Second-Order Energy Correction

Second order is where perturbation theory starts telling you things you did not already suspect. First order was an average over a frozen state; second order is the energy of adaptation — the bill (or refund) for the wavefunction's first-order lean. Its formula is the most quoted line of the approximation chapter:

E_n^{(2)}=\sum_{m\ne n}\frac{|H'_{mn}|^2}{E_n^{(0)}-E_m^{(0)}}

Every level borrows strength from every other, weighted by coupling squared over gap — and three structural theorems ride on the formula's shape. The sign theorem: for the ground state every denominator is negative, so $E_0^{(2)}<0$ always — adaptation can only help the floor (verified on random Hamiltonians; the variational echo made precise). The repulsion theorem: each pair of levels exchanges equal and opposite second-order pushes, so close levels shove each other apart — the mechanism whose exact two-level version the chapter opener proved, and whose statistical afterlife is this article's portrait. And the staggered rule formalized: energy at second order needs the state only at first — the " $2n{+}1$ " bookkeeping that makes high-order perturbation theory affordable. The showcase is the oscillator in a uniform field, inherited from the previous serial's null result: only two terms survive the sum ( $m=n\pm1$ ), they evaluate to $E^{(2)}=-q^2\mathcal{E}^2/2m\omega^2$ for every $n$ — and completing the square shows this is the exact answer: all higher orders vanish identically (verified to $10^{-10}$ against diagonalization). The rare convergent case, solved twice. The delta-bumped well supplies the opposite lesson: its mode sum converges slowly — tail $\propto1/M$ , measured and matched to its predicted constant — because a delta couples to every level equally, and the closure shortcut that tames tamer problems honestly diverges here. Between the exact case and the stubborn case sits the working physicist's craft, and the atom-in-a-field payoff — polarizability — is stated in one line with its owner named down the chapter.

Key takeaways

THE formula: $E_n^{(2)}=\sum_{m\ne n}\dfrac{|H'_{mn}|^2}{E_n^{(0)}-E_m^{(0)}}$ — derived by projecting the tower's second floor; needs the state only at first order.
Sign theorem: ground state has all denominators negative ⇒ $E_0^{(2)}<0$ unconditionally (verified on random Hamiltonians ×3) — second order always lowers the floor; the variational reading in one line.
Repulsion: pair $(n,m)$ contributes $+|H'_{mn}|^2/\Delta_{nm}$ to one and the exact negative to the other — equal-and-opposite pushes; close levels repel (the two-level exact version consumed from the chapter opener; the concept card tells the statistics).
Oscillator + uniform field (cashing the previous serial's $E^{(1)}=0$ ): only $m=n\pm1$ contribute; $E^{(2)}=-q^2\mathcal{E}^2/2m\omega^2$ all $n$ — and completing the square proves it exact: diagonalization agrees to $10^{-10}$ , higher orders identically zero.
Delta-bumped well ( $n=1$ ): $E^{(2)}=-0.00911$ (mode sum, $\alpha=0.3$ ); PT1+PT2 vs grid-exact differ by $9\times10^{-5}=\mathcal{O}(\alpha^3)$ ✓; convergence is slow — tail after $M$ terms $=-2\alpha^2/\pi^2M$ (measured to 15%): deltas couple to everyone.
Closure/Unsöld estimate: $E^{(2)}\approx-(\langle H'^2\rangle-\langle H'\rangle^2)/\overline{\Delta E}$ — exact when one gap dominates; diverges for the delta (the sum rule's honest failure, exhibited).
Sum rule behind it: $\sum_{m\ne n}|H'_{mn}|^2=\langle H'^2\rangle_n-\langle H'\rangle_n^2$ (completeness; verified to $10^{-10}$ on random matrices).
Atomic polarizability ( $\alpha=\tfrac92a_0^3$ for hydrogen) = this formula applied to an atom in a field — stated once, owned by Quadratic Stark Effect and Polarizability.
Key idea: second order prices the wavefunction's lean — always downward for the ground state, always apart for close pairs, and exactly summable precisely when the perturbation couples to few states.

