Non-Degenerate Time-Independent Perturbation Theory

A new chapter opens with a confession: the exactly solvable problems are finished. Twenty serials of central potentials closed the list that began with the infinite square well — box, oscillator, hydrogen, free space — and almost nothing in nature is on it. An atom in an electric field, a nucleus with a non-spherical shape, an electron feeling relativity's first whisper: each is a solvable Hamiltonian plus a small correction, and the machinery for living off that "plus" is perturbation theory — the approximation chapter's foundation and quantum mechanics' single most-used computational tool. This article builds the machinery itself, deliberately stopping short of its payoffs. Write $\hat{H}=\hat{H}_0+\lambda\hat{H}'$ , expand energies and states in powers of the bookkeeping dial $\lambda$ , and the Schrödinger equation shatters into a hierarchy — one equation per order, each solvable using only $\hat{H}_0$ 's known spectrum. Set up honestly, the hierarchy demands two disclosures most first courses mumble through: a normalization choice (we prove the convenient one is legitimate), and the assumption that no other unperturbed level shares the energy — the non-degeneracy clause that gives the article its name and the next serial its mandate. The laboratory here is the humble $2\times2$ matrix: the two-level system is rich enough to exhibit everything — the series' first terms, the level repulsion that no real coupling can evade, and, decisively, the series' breakdown. The exact eigenvalues have a branch point at imaginary coupling $\lambda=\pm i\Delta/2v$ , so the perturbation series has a finite, computable radius of convergence $\Delta/2|v|$ : verified numerically, errors shrink inside and the series diverges outside — with partial sums exploding from $10^{1}$ to $10^{7}$ between orders 10 and 30 just past the radius. First-order and second-order payoff formulas are stated in single lines with their owners named — the next two serials derive and spend them; this article's product is the hierarchy, its assumptions, and an educated respect for where the expansion ends.

Key takeaways

Setup: $\hat{H}=\hat{H}_0+\lambda\hat{H}'$ with $\hat{H}_0$ solved ( $\hat{H}_0|n^{(0)}\rangle=E_n^{(0)}|n^{(0)}\rangle$ , complete); expand $E_n=\sum_k\lambda^kE_n^{(k)}$ , $|n\rangle=\sum_k\lambda^k|n^{(k)}\rangle$ ; collect powers → the hierarchy.
Order $k$ : $(\hat{H}_0-E_n^{(0)})|n^{(k)}\rangle=(E_n^{(1)}-\hat{H}')|n^{(k-1)}\rangle+E_n^{(2)}|n^{(k-2)}\rangle+\cdots$ — each order needs only lower ones: recursive, mechanical.
Two disclosed assumptions: intermediate normalization $\langle n^{(0)}|n\rangle=1$ (a legal gauge choice, proven) and non-degeneracy of $E_n^{(0)}$ (the divorce clause — its failure is Degenerate Perturbation Theory's subject, named).
Payoffs stated with owners: $E^{(1)}=\langle n^{(0)}|\hat{H}'|n^{(0)}\rangle$ → First-Order Energy and State Corrections; the sum-over-states $E^{(2)}$ → Second-Order Energy Correction.
Validity currency: $|\lambda H'_{mn}|\ll|E_n^{(0)}-E_m^{(0)}|$ — smallness is measured against gaps, pair by pair, not globally.
Two-level laboratory: exact $E_\pm=\bar{E}\pm\tfrac12\sqrt{\Delta^2+4\lambda^2v^2}$ ; series matches ( $E_-\approx E_1-\lambda^2v^2/\Delta$ , odd orders vanish — verified); gap grows monotonically with $|\lambda|$ : level repulsion is exact, not perturbative.
The series has a finite radius $|\lambda|<\Delta/2|v|$ (branch point at $\pm i\Delta/2v$ , computed): errors $0.0005/0.007/0.03$ at $\lambda=0.5/1.0/1.5$ (radius $1.67$ ), divergence beyond — measured, not asserted.
Key idea: perturbation theory is a hierarchy of solvable equations built on a solved problem — powerful exactly as far as the nearest gap, and not one step farther.

