Spectral Decomposition of Time Evolution

Two articles ago the completeness relation assembled an operator from its eigendata and stopped, deliberately, at a pair of drifting eigenphases. The previous articles built the time-evolution operator and met the states it leaves alone. This article is the payoff: the working craft of quantum dynamics, the method you reach for whenever a time-independent Hamiltonian meets an initial state. The method is almost insolently simple. Diagonalize $\hat H$ once. Every energy eigenstate is a clock that only turns a phase, at its own rate $E_n/\hbar$ . Any state is a chord struck across these clocks, and evolving it costs three cheap steps: project, phase-rotate, resum. No differential equation is ever solved again — separation of variables promised this in the wave chapter; here the promise becomes an operator identity, $\hat U(t) = \sum_n e^{-iE_n t/\hbar}|n\rangle\langle n|$ . What you buy is an entire physics of interference between clocks. One populated level: nothing moves, ever. Two levels: one beat, at the Bohr frequency $(E_2 - E_1)/\hbar$ — only energy differences are audible, which is why spectroscopy works. Three levels in a box: a woven carpet of beats that dephases into apparent mess and then — because the box spectrum is commensurate — snaps back exactly, at a revival time you can compute to the femtosecond. You will compute all of it here, watch a selection rule silence one of the three beats, and leave with the algorithm behind everything from quantum-beat spectroscopy to the quantum simulators of 2026.

Key takeaways

For time-independent $\hat H$ : $\hat U(t) = \sum_n e^{-iE_n t/\hbar}\,|n\rangle\langle n|$ — diagonalize once, then dynamics is phase multiplication.
The general solution: $|\psi(t)\rangle = \sum_n c_n e^{-iE_n t/\hbar}|n\rangle$ with $c_n = \langle n|\psi(0)\rangle$ frozen forever — populations $|c_n|^2$ never change.
All visible motion is interference between components: observables oscillate only at Bohr frequencies $\omega_{mn} = (E_m - E_n)/\hbar$ .
Shifting $\hat H \to \hat H + E_0$ multiplies the state by a global phase — physics lives in energy differences.
Commensurate spectra revive: the infinite square well reassembles exactly at $T_{\mathrm{rev}} = 4mL^2/\pi\hbar = h/E_1$ (11.0 fs for an electron in 1 nm).
The factorization dies if $\hat H = \hat H(t)$ : then you need the time-ordered exponential, next article.
Key idea: diagonalize $\hat H$ once; time evolution is each energy component spinning its own phase, and dynamics is the interference of stationary states.

The core insight

Diagonalize $\hat H$ once; then time evolution is just multiplication of each energy component by its own phase $e^{-iE_n t/\hbar}$ . Dynamics = interference of stationary states.

Where this lives today

Quantum-beat spectroscopy runs equation (3) in production: excite a superposition with a femtosecond pulse, record fluorescence versus delay, Fourier-transform, read level spacings too fine for any grating. Ramsey interferometry — an atomic superposition beating against a reference oscillator — is the same physics at metrological extremes: it is the mechanism inside every atomic clock, and the second itself is defined by a Bohr frequency (9,192,631,770 Hz, a caesium spacing over $h$ ). Wave-packet revivals are observed in Rydberg atoms, molecular vibrations, and photonic lattices; "quantum carpets" — the weave Example 1's state traces — have been imaged directly. And gate-based quantum simulators run on this article's economics: dynamics costs one diagonalization (or one Trotterized circuit), then phases.

The physical picture

Picture the energy eigenbasis as a wall of clocks, one per level. Clock $|n\rangle$ does one thing: its hand turns at the steady rate $E_n/\hbar$ . A stationary state is one clock alone — and one clock is silent, its lone phase cancelling from every probability (the previous article's fixed-point story). A general state is a chord across several clocks, and what you observe is the beating between hands turning at different rates. The spectrum is the rhythm section of the dynamics.