The core insight

The second-order energy $\sum_{m\ne n}|H'_{mn}|^2/(E_n^{(0)}-E_m^{(0)})$ is the energetic price of first-order mixing: every partner level pushes with strength coupling-squared-over-gap, downward on the ground state without exception, pairwise equal-and-opposite everywhere — exact for the field-displaced oscillator, slowly convergent for delta-like perturbations, and the engine behind every polarizability.

Where this lives today

Second-order energies are the quantitative backbone of atomic-clock error budgets: blackbody-radiation shifts, quadratic Zeeman shifts, and lattice-light shifts in optical clocks are all equation (1) evaluated to metrological precision — clocks good to $10^{-18}$ are second-order perturbation theory audited to that level. Quantum chemistry's dispersion forces are the same formula twice: van der Waals attraction between molecules is a mutual second-order lowering (each molecule's virtual excitations polarizing the other — the $-C_6/R^6$ law's origin, its serial-owners in the atomic-interaction lanes). And level-repulsion statistics have become an experimental instrument in their own right: microwave-billiard cavities, quantum-dot conductance, and many-body-localization experiments all measure spacing distributions against the Wigner surmise to diagnose whether their dynamics is integrable or chaotic — the portrait's theorem, in daily laboratory service.

The physical picture

First order audited a frozen cloud; second order lets the cloud flinch. The perturbation tilts the landscape, the state leans (the previous serial's mixing formula), and the leaned state samples the perturbation differently than the frozen one did — by an amount quadratic in the perturbation, since both the lean and the resampling are linear. That quadratic flinch-energy is $E^{(2)}$ , and its sign logic is mechanical: a state leans toward configurations that lower its energy when it can, so the adaptable ground state always gains. Excited states are caught in a crowd — pushed up by leaners below, down by leaners above — and the net is a tug-of-war the formula referees gap by gap.

Figure

Second order as pairwise pushes: equal and opposite, strongest across small gaps

Hold two cautions from the picture. The pushes are quadratic in coupling — weak perturbations produce very weak second-order effects, which is why $E^{(1)}=0$ situations (last serial's oscillator) have their leading physics at this small scale. And the pushes diverge as gaps close: the formula contains its own obituary, written in its denominators — near-degeneracy inflates terms beyond the perturbative budget, and the machinery must be rebuilt (Degenerate Perturbation Theory, named once here and owed everything there).

The formula and its three theorems

Climb to the tower's second floor (Non-Degenerate Time-Independent Perturbation Theory's hierarchy, consumed): the order- $\lambda^2$ equation, projected onto $\langle n^{(0)}|$ , gives $E_n^{(2)}=\langle n^{(0)}|\hat{H}'|n^{(1)}\rangle$ — the perturbation evaluated between the frozen state and the lean. Insert the first-order state of First-Order Energy and State Corrections (its formula, consumed):

E_n^{(2)}=\sum_{m\ne n}\frac{|H'_{mn}|^2}{E_n^{(0)}-E_m^{(0)}} \qquad (1)

where: hermiticity turned $H'_{nm}H'_{mn}$ into $|H'_{mn}|^2$ — manifestly non-negative numerators, so every term's sign is its denominator's: partners below push up, partners above push down. The formula needs the state only to first order — the first instance of the general staggered rule (" $2n+1$ ": the state to order $k$ determines the energy to order $2k+1$ ; stated here, used at its $k=1$ instance, not proven in general).

Derivation AProjection, the sign theorem, and pairwise repulsion

We derive (1) honestly from the tower, then prove the two structural theorems that make it quotable without computation: the ground state always drops, and level pairs always spread. Both are verified numerically on random Hamiltonians — theorems should survive adversarial matrices.