The core insight

Splitting $\hat{H}=\hat{H}_0+\lambda\hat{H}'$ and matching powers of $\lambda$ turns one unsolvable eigenproblem into a recursive tower of solvable ones, valid while coupling matrix elements stay small against unperturbed energy gaps — with the two-level system as the exactly solvable laboratory exhibiting the series' success, its level-repulsion signature, and its finite radius of convergence.

Where this lives today

Perturbation theory is the most-executed algorithm in computational science's quantum corner: quantum chemistry's MP2 (Møller–Plesset second order) — the workhorse correction beyond mean-field — is precisely this article's hierarchy applied to molecular Hamiltonians, run millions of times daily in drug and materials pipelines. Precision QED pushes the same grammar to heroic order: the electron's magnetic moment, computed through fifth order in the fine-structure constant against exactly this kind of series, agrees with experiment to twelve digits — history's most accurate perturbative prediction. And the honesty card became its own discipline: resurgence theory, an active research program, mines the divergent tails of asymptotic perturbation series for the non-perturbative physics (tunneling, instantons) hiding in them — the two-level model's imaginary branch point, grown up.

The physical picture

Think of $\hat{H}_0$ 's spectrum as a ladder bolted to a wall, and $\lambda\hat{H}'$ as a hand slowly pushing on the ladder. For small pushes, each rung shifts smoothly and each eigenstate leans slightly toward its neighbors — deformations you can track rung by rung, in powers of the push. That trackability has one enemy: rungs that touch. Where two unperturbed levels are close, an infinitesimal push mixes them completely (not slightly), and the smooth story fails. Perturbation theory is the mathematics of the smooth regime, with the gap structure of the spectrum as its map of danger zones.

Figure

The perturbation dial: a solved spectrum deforming smoothly — until rungs approach

The second picture is bookkeeping made visible: $\lambda$ is a dial, not (usually) a physical field. We install it so that powers of $\lambda$ can be collected — a filing system for corrections — and set $\lambda=1$ at the end, when "order in $\lambda$ " has done its job of meaning "order in smallness." Some perturbations carry a genuine physical dial (field strengths in the Stark and Zeeman serials ahead); for the rest, $\lambda$ is scaffolding, removed after construction.

The hierarchy

Assume the unperturbed problem is fully solved: $\hat{H}_0|n^{(0)}\rangle=E_n^{(0)}|n^{(0)}\rangle$ , with $\{|n^{(0)}\rangle\}$ a complete orthonormal set (the expansion license of Complete Orthonormal Bases, consumed) and — the standing assumption of this article — the level $E_n^{(0)}$ under study shared by no other state. Seek the deformed pair $(E_n,|n\rangle)$ as power series:

E_n=E_n^{(0)}+\lambda E_n^{(1)}+\lambda^2E_n^{(2)}+\cdots,\qquad |n\rangle=|n^{(0)}\rangle+\lambda|n^{(1)}\rangle+\lambda^2|n^{(2)}\rangle+\cdots \qquad (1)

where: the ansatz itself is the article's deepest assumption — that eigenvalues and eigenvectors are analytic in the dial near $\lambda=0$ . For finite matrices this is a theorem away from degeneracies; for realistic Hamiltonians it can fail in instructive ways (the series is often asymptotic rather than convergent — the honesty card below), and the two-level laboratory will exhibit the failure's cleanest mechanism.

Derivation ACollecting powers: the order-by-order tower and the normalization gauge

We insert the series (1) into $(\hat{H}_0+\lambda\hat{H}')|n\rangle=E_n|n\rangle$ , collect each power of $\lambda$ , and prove that the convenient normalization $\langle n^{(0)}|n\rangle=1$ is a legitimate choice rather than an extra assumption. The result is the recursive tower every later perturbation article climbs.

Step 1. Substitute and equate coefficients of $\lambda^k$ (legitimate because a power series vanishing identically has all-zero coefficients):

(\hat{H}_0-E_n^{(0)})|n^{(k)}\rangle=(E_n^{(1)}-\hat{H}')|n^{(k-1)}\rangle+E_n^{(2)}|n^{(k-2)}\rangle+\cdots+E_n^{(k)}|n^{(0)}\rangle \qquad (2)

Order zero returns the unperturbed problem (consistency); every higher order is an inhomogeneous linear equation for $|n^{(k)}\rangle$ whose right side contains only lower-order data — the tower is recursive, and climbing it is mechanical.

