eigenfunctions span the space of admissible states, so every initial condition can be expanded this way — the doctrine behind that guarantee lives with the Formalism chapter's Complete Orthonormal Bases. Orthogonality: distinct eigenfunctions are mutually orthogonal, which is what lets you extract each $c_n$ from $\Psi(x,0)$ by an integral — the "Fourier trick," performed with full machinery on a concrete spectrum in Infinite Square Well. Here we take the coefficients as given and ask what the physics of (1) is. The physics is in the exponents. Each component carries its own clock; observable quantities involve $|\Psi|^2$ , where clocks meet other components' conjugate clocks and survive as relative phases $e^{-i(E_m - E_n)t/\hbar}$ . Everything that moves, moves at difference frequencies.

Go deeper · Derivation C · 1 minThe two-state beat — where visible motion comes from

Derivation CThe two-state beat — where visible motion comes from

The simplest state that actually does something: an equal blend of two real eigenfunctions, $\Psi = \frac{1}{\sqrt2}(\psi_1 e^{-iE_1t/\hbar} + \psi_2 e^{-iE_2t/\hbar})$ . We square it honestly and watch the cross term — the only term that remembers the time — oscillate at the Bohr frequency. This four-line computation is the quantitative version of the beat picture promised by Time-Dependent Schrödinger Equation, and it is the ancestor of every transition frequency in spectroscopy. Assume $\psi_1, \psi_2$ real and orthonormal (bound eigenfunctions can always be chosen real — the freedom is proven in Time-Independent Schrödinger Equation in One Dimension).

Step 1. Square the sum. $|\Psi|^2 = \tfrac12\left[\psi_1^2 + \psi_2^2 + \psi_1\psi_2\left(e^{i(E_1-E_2)t/\hbar} + e^{-i(E_1-E_2)t/\hbar}\right)\right]$ . Why: the diagonal terms lose their clocks as in the stationary case; each cross term pairs one clock with the other's conjugate, leaving the difference winding.

Step 2. Collect the cosine.

|\Psi(x,t)|^2 = \tfrac12\left(\psi_1^2 + \psi_2^2\right) + \psi_1\psi_2\,\cos(\omega_{21}t), \qquad \omega_{21} = \frac{E_2 - E_1}{\hbar} \qquad (2)

Why: the two exponentials are conjugates, and their sum is $2\cos(\omega_{21}t)$ . The density is a static pedestal plus a breathing term whose spatial pattern is the product $\psi_1\psi_2$ — positive where the modes agree, negative where they disagree, which is what makes the density lean side to side.

Step 3. Read the rate. Only the energy difference appears. Shift every energy by a constant ( $E_n \to E_n + V_0$ ) and $\omega_{21}$ is untouched — absolute energies are gauge, differences are physics. Why: this is why spectra measure level gaps, never levels, and why the "global phase is unphysical" doctrine of the wavefunction articles is consistent with phases driving all dynamics: only relative phase is real.

Step 4. Note the recipe's constancy. The weights $\tfrac12, \tfrac12$ never appeared with a time dependence: energy statistics are frozen even while position statistics dance. Why: each clock has unit modulus, so $|c_n e^{-iE_nt/\hbar}|^2 = |c_n|^2$ forever — conservation of the energy distribution, superposition edition.

Limiting cases

Case	Result	Meaning
single eigenstate, any $n$	$\lvert\Psi\rvert^2$ and all $\langle Q\rangle$ static; $\sigma_H = 0$	the sealed snow globe: sharp energy, frozen statistics
two states, equal weights	pedestal $+\ \psi_1\psi_2\cos(\omega_{21}t)$	density breathes at the Bohr frequency — the atom of quantum dynamics
many states, spread of energies	wavepacket motion, dispersal, revivals	rich dynamics from many beating clocks; the free-particle version is Free Particle and Wave Packets
$E_n$ shifted by constant $V_0$	all $\omega_{mn}$ unchanged	only energy differences are observable — global phase is gauge
$\hat H = \hat H(t)$	no stationary states exist	the driver keeps changing the mechanism; Quantum Dynamics chapter tools