Step 1. The tower's second-order equation reads $(\hat{H}_0-E_n^{(0)})|n^{(2)}\rangle=(E^{(1)}-\hat{H}')|n^{(1)}\rangle+E^{(2)}|n^{(0)}\rangle$ . Project onto $\langle n^{(0)}|$ : the left side dies (hermiticity on the eigenbra, as at first order), and intermediate normalization kills $\langle n^{(0)}|n^{(1)}\rangle$ , leaving $E^{(2)}_n=\langle n^{(0)}|\hat{H}'|n^{(1)}\rangle$ . Substitute the mixing sum: equation (1). (Cross-check against exactly solvable truth: on the two-level model, (1) gives $-v^2/\Delta$ for the lower level — equal, symbolically, to the $\lambda^2$ coefficient of the exact eigenvalue, consumed from the chapter opener's laboratory.)

Step 2. Sign theorem. For the ground state, $E_0^{(0)}-E_m^{(0)}<0$ for every $m$ : each term of (1) is $\le0$ , so $E_0^{(2)}\le0$ , with equality only if the perturbation couples the ground state to nothing. No assumption about $\hat{H}'$ beyond hermiticity. Verified adversarially: three random Hamiltonian pairs, $E_0^{(2)}=-129.3,\,-10.6,\,-4.2$ — negative every time, as the theorem demands. Why this echoes the variational principle: the frozen evaluation $E^{(0)}+E^{(1)}$ is the energy of a trial state (the unperturbed one) under the full Hamiltonian — an upper bound for the ground level. Adaptation must therefore only lower it, and $E^{(2)}<0$ is that mandate's first installment. The bound machinery itself is The Variational Principle's, one line, owner named.

Step 3. Repulsion theorem. Isolate the pair $(n,m)$ : level $n$ receives $|H'_{mn}|^2/(E_n-E_m)$ and level $m$ receives the same numerator over the opposite denominator — equal and opposite, exactly. Second-order pushes are internal forces of the spectrum: they cancel in the trace (sum over all $E^{(2)}$ of the pushes is zero pair by pair) and they always increase each pair's separation. For well-separated levels this is a whisper; as a gap closes it becomes the dominant term of both levels — the avoided crossing, whose exact two-level anatomy the chapter opener proved and whose statistical career the portrait below recounts. Verified: two-level cross-check symbolic; random-matrix sign tests ×3; the closure sum rule of Step 1's machinery below (article verifier).

Go deeper · Derivation B · 2 minThe exactly summable case: an oscillator in a uniform field

Derivation BThe exactly summable case: an oscillator in a uniform field

We evaluate (1) for $\hat{H}'=q\mathcal{E}\hat{x}$ on the harmonic oscillator — the system the previous serial left at $E^{(1)}=0$ — and then prove the second-order answer is exact by solving the problem in closed form. One perturbation, two complete solutions, agreeing: the chapter's cleanest audit.

Step 1. The sum collapses. $\hat{x}$ connects $|n\rangle$ only to $|n\pm1\rangle$ (Ladder Operators and the Algebraic Oscillator's matrix elements, consumed): $|\langle n{+}1|\hat{x}|n\rangle|^2=(n{+}1)\hbar/2m\omega$ and $|\langle n{-}1|\hat{x}|n\rangle|^2=n\hbar/2m\omega$ . Two terms, gaps $\mp\hbar\omega$ :

E_n^{(2)}=q^2\mathcal{E}^2\Big[\frac{(n{+}1)\hbar/2m\omega}{-\hbar\omega}+\frac{n\hbar/2m\omega}{+\hbar\omega}\Big]=-\frac{q^2\mathcal{E}^2}{2m\omega^2} \qquad (2)

— the $n$ -dependence cancels between the two terms (verified symbolically): every level drops by the same amount. A uniform shift again, as with the slanted floor at first order — but now a physical one, quadratic in the field.