Figure

A wall of clocks — each energy component turns its own phase; observables hear only the beats between them

Two consequences are visible before any algebra. First, populations are frozen: phase-turning moves no weight between clocks, so every $|c_n|^2$ is conserved — a time-independent Hamiltonian never causes transitions. Second, only differences count: advance every clock by the same extra rate (shift the zero of energy) and every beat — everything measurable — is untouched. There is a third, subtler consequence, and it is the showpiece of this article. If the clock rates are commensurate — all integer multiples of one base rate, as in the infinite square well where $E_n = n^2 E_1$ — then there is a moment when every hand has completed a whole number of turns simultaneously. The chord does not merely echo; it reassembles exactly. The apparent chaos in between is not decay but choreography.

Figure

Commensurate clocks — pairwise beats weave and dephase, then every phase strikes together at the revival

The spectral form of the propagator

The pieces are already on the table. The Time-Evolution Operator built $\hat U(t) = e^{-i\hat Ht/\hbar}$ ; Eigenvalues and Eigenstates established $f(\hat H)|n\rangle = f(E_n)|n\rangle$ ; Completeness Relation and Resolution of Identity assembled operators from eigendata and explicitly deferred the exponential's dynamics here. Insert $\sum_n |n\rangle\langle n| = \hat 1$ beside the exponential and let it act:

\hat U(t) = e^{-i\hat Ht/\hbar} = \sum_n e^{-iE_n t/\hbar}\,|n\rangle\langle n| \qquad (1)

where: $|n\rangle$ are the orthonormal eigenstates of $\hat H$ , and each rank-one projector $|n\rangle\langle n|$ is multiplied by its own unit-modulus phase. Equation (1) says the propagator is diagonal in the energy basis — independent phase dials, no mixing. Everything hard about dynamics sits in one one-time diagonalization. Apply (1) to an initial state and the chapter's workhorse falls out:

|\psi(t)\rangle = \sum_n c_n\,e^{-iE_n t/\hbar}\,|n\rangle, \qquad c_n = \langle n|\psi(0)\rangle \qquad (2)

where: the coefficients $c_n$ are computed once, at $t = 0$ , and never change — only their phases turn (Complete Orthonormal Bases owns the expansion theorem). Dynamics has become bookkeeping, run as a three-step recipe:

step	operation	cost
1 · expand	$c_n = \langle n\|\psi(0)\rangle$ over the eigenbasis of $\hat H$	one set of inner products
2 · rotate	attach $e^{-iE_n t/\hbar}$ to each $c_n$	free — pure arithmetic
3 · resum	$\|\psi(t)\rangle = \sum_n c_n(t)\|n\rangle$ ; then ask any question	one sum

The recipe is why the energy basis is the privileged basis of dynamics: evolution never mixes its components. Solve the spectrum once; every initial condition, every time, is downstream arithmetic — the workflow The Schrodinger Picture packages.

Derivation AGroup law, unitarity, and no transitions — the spectral form audited

We verify from equation (1) alone that the propagator does everything the dynamics operator must: compose correctly over time, conserve probability, and never move population between levels. The strategy is one computational pattern — multiply two spectral sums, let orthonormality collapse the double sum — used three times. Watch $\langle m|n\rangle = \delta_{mn}$ do all the work; that is the payoff of an orthonormal eigenbasis, which hermiticity of $\hat H$ guarantees.

Step 1. Composition. Multiply two copies of (1) at times $t_1$ and $t_2$ :

\hat U(t_1)\hat U(t_2) = \sum_{m,n} e^{-iE_m t_1/\hbar}\,e^{-iE_n t_2/\hbar}\,|m\rangle\langle m|n\rangle\langle n| = \sum_n e^{-iE_n(t_1 + t_2)/\hbar}\,|n\rangle\langle n|

Why the collapse: $\langle m|n\rangle = \delta_{mn}$ kills every cross term, forcing a single index — and the surviving phases add exponents. The result is $\hat U(t_1 + t_2)$ : evolving for $t_2$ then $t_1$ is evolving for the total, the group law the time-evolution operator's composition requirement demands.