Common trap“Stationary means the particle is sitting still”

Three separate confusions hide in that sentence; kill them in order. The wavefunction is not still — its phase rotates at $E/\hbar$ eternally; what is still is every probability extracted from it. The particle is not at rest — a stationary state generally has $\langle p^2\rangle > 0$ and plenty of kinetic energy; the box eigenstates of Infinite Square Well are stationary yet momentum measurements scatter over $\pm\pi\hbar n/L$ . Flow is not forbidden — a plane wave, and the current-carrying ring states you will meet in the angular-momentum chapter, are stationary states with nonzero probability current: a river in steady state is unchanging and flowing. The safe reading of "stationary" is: the state's statistics have no time dependence. And its mirror error deserves equal fire: a superposition of stationary states is NOT stationary — the beat term in equation (2) is precisely its non-stationarity, and forgetting the cross term is the classic way to lose the physics of a two-level system.

Worked examples

1 · The beat of a box superposition (confidence builder)

An electron in a $L = 1.0\ \mathrm{nm}$ infinite square well (spectrum $E_n = n^2E_1$ , $E_1 = 0.376\ \mathrm{eV}$ , per Infinite Square Well) is prepared in the equal mix $\Psi = \frac{1}{\sqrt2}(\psi_1 + \psi_2)$ at $t = 0$ . Find the beat (Bohr) frequency and period of the density's sloshing, and the probabilities of each energy outcome.

Step 1. Energy gap: $E_2 - E_1 = (4-1)E_1 = 3\times0.376 = 1.128\ \mathrm{eV} = 1.807\times10^{-19}\ \mathrm{J}$ .

Step 2. Bohr frequency: $\omega_{21} = \dfrac{E_2 - E_1}{\hbar} = \dfrac{1.807\times10^{-19}}{1.055\times10^{-34}} = \boxed{1.71\times10^{15}\ \mathrm{rad/s}}$ .

Step 3. Period of the slosh: $T = \dfrac{2\pi}{\omega_{21}} = \boxed{3.67\ \mathrm{fs}}$ — the density leans left, then right, and returns every $3.67$ femtoseconds, forever (no friction in the equation).

Step 4. Energy statistics: outcomes $E_1$ or $E_2 = 1.504\ \mathrm{eV}$ only, each with probability $|c_n|^2 = \boxed{\tfrac12}$ — and these numbers hold at every $t$ : the slosh never transfers weight between levels. A photon emitted in a real $2\to1$ decay would carry $1.128\ \mathrm{eV}$ ( $\lambda = hc/\Delta E \approx 1.10\ \mathrm{\mu m}$ , near infrared), oscillating at exactly the beat frequency — spectroscopy is beats made audible.

2 · An unequal pour (exam level)

Same well, but the state is $\Psi = \frac{\sqrt3}{2}\psi_1 + \frac12\psi_2$ (check: $\frac34 + \frac14 = 1$ , normalized by orthonormality). Find the probability of each energy outcome, the mean of many energy measurements, and the relative amplitude of the density's breathing compared with Example 1.

Step 1. Probabilities: $P(E_1) = \left(\tfrac{\sqrt3}{2}\right)^2 = \boxed{0.75}$ , $P(E_2) = \left(\tfrac12\right)^2 = \boxed{0.25}$ . Born statistics on the energy menu: nothing but $E_1$ or $E_2$ ever appears, in odds 3 : 1.

Step 2. Mean energy, as the probability-weighted average of outcomes: $\bar E = 0.75\,E_1 + 0.25\,E_2 = 0.75(0.376) + 0.25(1.504) = \boxed{0.658\ \mathrm{eV}}$ — a number between the levels that no single measurement ever returns. (Its operator formulation, $\langle H\rangle = \int\Psi^*\hat H\Psi\,dx$ , is the next-but-one article's subject, Expectation Values.)