Step 2. The normalization gauge. The eigenvalue equation fixes $|n\rangle$ only up to a $\lambda$ -dependent complex factor. Use that freedom: demand $\langle n^{(0)}|n\rangle=1$ at every order — intermediate normalization — which by (1) says $\langle n^{(0)}|n^{(k)}\rangle=0$ for all $k\ge1$ : corrections live entirely in the orthogonal complement of the unperturbed state. Why this is legal and what it costs: legality — given any solution, multiply by the factor $1/\langle n^{(0)}|n\rangle$ (nonzero for small $\lambda$ by continuity) and the condition holds; no physics is touched because rays, not vectors, are states. Cost — the resulting $|n\rangle$ is not unit-norm beyond first order ( $\langle n|n\rangle=1+\mathcal{O}(\lambda^2)$ ); observables computed from it need a final renormalization. The alternative convention (unit norm at every order) buys a normalized state at the price of uglier equations; the two agree on all physics, and knowing that is what prevents convention-mixing errors between textbooks.

Step 3. The solvability audit — where non-degeneracy earns its place in the title. Equation (2)'s operator $(\hat{H}_0-E_n^{(0)})$ annihilates $|n^{(0)}\rangle$ : it is not invertible. An inhomogeneous equation with a non-invertible operator is solvable only if the right side is orthogonal to the offending null space — project (2) onto $\langle n^{(0)}|$ and the left side dies, leaving a solvability condition that determines $E_n^{(k)}$ at each order (its first two instances are the payoff formulas owned by the next two serials). On the orthogonal complement, the operator inverts cleanly — provided no other state shares the eigenvalue $E_n^{(0)}$ . A degenerate partner would enlarge the null space, impose additional solvability conditions the naive series cannot satisfy, and force the reorganization that is Degenerate Perturbation Theory's whole subject (named; nothing more here). Verified: the tower's first orders evaluated on the two-level model reproduce the exact eigenvalues' Taylor coefficients — including the vanishing of odd orders when $\hat{H}'$ is purely off-diagonal (article verifier, symbolic).

The payoffs, stated once each with their owners: projecting order one onto $\langle n^{(0)}|$ gives $E_n^{(1)}=\langle n^{(0)}|\hat{H}'|n^{(0)}\rangle$ — derived, interpreted, and spent in First-Order Energy and State Corrections; carrying the tower one floor higher yields the sum-over-states second-order shift — the subject, sign theorem and all, of Second-Order Energy Correction. This division is deliberate: the present article owns the machine; its two successors own the first two products. What controls the machine's accuracy? Project (2) onto an unperturbed state $\langle m^{(0)}|$ , $m\ne n$ , and every order manufactures denominators $E_n^{(0)}-E_m^{(0)}$ against numerators built from $H'_{mn}=\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle$ . The expansion's true small parameter is therefore the ratio:

\left|\frac{\lambda\,H'_{mn}}{E_n^{(0)}-E_m^{(0)}}\right|\ll1\quad\text{for every }m\ne n \qquad (3)

where: smallness is a pairwise statement measured in the local currency of gaps — a perturbation "small" against one gap can be catastrophically large against another, which is why the same field is a gentle nudge for one level and a demolition for a near-degenerate pair. There is no single dimensionless " $\lambda$ is small" in nature; there is a matrix of ratios, and validity means all of them are small.

Go deeper · Derivation B · 2 minThe two-level laboratory: series, repulsion, and the edge of convergence

Derivation BThe two-level laboratory: series, repulsion, and the edge of convergence

We solve $H=\begin{pmatrix}E_1&\lambda v\\\lambda v&E_2\end{pmatrix}$ exactly (real $v$ , gap $\Delta=E_2-E_1>0$ ), test the perturbation series against truth, extract the level-repulsion theorem, and locate — exactly — where the series stops converging. Everything checkable is checked; this model is the chapter's tuning fork.