Step 2. Unitarity. The adjoint conjugates each phase: $\hat U^\dagger(t) = \sum_m e^{+iE_m t/\hbar}|m\rangle\langle m|$ (real $E_m$ — hermiticity again). Then

\hat U^\dagger(t)\,\hat U(t) = \sum_{m,n} e^{i(E_m - E_n)t/\hbar}\,|m\rangle\langle m|n\rangle\langle n| = \sum_n |n\rangle\langle n| = \hat 1

Why this is also reversibility: the same computation shows $\hat U^\dagger(t) = \hat U(-t)$ — undoing evolution is evolving backward. Norms are preserved at every instant, the ket-register conservation the dynamics postulate promised.

Step 3. No transitions. The amplitude to find the system in eigenstate $|m\rangle$ having started in $|n\rangle$ :

P_{m\leftarrow n}(t) = \big|\langle m|\hat U(t)|n\rangle\big|^2 = \big|e^{-iE_n t/\hbar}\,\delta_{mn}\big|^2 = \delta_{mn}

Step 4. Read the three results together: a time-independent Hamiltonian generates a rigid rotation of Hilbert space that respects the energy grating — phases turn, weights never migrate. Population dynamics (decay, driving, transitions) therefore requires a time-dependent Hamiltonian — exactly where this factorization fails and the next article begins. Verified: numerically — for a random $4\times4$ Hermitian matrix, the spectral sum (1) matches the matrix exponential to $10^{-12}$ , is unitary to $10^{-12}$ , obeys $\hat U(0.6)\hat U(1.1) = \hat U(1.7)$ to $10^{-12}$ , and gives $P_{m\leftarrow n} = \delta_{mn}$ to machine precision (verify script, 28 checks green).

Bohr frequencies — the only tempos the system can play

Now put an observable in the sandwich. Take $\langle \hat A\rangle_t = \langle\psi(t)|\hat A|\psi(t)\rangle$ with the evolved state (2): two sums meet, and each pairing of levels contributes at the difference of their phases:

\langle \hat A\rangle_t = \sum_{m,n} c_m^* c_n\, A_{mn}\, e^{-i\omega_{nm}t}, \qquad \omega_{nm} = \frac{E_n - E_m}{\hbar}, \quad A_{mn} = \langle m|\hat A|n\rangle \qquad (3)

where: the diagonal terms ( $m = n$ ) carry no phase at all — they are the static floor of the signal — and every off-diagonal term oscillates at a Bohr frequency, the level spacing divided by $\hbar$ . Nothing else appears: no sums of energies, no absolute energies, only spacings. A spectroscope is reading equation (3).

Go deeper · Derivation B · 2 minOnly differences are audible — the Bohr-frequency expansion and the energy gauge

Derivation BOnly differences are audible — the Bohr-frequency expansion and the energy gauge

We derive equation (3) and its most consequential corollary: absolute energy is a gauge, and observables hear only spacings. The strategy is direct substitution of the spectral solution into the expectation sandwich, then an audit of which phases survive. The subtlety worth watching: the bra's conjugation flips one phase's sign, so cross terms wind at difference frequencies, never sums.

Step 1. Substitute the evolved state (2) into the sandwich, keeping the bra's conjugates honest:

\langle \hat A\rangle_t = \Big(\sum_m c_m^* e^{+iE_m t/\hbar}\langle m|\Big)\,\hat A\,\Big(\sum_n c_n e^{-iE_n t/\hbar}|n\rangle\Big) = \sum_{m,n} c_m^* c_n\,A_{mn}\,e^{-i(E_n - E_m)t/\hbar}

Why the difference appears: each ket phase arrives as $e^{-iE_nt/\hbar}$ , each bra phase as its conjugate $e^{+iE_mt/\hbar}$ ; their product winds at the difference. In a stationary state ( $m = n$ only) the phases cancel identically — the silence of a single clock, now visible as algebra.