Step 2. The exact solve. Complete the square: $\tfrac12m\omega^2x^2+q\mathcal{E}x=\tfrac12m\omega^2(x+q\mathcal{E}/m\omega^2)^2-q^2\mathcal{E}^2/2m\omega^2$ (verified symbolically). The Hamiltonian is an identical oscillator on a displaced seat with its whole ladder lowered by exactly (2): the perturbation series terminates at second order — $E^{(3)}=E^{(4)}=\cdots=0$ identically. Numerical diagonalization agrees to $10^{-10}$ at two field strengths, three levels each (computed). Why termination happens here and almost nowhere else: the exact energy is a polynomial in the coupling (degree 2), so its Taylor series is itself. Generic perturbations produce infinite, usually asymptotic series (the chapter opener's honesty card); linear coupling to a quadratic potential is the textbook's one perfect customer. The state, note, is NOT unchanged — it is displaced, polarized at first order exactly as the previous serial's null-result discussion promised; only the energy's series is finite.

Step 3. Read (2) as physics: the induced dipole is $p=q^2\mathcal{E}/m\omega^2$ (differentiate the energy: $-\partial E/\partial\mathcal{E}$ ), linear in the field — this is polarizability, $\alpha_{\text{osc}}=q^2/m\omega^2$ , the oscillator's version of the atomic response. For real atoms the same formula (1) with Coulomb eigenstates gives the famous hydrogen value $\alpha=\tfrac92a_0^3$ — stated in this one line only: the calculation, the field-induced spectroscopy, and the polarizability concept belong BY NAME to Quadratic Stark Effect and Polarizability (the base article's hydrogen-polarizability estimate is relocated there per the chapter's application bank; edition note in the provenance). Verified: two-term sum symbolic (all $n$ ); completing the square symbolic; diagonalization to $10^{-10}$ ×6 (article verifier).

One more tool and one more honest failure. The closure (Unsöld) estimate replaces every gap by a typical $\overline{\Delta E}$ : $E^{(2)}\approx-\dfrac{\langle H'^2\rangle_n-\langle H'\rangle_n^2}{\overline{\Delta E}}$ , using the completeness sum rule

\sum_{m\ne n}|H'_{mn}|^2=\langle n|\hat{H}'^2|n\rangle-\langle n|\hat{H}'|n\rangle^2 \qquad (3)

where: the rule is pure completeness (Completeness and Resolutions of the Identity's license: insert $\sum_m|m\rangle\langle m|$ into $\langle H'^2\rangle$ — verified to $10^{-10}$ on random matrices), and the estimate is exact when a single gap dominates (the two-level case, trivially). Its honest failure: for the delta-bump well, $|H'_{m1}|^2$ does not decay with $m$ (every odd level couples equally — computed), so the left side of (3) diverges and the closure estimate is meaningless — a perturbation must be smooth for closure to be kind. The full sum (1) still converges (denominators grow like $m^2$ ), but slowly: tail after $M$ terms $=-2\alpha^2/\pi^2M$ , measured against its predicted constant at $M=41$ and $201$ (within 15%). Deltas are the worst case; smooth perturbations converge in a handful of terms.

Figure

The exact customer: a uniform field displaces the parabola and lowers the whole ladder by one fixed amount

Limiting cases

case	result	meaning
ground state	$E^{(2)}<0$ always	sign theorem — adaptation helps the floor
one dominant gap	(1) → two-level formula $-\lambda^2v^2/\Delta$	closure estimate exact; the opener's lab recovered
$\hat{H}'$ couples to few states	rapid or terminating sum	oscillator-in-field: exact at second order
delta-like $\hat{H}'$	$\|H'_{mn}\|^2$ flat in $m$ : slow $1/M$ tail	closure diverges; sum survives on denominators alone
gap $\to0$ with coupling on	term $\to\infty$	the formula's self-diagnosis → degenerate machinery
top of a finite spectrum	all denominators positive: $E^{(2)}>0$	the "anti-ground" mirror of the sign theorem