Step 3. The breathing term generalizes equation (2) to $2c_1c_2\,\psi_1\psi_2\cos(\omega_{21}t)$ with $2c_1c_2 = 2\cdot\tfrac{\sqrt3}{2}\cdot\tfrac12 = \tfrac{\sqrt3}{2} \approx 0.866$ , versus $2\cdot\tfrac{1}{\sqrt2}\cdot\tfrac{1}{\sqrt2} = 1$ for the equal mix. The slosh runs at the same frequency but $\boxed{87\%}$ of the amplitude. Equal pours interfere hardest; as either weight approaches 1 the state approaches a sealed globe and the breathing dies out smoothly.

Step 4. Sanity checks: probabilities sum to 1 ✓; $\bar E$ lies between $E_1$ and $E_2$ ✓; setting $c_1 = c_2 = \tfrac{1}{\sqrt2}$ recovers Example 1 ✓.

Origins · 1 minNiels Bohr and how this was discovered

Niels Bohr

1885–1962 · Copenhagen

"Stationary state" is Bohr's phrase, and it was an act of daring before it was a definition. In his 1913 trilogy the 27-year-old Dane postulated — against everything Maxwell's electrodynamics guaranteed — that an atom possesses special states in which it simply does not radiate, and that light appears only in jumps between them, at frequency $\nu = (E_2 - E_1)/h$ . He could not say why such states held still; he asserted that they did, because Balmer's spectral lines said something did. Thirteen years later wave mechanics kept his noun and replaced his mechanism: the stillness is the unit-modulus phase factor of this article, and his jump frequency is exactly the beat $\omega_{21}$ of equation (2) — derived, not decreed. Bohr's institute in Copenhagen then became the clearing-house where the new mechanics was argued into coherence, and Bohr himself its most relentless interrogator. His planetary model of the atom did not survive; his conviction that atoms have a discrete menu of non-radiating states, and that observable physics lives in the differences, is on every page of this chapter. The model's own story belongs to the optics track.

History

1913

— Bohr postulates non-radiating "stationary states" and transition frequencies ν = ΔE/h, fitting hydrogen's Balmer series.

1916

— Sommerfeld generalizes the quantization rules; the stationary-state menu acquires fine structure.

1925

— Matrix mechanics encodes observable physics entirely in transition quantities between stationary states.

1926

— Schrödinger derives the stationary states as eigenfunctions and shows superpositions beat at Bohr's frequencies.

1926

— Born's statistical reading makes the expansion weights probabilities, completing the modern picture of equation (1).

Going further

Texts. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §2.1 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 4 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.1 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. III · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 6. Key results for reference: substituting $\Psi_n = \psi_n e^{-iE_nt/\hbar}$ into the TDSE yields the TISE; the two-state beating density is $|\Psi|^2 = \tfrac12(|\psi_1|^2 + |\psi_2|^2) + \psi_1\psi_2\cos(\omega_{21}t)$ ; and $|c_n|^2$ is conserved by unitarity. Numerical values (ISW, $L = 1\ \mathrm{nm}$ , spectrum from Infinite Square Well): $E_2 - E_1 = 1.128\ \mathrm{eV}$ , $\omega_{21} = 1.71\times10^{15}\ \mathrm{rad/s}$ , beat period $3.67\ \mathrm{fs}$ , $\lambda = 1.10\ \mathrm{\mu m}$ ; unequal mix $(\sqrt3\psi_1 + \psi_2)/2$ : $P = 0.75/0.25$ , $\bar E = 0.658\ \mathrm{eV}$ , breathing amplitude ratio $\sqrt3/2$ .

Prerequisites. Time-Dependent Schrödinger Equation · Time-Independent Schrödinger Equation · The Wavefunction · Probability Density and Normalization · Born Statistical Interpretation. Leads to. Probability Current and the Continuity Equation · the one-dimensional laboratory (Infinite Square Well onward) · Expectation Values.

If the phase rotation is invisible, in what sense is it physically real?