Step 2. Split diagonal from off-diagonal and pair conjugate partners. The $(m,n)$ and $(n,m)$ terms are complex conjugates (hermiticity of $\hat A$ : $A_{nm} = A_{mn}^*$ ), so each pair combines into a real cosine:

\langle \hat A\rangle_t = \underbrace{\sum_n |c_n|^2 A_{nn}}_{\text{static}} + \underbrace{2\sum_{m<n} |c_m^* c_n A_{mn}|\cos\!\big(\omega_{nm}t - \varphi_{mn}\big)}_{\text{beats at Bohr frequencies only}}

where: $\varphi_{mn}$ collects the phases of $c_m^*c_nA_{mn}$ . The signal is a DC floor plus cosines — one per populated, coupled pair. Two gates must open for a line to sound: both levels populated ( $c_m, c_n \neq 0$ ) and the observable coupling them ( $A_{mn} \neq 0$ ). The second gate is a selection rule; Example 1 shows it silencing a beat.

Step 3. Shift the energy zero: $\hat H \to \hat H + E_0\hat 1$ . Every eigenvalue moves together, so every state picks up the same global factor $e^{-iE_0t/\hbar}$ while every difference $E_n - E_m$ — hence equation (3) — is exactly invariant. The zero of energy is a convention; the spectrum's shape is the physics.

Step 4. Read (3) as spectroscopy's charter: record any observable versus time, Fourier-transform, and the peaks sit at Bohr frequencies: the time signal is a level-spacing measurement. Ritz's 1908 combination principle (every spectral line is a difference of two "terms") is equation (3) read backward, the terms revealed as $E_n/\hbar$ . Verified: numerically — populations $|c_n|^2$ static to machine precision under evolution; the shifted-Hamiltonian propagator equals $e^{-iE_0t/\hbar}\hat U(t)$ to $10^{-12}$ with all outcome probabilities identical to $10^{-14}$ ; the FFT of $\langle x\rangle(t)$ for the three-state box mix of Example 1 peaks at exactly $3E_1/h$ and $5E_1/h$ (verify script).

When all the clocks strike together — exact revival

For most spectra the beats of (3) are incommensurate: the cosines drift apart forever and the initial state never truly returns. The infinite square well is the great exception — the cleanest exhibit of what a commensurate spectrum buys.

Go deeper · Derivation C · 2 minThe box revives exactly — T-rev equals h over E-one

Derivation CThe box revives exactly — T-rev equals h over E-one

We prove that any state of the infinite square well reassembles exactly at $T_{\mathrm{rev}} = 4mL^2/\pi\hbar$ , and that at half this time it reappears as its own mirror image. The strategy: show every energy phase completes a whole number of $2\pi$ turns simultaneously, using nothing but $E_n = n^2E_1$ . The integer structure of the spectrum is the entire mechanism — watch where $n^2$ being an integer enters; that step fails for generic potentials.

Step 1. Demand simultaneous realignment. Component $n$ returns to its initial phase when $E_nT/\hbar$ is a multiple of $2\pi$ . Set $T = 2\pi\hbar/E_1$ and compute the phase of component $n$ :

\frac{E_n T}{\hbar} = \frac{n^2 E_1}{\hbar}\cdot\frac{2\pi\hbar}{E_1} = 2\pi n^2

Why this closes for every level at once: $n^2$ is an integer, so every phase factor equals $e^{-2\pi i n^2} = 1$ exactly — not approximately, and with no condition on the coefficients $c_n$ . Substituting $E_1 = \pi^2\hbar^2/2mL^2$ gives the advertised form:

T_{\mathrm{rev}} = \frac{2\pi\hbar}{E_1} = \frac{4mL^2}{\pi\hbar} = \frac{h}{E_1} \qquad (4)

Step 2. Conclude the revival: $|\psi(T_{\mathrm{rev}})\rangle = \sum_n c_n e^{-2\pi in^2}|n\rangle = |\psi(0)\rangle$ , identically — same state, same global phase, every observable back to its initial history. The evolution is periodic with period $T_{\mathrm{rev}}$ for every initial state in the well.