Step 1. Exact diagonalization (characteristic polynomial of a $2\times2$ ):

E_\pm=\frac{E_1+E_2}{2}\pm\frac12\sqrt{\Delta^2+4\lambda^2v^2} \qquad (4)

(trace and determinant identities verified symbolically). Taylor-expand the lower root: $E_-=E_1-\lambda^2v^2/\Delta+\mathcal{O}(\lambda^4)$ — no first-order term (the perturbation is purely off-diagonal, so its diagonal average vanishes; indeed all odd orders vanish, verified through $\lambda^5$ ), and the second-order coefficient matches the general formula's prediction $|v|^2/(E_1-E_2)=-v^2/\Delta$ exactly.

Step 2. Level repulsion, exactly. The gap $E_+-E_-=\sqrt{\Delta^2+4\lambda^2v^2}$ is monotonically increasing in $|\lambda|$ (derivative verified symbolically): any real coupling pushes the pair apart, never together, and the levels cannot cross while $v\neq0$ . At second order this is the equal-and-opposite pairwise push the next-but-one serial will generalize; here it is exact to all orders — the no-crossing behavior whose statistical afterlife is told in that article's portrait card.

Step 3. The edge. The exact $E_\pm(\lambda)$ are branches of a square root; they collide (as analytic functions) where the radicand vanishes: $\lambda=\pm i\Delta/2v$ (computed) — off the real axis, at distance

\lambda_c=\frac{\Delta}{2|v|} \qquad (5)

A power series converges in a disk reaching the nearest singularity, so the perturbation series converges iff $|\lambda|<\Delta/2|v|$ — a finite radius set by gap over coupling, the validity currency (3) sharpened into an exact theorem for this model. Numbers ( $\Delta=1$ , $v=0.3$ , radius $1.67$ ): second-order errors $0.0005$ , $0.007$ , $0.030$ at $\lambda=0.5$ , $1.0$ , $1.5$ — creeping up as the edge nears — while just beyond the radius the series diverges: partial sums at $\lambda=2.5$ blow from error $2\times10^{1}$ at order 10 to $4\times10^{7}$ at order 30 (computed), even though the exact eigenvalues there are perfectly finite and smooth. Why the divergence is instructive rather than pathological: nothing bad happens on the real axis — the failure is caused by a singularity at imaginary coupling, invisible to the physical problem yet fatal to the series. That is the general mechanism behind divergent perturbation series in quantum mechanics: the anharmonic oscillator and the Stark problem diverge for exactly this reason (their singularities sit at negative/complex coupling, where the potential becomes unstable — prose pointers only: the quartic exhibit belongs to the next serial's examples; field problems to the Stark serials). The practical moral, stated once and honestly: perturbation series in real physics are typically asymptotic — optimal accuracy at some finite order, nonsense beyond — and the two-level model is the rare convergent case that still teaches the failure mode. *Verified: (4) via trace/determinant; series match + odd-order vanishing; monotone gap; branch points and radius; the error table; divergence/convergence of partial sums outside/inside the radius (article verifier).*

Figure

The convergence disk: a branch point at imaginary coupling caps the real-axis series

Limiting cases

case	result	meaning
$\lambda\to0$	hierarchy collapses to the solved problem	consistency floor
$H'_{nn}$ only (diagonal $\hat{H}'$ )	series terminates at first order	no mixing, no denominators — trivial exactness
off-diagonal $\hat{H}'$	$E^{(1)}=0$ ; story starts at second order	the two-level lab's structure; symmetry often enforces it
gap $\Delta\to0$ at fixed $v$	radius $\lambda_c\to0$ : series dies	degeneracy = zero convergence disk → next serial's machinery
$\hat{H}'$ commutes with $\hat{H}_0$	eigenstates unchanged; energies shift exactly by $\lambda H'_{nn}$	shared eigenbasis: perturbation theory not needed
infinite systems, generic	series asymptotic, not convergent	truncate at optimal order; honesty card, not failure

Common trapThe lambda-dial is bookkeeping — and "small perturbation" is not a property of the perturbation