Common trapThe denominator sign convention eats more homework than any other line in this chapter

Equation (1) carries $E_n^{(0)}-E_m^{(0)}$ — the level under study minus the partner — and half the literature writes the equivalent form with $E_m^{(0)}-E_n^{(0)}$ and an overall minus sign. Mixing the two conventions flips every second-order shift. Three audits that catch the error instantly: (i) the ground state must come out negative — if your $E_0^{(2)}>0$ , the sign convention is scrambled; (ii) the two-level limit must reproduce $-v^2/\Delta$ for the lower level; (iii) pairs must repel — if your calculation attracts two levels, a denominator is upside down. Second trap: forgetting that (1) requires intermediate normalization bookkeeping upstream — with unit normalization the formula is the same but the second-order state differs, and mixing conventions mid-calculation corrupts third order. Third: summing over a continuum requires the density-of-states measure (the free-particle chapter's grammar) — treating continuum states as if discrete undercounts the push, sometimes by orders of magnitude.

Worked examples

Example 1 — the delta bump closed out (the trio's running system). $\alpha=0.3$ (units $\hbar=2m=L=1$ ), ground state.

Step 1. Assemble from the trio's parts: $E^{(1)}=2\alpha=0.600$ (previous serial); the mode sum (1) over even... — over the coupled levels only: odd $m$ , $|H'_{m1}|^2=4\alpha^2$ , denominators $\pi^2(1-m^2)$ . Summed to $m=4000$ : $E^{(2)}=-0.00911$ (computed).

Step 2. Audit against grid-exact diagonalization ( $N=3000$ ): exact shift $0.590979$ vs PT1+PT2 $0.590886$ — discrepancy $9.3\times10^{-5}$ , i.e. $\mathcal{O}(\alpha^3)$ as the tower promises (computed). Perturbation theory earns four digits on a strong-ish bump.

Step 3. Watch the convergence honestly: partial sums $-0.00456$ ( $M{=}5$ ), $-0.00730$ ( $M{=}11$ ), $-0.00866$ ( $M{=}41$ ), $-0.00903$ ( $M{=}201$ ) — crawling, tail $=-2\alpha^2/\pi^2M$ (measured within 15% at two $M$ 's). Answer: $E^{(2)}=-0.00911$ ; total shift $0.5909$ matching exact to $\mathcal{O}(\alpha^3)$ ; and a measured lesson — delta perturbations pay their second-order bill in installments of $1/M$ .

Example 2 — the exact customer, end to end. Charge $q$ in an oscillator, field $\mathcal{E}$ on.

Step 1. First order: zero by parity (previous serial's referral, cashed). Second order: two terms, equation (2): $E^{(2)}=-q^2\mathcal{E}^2/2m\omega^2$ , all levels equally (symbolic).

Step 2. Exact check: completing the square displaces the well by $x_0=-q\mathcal{E}/m\omega^2$ and lowers it by exactly $q^2\mathcal{E}^2/2m\omega^2$ (symbolic); diagonalization at $q\mathcal{E}=0.1$ and $0.3$ (natural units) reproduces $E_n=n+\tfrac12-\tfrac{f^2}{2}$ to $10^{-10}$ for $n=0,1,2$ (computed): the series terminated — orders three and up are identically zero.

Step 3. Extract the response: induced dipole $p=-\partial E/\partial\mathcal{E}=q^2\mathcal{E}/m\omega^2$ ; polarizability $\alpha_{\text{osc}}=q^2/m\omega^2$ — stiffer traps polarize less, the harmonic version of why tightly bound electrons make small polarizabilities. The hydrogen number ( $\tfrac92a_0^3$ ) follows from the same machinery on Coulomb states — its serial's business. Answer: $E^{(2)}=-q^2\mathcal{E}^2/2m\omega^2$ exactly, all higher orders vanish (verified); the response is a field-independent polarizability — second-order perturbation theory caught telling the whole truth for once.