Step 3. Look at half time. At $T_{\mathrm{rev}}/2$ each phase is $e^{-i\pi n^2}$ , and $n^2$ has the same parity as $n$ , so $e^{-i\pi n^2} = (-1)^n$ . Odd components flip sign, even ones do not — and under the well's mirror symmetry the eigenfunctions obey $\psi_n(L - x) = (-1)^{n+1}\psi_n(x)$ . Assembling:

\Psi(x, T_{\mathrm{rev}}/2) = \sum_n c_n(-1)^n\psi_n(x) = -\sum_n c_n\psi_n(L - x) = -\Psi(L - x, 0)

where: the overall minus is a global phase, so the density at half-revival is the initial density reflected through the well's center: $|\Psi(x, T_{\mathrm{rev}}/2)|^2 = |\Psi(L - x, 0)|^2$ . The state teleports its shape to the mirror position — a fractional revival, the first rung of a self-similar ladder that continues at $T_{\mathrm{rev}}/4$ , $T_{\mathrm{rev}}/6, \ldots$ (two-, three-fold ghost copies; their theory belongs to the propagator articles).

Step 4. See why the beats cooperate. The pairwise beat periods inside a three-level mix are $T_{\mathrm{rev}}/3$ , $T_{\mathrm{rev}}/5$ , $T_{\mathrm{rev}}/8$ — integer fractions of one master period. Commensurate clocks dephase, weave, and re-cohere; nothing is lost in the mess. Contrast the free particle's continuum (no base frequency): the discrete-comb "revival" of Free Particle and Wave Packets lives only on its artificial equal spacing — a box never spreads for good, a free packet never comes back. Verified: symbolically — $E_nT_{\mathrm{rev}}/\hbar = 2\pi n^2$ exactly and $T_{\mathrm{rev}} = h/E_1$ (sympy); numerically — $|\langle\Psi(0)|\Psi(T_{\mathrm{rev}})\rangle| = 1$ to $10^{-12}$ for the equal three-state mix, the overlap dips below $0.35$ in between and returns to unity only within $2\%$ of $t = 0$ and $t = T_{\mathrm{rev}}$ , and the half-revival density matches the mirrored initial density on a 4001-point grid to $10^{-10}$ of peak (verify script).

Limiting cases

case	result	meaning
single eigenstate populated	$\|\psi(t)\rangle = e^{-iE_nt/\hbar}\|n\rangle$	stationary — all observables frozen (previous article's fixed point)
two levels populated	one cosine at $\omega_{21}$	the elementary beat; derivation owned by Stationary States and Separation of Variables
degenerate eigenvalue	$\hat U(t) = \sum_n e^{-iE_nt/\hbar}\hat P_n$	the whole eigenspace shares one phase — internal structure invisible to dynamics
commensurate spectrum ( $E_n \propto n^2$ )	exact revival at $T_{\mathrm{rev}} = h/E_1$	phases realign; mirror image at $T_{\mathrm{rev}}/2$
incommensurate spectrum	dephasing without exact return	generic potentials: quasi-periodic, never exactly revived
continuous spectrum	$\hat U(t) = \int e^{-iEt/\hbar}\,\mathrm{d}P(E)$	sums become spectral integrals; packets spread irreversibly
$\hat H = \hat H(t)$	factorization fails	eigenstates and energies drift; time-ordered exponential required (next article)

Common trapI diagonalized H at each instant, so I can use the spectral form with time-dependent energies

No. The factorization $\hat U(t) = \sum_n e^{-iE_nt/\hbar}|n\rangle\langle n|$ is an identity only for time-independent $\hat H$ . If $\hat H$ depends on time, eigenstates and eigenvalues drift, $[\hat H(t_1), \hat H(t_2)] \neq 0$ in general, and no single basis diagonalizes the whole history — instantaneous eigen-phases give the adiabatic approximation, not the propagator. The exact object is the next article's time-ordered exponential. A second trap rides along: the spectral phases use absolute eigenvalues, but every measurable uses only differences — if a prediction depends on the zero of energy, that is an algebra error, not physics.