Real the way potential energy is real: unobservable absolutely, decisive relatively. One stationary state's phase can be redefined at will — that is the global-phase gauge freedom. But the ratio of two components' phases in a superposition is nailed to observable fact: it sets where the density leans in equation (2), and interference experiments read it out directly. The clean statement: phases are unobservable one at a time, and all-important two at a time. A theory with truly no phases (just densities evolving) could not produce the beat term at all — that cross term is the experimental proof the rotation is really happening.

Why do superpositions of different energies exist at all — doesn't energy conservation forbid an indefinite energy?

Conservation constrains dynamics, not states. The energy distribution of an isolated system is frozen — Step 4 of the beat derivation showed $|c_n|^2$ never changes — and that is precisely what energy conservation means when the energy is statistical. Nothing in the theory demands the distribution be a single spike. Preparing such superpositions is routine: shine a short pulse on an atom (a short pulse is a spread of frequencies, hence energies), or suddenly change a trap so the old ground state is a blend of the new eigenstates. What you cannot do is watch $|c_n|^2$ drift while the system is isolated.

Do stationary states actually exist in the lab, or are they an idealization like plane waves?

Bound-state members of the menu are as real as anything in physics — an unperturbed atom in its ground state simply is $\psi_1$ , and it will sit there indefinitely. Excited stationary states are honest but slightly compromised: coupling to the electromagnetic field (not in our Hamiltonian) gives them finite lifetimes, so "stationary" holds on timescales short of the decay time — nanoseconds for typical atomic transitions, which is an eternity compared with the femtosecond beats of this article. The truly ideal members are the continuum ones: a plane wave is a stationary state and is unrealizable, per the normalizability discussion in Bound versus Scattering States.

Where do the coefficients cₙ actually come from for a given initial state?

From orthogonality, by an integral: $c_n = \int\psi_n^*(x)\,\Psi(x,0)\,dx$ . The logic — multiply the expansion by one eigenfunction's conjugate, integrate, and watch every term but one die by orthogonality — is called the Fourier trick, and it is performed with the full machinery (orthonormality integrals included) on the box spectrum in Infinite Square Well. This article deliberately treats the $c_n$ as given, because the physics of the expansion — clocks, beats, frozen weights — does not care how the recipe was obtained.

Is the ground state also 'turning its phase'? Nothing lower exists for it to beat against.

Yes — $e^{-iE_1t/\hbar}$ turns like all the others, and in isolation that rotation is pure gauge: with no partner to beat against, no observable knows about it. This is why a universe containing only a ground-state atom has literally no internal clock ticking observably. The moment you superpose anything else — another level, another atom's state — the ground state's clock re-enters the physics through differences. Energy zero-points are conventions; the hydrogen binding energy $-13.6\ \mathrm{eV}$ matters only because ionized states sit $13.6\ \mathrm{eV}$ away.

The beat frequency looks like the classical oscillation frequency. Is that a coincidence?

It is the correspondence principle doing its job. For highly excited box states, neighbouring levels are spaced by $E_{n+1} - E_n \approx dE/dn$ , and the beat frequency of adjacent levels approaches the classical round-trip frequency of a particle bouncing at that energy — you can check this scaling on the ISW spectrum. A superposition of many adjacent high levels beats itself into a localized lump that bounces classically. For low quantum numbers the match is loose (the $n = 1,2$ beat of Example 1 is only within a factor of order one of the classical bounce), which is the correspondence principle's honest fine print: classicality emerges in the limit, not at the bottom of the ladder.

Why is so much made of energy eigenstates rather than, say, momentum or position eigenstates?

Because the Hamiltonian runs the clock. Expanding in eigenstates of any observable is legal, but only the energy expansion evolves by simple phase-turning — that is the content of equation (1), and it exists because $\hat H$ itself generates the dynamics. Expand in momentum eigenstates and each component still evolves simply only for a free particle (where momentum eigenstates are energy eigenstates); add a potential and the momentum components mix. Energy eigenstates are the unique basis in which time evolution is diagonal — bookkeeping instead of work. Other bases repay their cost elsewhere: position for reading densities, momentum for scattering, as later chapters exploit.