Origins · 1 minLevel Repulsion and how this was discovered

Level Repulsion

1929–1957 · Berlin, Princeton

The pairwise-push theorem of this article has a biography of its own — it began as a curiosity of matrix algebra and ended as the spectroscopy of chaos. In 1929 John von Neumann and Eugene Wigner (whose person-cards serve elsewhere in this track — the self-adjointness band and three others; this card is the theorem's) proved the no-crossing rule: tune one parameter of a generic Hamiltonian and two levels of the same symmetry will not cross — the off-diagonal coupling would have to vanish simultaneously with the gap, two conditions on one knob. Second-order repulsion is the perturbative shadow of that theorem; the two-level square root is its exact core. The idea's second act came in 1950s Princeton, when Wigner, staring at forests of neutron-resonance levels in heavy nuclei — hundreds of levels, hopelessly beyond computation — proposed treating the Hamiltonian's matrix elements as random and asking only for statistics. The 1957 "Wigner surmise" predicted the spacing distribution of neighboring levels: small gaps suppressed linearly — repulsion, now as a probability law — in stark contrast to the clumping of uncorrelated (Poisson) spectra. It fit the nuclear data, launched random-matrix theory (Dyson's threefold-way classification extended it — his person-card serves at the series article), and eventually became the working definition of quantum chaos: integrable systems show Poisson spacings, chaotic ones show Wigner repulsion. A theorem about two levels declining to meet, scaled up to a census of thousands, now diagnoses everything from quantum dots to the zeros of the Riemann zeta function — the strangest client list any perturbation-theory corollary ever assembled.

History

1929

— von Neumann and Wigner prove the no-crossing theorem for levels of shared symmetry

1930

— Second-order (quadratic) Stark shifts computed for hydrogen-like atoms by the new wave mechanics

1955

— Wigner introduces random matrices for heavy-nucleus spectra

1957

— The Wigner surmise: level-spacing statistics with built-in repulsion

1984

— Bohigas, Giannoni and Schmit conjecture repulsion statistics as the signature of quantum chaos

Going further

Griffiths & Schroeter §7.1–7.2 — the second-order formula and the well examples at source register.
Sakurai & Napolitano §5.1 — with the closure discussion and applications forward.
Shankar Ch. 17 — the staggered ("2n+1") bookkeeping made explicit.
Mehta, Random Matrices — where the portrait's statistics became a discipline.

Provenance note: Master formula and derivation verified against Griffiths §6.1–6.2, Sakurai & Napolitano §5.1, Shankar Ch. 17, and Cohen-Tannoudji Vol. II Ch. XI. A verifier should confirm: (1) the order- $\lambda^2$ projection yielding $E_n^{(2)}=\langle n^{(0)}|H'|n^{(1)}\rangle$ ; (2) the sign argument for $E_0^{(2)}\le 0$ ; (3) the two-level expansion of $E_-$ matching the single-term formula; (4) the hydrogen ground-state polarizability $\alpha=\tfrac{9}{2}a_0^3$ (Bethe–Salpeter; Griffiths Problem 6.36). — Items (1)–(3) are executed in this article's verify script (projection cross-checked on the two-level model symbolically; sign theorem on random Hamiltonians ×3 plus the structural argument; the $-v^2/\Delta$ match; plus the exact displaced-oscillator case to $10^{-10}$ , the delta-bump mode sum against grid diagonalization with its measured $1/M$ tail, and the closure sum rule to $10^{-10}$ ). Item (4), the hydrogen polarizability, is deferred BY NAME with the whole field-response application to Quadratic Stark Effect and Polarizability per the chapter's application bank — the base article's hydrogen-polarizability derivation sketch is RELOCATED there (edition note; this article keeps the oscillator's exactly solvable polarizability as its owned exhibit and states the hydrogen number once). Edition note: the level-repulsion two-level derivation resident in the chapter opener is consumed here by citation (its laboratory, this article's theorem) — division per the chapter map, provenance notes both ends.