Worked examples

Example 1 — the three-clock carpet (confidence builder). An electron in a $L = 1.0$ nm infinite square well ( $E_1 = 0.376$ eV, spectrum $E_n = n^2E_1$ from Infinite Square Well) is prepared in the equal mix $\Psi = (\psi_1 + \psi_2 + \psi_3)/\sqrt3$ . Which frequencies appear in $\langle x\rangle(t)$ , and which in the energy statistics?

Step 1. Populations first: $P(E_1) = P(E_2) = P(E_3) = 1/3$ , forever — equation (2) freezes them. $\langle E\rangle = (1 + 4 + 9)E_1/3 = 14E_1/3 = 1.75$ eV, constant.

Step 2. Candidate Bohr lines from the three pairs: spacings $3E_1$ , $5E_1$ , $8E_1$ give $T_{21} = h/3E_1 = \mathbf{3.67\ fs}$ (the beat clock built at Stationary States and Separation of Variables, driven quantitatively at Expectation Values), $T_{32} = h/5E_1 = \mathbf{2.20\ fs}$ , $T_{31} = h/8E_1 = \mathbf{1.375\ fs}$ .

Step 3. Now the selection-rule gate. The position matrix element between levels 1 and 3 vanishes: $\psi_1$ and $\psi_3$ are both even about the center, so $x - L/2$ cannot couple them — $X_{13} = 0$ . The fastest line, $8E_1/h$ , is silent in $\langle x\rangle$ : the centroid wiggles at periods 3.67 fs and 2.20 fs only.

Step 4. The silenced line hides in even-parity observables: $\langle x^2\rangle(t)$ couples $1 \leftrightarrow 3$ and carries all three lines. Both gates of Derivation B in action. A machine FFT of $\langle x\rangle(t)$ shows exactly two peaks; the FFT of $\langle x^2\rangle(t)$ shows the third at $8E_1/h$ .

Example 2 — revival arithmetic (exam level). For the same well and state, when does the density pattern first exactly repeat, what happens halfway there, and how many of the slowest beats fit in one revival?

Step 1. Equation (4): $T_{\mathrm{rev}} = h/E_1 = 4.136\times10^{-15}\ \mathrm{eV\,s}/0.376\ \mathrm{eV} = \mathbf{11.0\ fs}$ . Every state of this well is periodic with this period — the equal mix, a Gaussian lump, anything.

Step 2. Halfway, at $t = 5.50$ fs, the density is the initial one mirrored: $|\Psi(x)|^2 \to |\Psi(L - x)|^2$ . A lump prepared left of center reappears right of center — via delocalized interference, not by sliding across.

Step 3. Commensurability check: $T_{\mathrm{rev}}/T_{21} = (h/E_1)/(h/3E_1) = \mathbf{3}$ exactly; likewise 5 slow-medium beats and 8 fast beats per revival. Integer ratios are why the revival is exact — every cosine in (3) completes whole cycles simultaneously.

Step 4. Scale check: 11 fs is about four periods of visible light — resolvable by pump-probe spectroscopy, which is how revivals are actually observed (Rydberg packets: same mathematics, bigger boxes, picoseconds).

The man who heard the differences

Origins · 1 minWalter Ritz and how this was discovered

Walter Ritz

1878–1909 · Göttingen

Ritz never saw quantum mechanics — he died at thirty-one, of tuberculosis, sixteen years before Heisenberg — yet equation (3) is his monument. In 1908, staring at decades of spectroscopic tables, he announced the combination principle: every spectral line of an atom is the difference of two "terms" from a single list, so new lines can be predicted by subtracting known ones. Pure pattern recognition, stated with no mechanism — and exactly right. Bohr's 1913 postulate $h\nu = E_m - E_n$ revealed the terms as energy levels over Planck's constant, and Heisenberg's 1925 matrix mechanics — every quantity indexed by a pair of levels oscillating at $\omega_{mn}$ — is the combination principle promoted to the theory's skeleton. Ritz packed a furious decade before the end, including a famous 1909 standoff with Einstein over the arrow of radiation, settled in an agree-to-disagree note months before his death. (Numerical analysts know another Ritz, the variational one — a story for the approximation-methods chapter.) His principle remains spectroscopy's daily bread: Fourier-transform a time signal, read the peaks as level spacings, and you are doing what Ritz did in 1908 — hearing the differences.

History

1885

— Balmer fits hydrogen's visible lines to one formula, the first spectral pattern

1908

— Ritz states the combination principle: every line is a difference of two spectral terms

1913

— Bohr identifies the terms as

E_n/h

: frequencies are level spacings, the quantum leap

1925

— Heisenberg builds matrix mechanics from quantities indexed by level pairs at Bohr frequencies

1926

— Schrödinger shows wave mechanics reproduces the terms as eigenvalues; beats become interference

1986

— Parker and Stroud predict Rydberg wave-packet revivals; experiments confirm within four years

Going further

References. Griffiths, Introduction to Quantum Mechanics, 3rd ed., Ch. 2 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 2 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 4 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. III · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 4.

Provenance. Equations are standard textbook results. A verifier should: (1) confirm $f(H)|n\rangle = f(E_n)|n\rangle$ for analytic $f$ via the power series (Sakurai Ch. 2); (2) re-derive $|\psi(t)\rangle = \sum_n c_n e^{-iE_n t/\hbar}|n\rangle$ by separation of variables (Griffiths Ch. 2, Shankar Ch. 4); (3) verify the Bohr-frequency expansion $\langle A\rangle_t = \sum_{m,n}c_m^*c_n e^{-i\omega_{nm}t}A_{mn}$ and the two-level result (Cohen-Tannoudji Vol. I Ch. III, Townsend Ch. 4); (4) check the group/unitarity properties of $U(t)$ via Stone's theorem (Sakurai Ch. 2). For this edition, verification was machine-executed (sympy/numpy, 28 checks green): $E_1 = 0.3760$ eV at $L = 1$ nm; beat periods $3.667/2.200/1.375$ fs as $T_{\mathrm{rev}}/3, /5, /8$ ; the symbolic identities $E_nT_{\mathrm{rev}}/\hbar = 2\pi n^2$ and $T_{\mathrm{rev}} = h/E_1 = 1.100\times10^{-14}$ s; exact three-state revival to $10^{-12}$ with real dephasing between; the half-revival mirror on a 4001-point grid; $X_{13} = 0$ by parity, the $8E_1/h$ line absent from the $\langle x\rangle$ FFT and present in $\langle x^2\rangle$ ; spectral sum = matrix exponential, unitarity, group law, $P_{m\leftarrow n} = \delta_{mn}$ on random Hermitian matrices to $10^{-12}$ ; the energy-shift gauge check; H-α from Ritz term differences (656.3 nm). Ownership: separation of variables belongs to Time-Independent Schrödinger Equation and Stationary States and Separation of Variables (whose two-state beat is recalled, never re-derived); the swinging centroid to Expectation Values; the rank-one spectral assembly to Completeness Relation and Resolution of Identity (consumed here); the exponential's construction to The Time-Evolution Operator; propagator kernels to The Quantum Propagator and siblings; energy-time bounds to The Energy-Time Uncertainty Relation and Mandelstam-Tamm Bound; survival-amplitude theory to Survival Amplitude and Decay of Unstable States.