Where does the third-order formula fit, and why wasn't it derived?

$E^{(3)}_n=\sum_{m,k\ne n}\frac{H'_{nm}H'_{mk}H'_{kn}}{(E_n-E_m)(E_n-E_k)}-E^{(1)}_n\sum_{m\ne n}\frac{|H'_{mn}|^2}{(E_n-E_m)^2}$ — still built from the first-order state (the staggered rule's next dividend). It was exercised implicitly: for the two-level model with off-diagonal $\hat{H}'$ , both terms vanish (odd orders die, verified in the opener's laboratory). Deriving it in general adds bookkeeping, not ideas; texts that need it (quantum chemistry's MP3) quote exactly this form.

Can E⁽²⁾ ever be positive for a physical ground state?

Not for the true ground state of the full problem — the theorem is unconditional. Apparent counterexamples dissolve on inspection: a metastable state treated as "the" state (it has partners below — some denominators positive); a symmetry-restricted sector (the "ground state of the odd sector" can be pushed up by even levels only if the perturbation breaks the symmetry — but then the sectors were the wrong basis); or a sign-convention accident (the warning box's first audit). If your ground-state $E^{(2)}$ is positive, one of these three is happening.

How do continuum states enter the sum in practice?

As an integral with the density-of-states measure: $\sum_m\to\int dE\,\rho(E)$ over δ-normalized continuum states (the free-particle chapter's box-bridge grammar). For a bound level under a continuum, all continuum denominators are negative — the continuum pushes bound levels down, often dominating the sum (atomic polarizabilities take most of their value from continuum contributions, a standard trap when truncating to discrete states only). When the level sits inside a continuum, denominators change sign across resonance and the naive sum acquires a principal-value/width structure — the doorway to decay physics owned by the time-dependent chapter.

Is there a variational way to get E⁽²⁾ without summing over states?

Yes — the Dalgarno–Lewis trick, one honesty line: solve the single inhomogeneous differential equation $(\hat{H}_0-E^{(0)})|\phi\rangle=-(\hat{H}'-E^{(1)})|n^{(0)}\rangle$ for the first-order state directly (no spectrum needed), then $E^{(2)}=\langle n^{(0)}|\hat{H}'|\phi\rangle$ . The hydrogen polarizability's exact $\tfrac92a_0^3$ is classically obtained this way — in the serial that owns it. The method converts an infinite sum into one ODE: worth knowing exists before you grind a sum that has a closed form.

Why is the oscillator-in-field shift the same for every level when excited states are more polarizable in real atoms?

Because the oscillator is linear: its restoring force is field-independent and its response function has one frequency. Real atoms are anharmonic ladders — excited states are bigger, softer, closer to their neighbors (smaller denominators), so their polarizabilities grow steeply with excitation (Rydberg polarizabilities scale like $n^7$ ). The oscillator's democratic $-q^2\mathcal{E}^2/2m\omega^2$ is the exception that calibrates: whenever a computed response doesn't depend on the state, suspect an underlying harmonic structure — and vice versa.

Do the second-order pushes really cancel in the trace — and does that matter?

Pair by pair, exactly: the $(n,m)$ push enters $E^{(2)}_n$ and $E^{(2)}_m$ with opposite signs, so $\sum_nE^{(2)}_n=0$ for any finite spectrum with the full sum kept. It matters as a computational audit (a nonzero total signals dropped terms or convention mix-ups) and as physics: second order redistributes energy across a spectrum without creating any — perturbations shift the center of gravity only at first order ( $\sum E^{(1)}=\text{Tr}\,\hat{H}'$ ). Spectroscopists use exactly this: line centroids move at first order, line spreads at second.