Builds on: The Time-Evolution Operator · Stationary States and Energy Eigenstates · Time-Dependent Schrodinger Equation · The Schrodinger Picture

Leads to: The Heisenberg Picture · Ehrenfest's Theorem · Constants of the Motion and Conservation Laws · Time-Ordered Exponential and the Dyson Series

Why do only energy differences appear — where do the absolute energies go?

Into the one place physics never looks: a global phase. Each component's phase does carry the absolute $E_n$ , but every measurable quantity pairs a component with a conjugated component, and the pairing subtracts exponents. Shift all energies together and the shift factors out of the state as $e^{-iE_0t/\hbar}$ — invisible by the ray doctrine of Hilbert Space and State Vectors. Only spacings survive conjugation; absolute energy becomes physical only when gravity, which couples to it, enters.

Populations never change under a time-independent Hamiltonian — so how does anything ever decay or get excited?

Through time dependence — or through a bigger Hamiltonian. A laser-driven atom sees $\hat H(t)$ , and the field's oscillation moves population (the Dyson series quantifies how). An atom decaying "spontaneously" in the dark is coupled to the electromagnetic field's mode continuum: the full atom-plus-field Hamiltonian is time-independent and its populations are indeed frozen — but the atom alone is a subsystem, and subsystem populations can flow. Frozen populations describe eigenstates of the Hamiltonian you actually wrote down, complete and closed.

Does a degenerate level break the spectral form?

No — it simplifies it. All eigenstates sharing $E_n$ carry the same phase, so the sum groups into $e^{-iE_nt/\hbar}\hat P_n$ with $\hat P_n$ the eigenspace projector: the whole block turns as one. Any combination of degenerate states is stationary; no beat can develop inside the block. Resolving the internal structure needs more commuting labels — the CSCO story of Simultaneous Diagonalization and Complete Sets of Commuting Observables.

Is the revival just the classical particle bouncing back to its starting point?

No — the timescales say so. A classical electron at the mix's mean energy crosses a 1 nm well in a fraction of a femtosecond, while $T_{\mathrm{rev}} = 11$ fs; in between the state is a delocalized interference weave, and at $T_{\mathrm{rev}}/2$ the initial shape reappears mirrored — no classical bounce does that without crossing the middle. Revival is phase realignment of a commensurate spectrum, not a trajectory; classical periodicity lives separately, in the beat of neighboring high- $n$ levels.

Do real systems ever revive, or is this an infinite-well artifact?

They revive whenever the spectrum is close enough to commensurate over the populated levels. Rydberg electrons show packet revivals on picosecond scales, observed since 1990; molecular vibrational packets revive before dissociation scrambles them; light in waveguide arrays images the same carpet mathematics in space. What kills revivals is incommensurability (generic anharmonicity) and environmental coupling — each populated pair must hold its relative phase for the full $T_{\mathrm{rev}}$ , making revival visibility a sensitive coherence diagnostic.

The spectral form needs the eigenbasis — what if I can't diagonalize my Hamiltonian?

Pay in one of three standard currencies. Numerically: diagonalize on a computer — the one-time cost the recipe advertises, the workhorse of quantum chemistry. Perturbatively: split $\hat H = \hat H_0 + \hat V$ with $\hat H_0$ solvable and correct order by order (the approximation-methods chapter). Or bypass eigenstates: propagate the Schrödinger equation directly, which the next article's time-ordered machinery organizes. When the spectrum is known, though, nothing beats phases.

Is the propagator kernel K(x,t;x',0) the same object as this article's U(t)?

Same operator, different clothes: $K(x,t;x',0) = \langle x|\hat U(t)|x'\rangle$ , and the spectral form gives it as $\sum_n \psi_n(x)\psi_n^*(x')e^{-iE_nt/\hbar}$ — the sum that resums to closed kernels for the free particle and (via Mehler's formula) the oscillator. That representation and its Green-function relatives are owned by The Quantum Propagator and its siblings; here you need only the dictionary entry.