Two conceptual traps that produce real errors. First: $\lambda$ is usually scaffolding. Physical statements live at $\lambda=1$ ; "first order in $\lambda$ " is code for "first order in the physically small ratio (3)." Treating $\lambda$ as a physical knob invites nonsense like "convergence for $\lambda<1$ " — the radius is $\Delta/2|v|$ , which may be far above or far below 1; the dial's numerical value means nothing until the split $\hat{H}_0+\lambda\hat{H}'$ is specified. Second, and worse in practice: smallness is a relationship between the perturbation and the spectrum's gaps. The same $\hat{H}'$ is perturbative for a level with distant neighbors and non-perturbative for a near-degenerate pair — one calculation, two regimes, and the formulas will not warn you: they simply produce huge terms (or small ones that lie). Audit the ratios (3) before trusting any perturbative number; when one approaches unity, you owe the problem the degenerate machinery, named below.

Worked examples

Example 1 — the two-level audit table. ( $\Delta=1$ , $v=0.3$ ; the laboratory run to numbers.)

Step 1. Radius first: $\lambda_c=\Delta/2v=1.667$ — the map of trustworthiness, drawn before any series is summed.

Step 2. Compare second-order series to exact (computed): at $\lambda=0.5$ , exact $-0.02202$ vs series $-0.02250$ (error $5\times10^{-4}$ ); at $\lambda=1.0$ : $-0.08310$ vs $-0.09000$ (error $7\times10^{-3}$ ); at $\lambda=1.5$ : $-0.17268$ vs $-0.20250$ (error $3\times10^{-2}$ ). The series overestimates the push (fourth order is positive here) and degrades smoothly as the edge approaches.

Step 3. Cross the edge: at $\lambda=2.5$ the exact answer is a tame $-0.29$ -ish number, but the series partial sums explode — order 10 misses by $2\times10^{1}$ , order 30 by $4\times10^{7}$ (computed). Answer: inside the radius, second order earns 3-digit accuracy at half the radius and degrades predictably; outside, more terms make it worse — the defining behavior to recognize before it recognizes you.

Example 2 — setting up the ISW with a delta bump (the chapter's running system). An infinite square well of width $L$ gains $H'=\alpha\,\delta(x-L/2)$ . Prepare the perturbative ingredients — and only the ingredients.

Step 1. Matrix elements from the known well states $\psi_n=\sqrt{2/L}\sin(n\pi x/L)$ (Infinite Square Well's, consumed): $H'_{mn}=\alpha\psi_m(L/2)\psi_n(L/2)=\tfrac{2\alpha}{L}\sin\tfrac{m\pi}{2}\sin\tfrac{n\pi}{2}$ — computed exactly: $H'_{11}=2\alpha/L$ ; $H'_{22}=0$ (the $n=2$ state has a node at the bump — it cannot feel it); $H'_{13}=-2\alpha/L$ (odd-odd coupling with an honest sign); $H'_{12}=0$ (parity-forbidden: even states decouple entirely).

Step 2. Audit validity: the relevant ratios are $|H'_{m1}|/(E_1-E_m)\sim\tfrac{2\alpha/L}{(m^2-1)E_1}$ — small for $\alpha\ll LE_1$ , shrinking like $1/m^2$ down the tower: the series is safe, and dominated by its first few terms.

Step 3. Stop, by design. The evaluations — $E^{(1)}_n$ for the slanted and bumped well, the mixed wavefunction, the second-order sum with its convergence-in- $m$ table — are the worked examples of the next two serials, which take exactly this prepared system as input. Answer: $H'_{mn}=\tfrac{2\alpha}{L}\sin\tfrac{m\pi}{2}\sin\tfrac{n\pi}{2}$ ; even states immune, odd states coupled at fixed magnitude with alternating signs; ratios audited small — the machine is loaded, and the successors fire it.

Origins · 1 minLord Rayleigh and how this was discovered

Lord Rayleigh

1842–1919 · Cambridge and Terling Place, England

The "Rayleigh" in Rayleigh–Schrödinger perturbation theory was earned four decades before quantum mechanics existed. (His wave-mechanics side — the group-velocity arithmetic that de Broglie's matter waves inherited — is a different portrait in this track's free-particle chapter; today's card is about mistuned bells.) In The Theory of Sound (1877–78, written partly on a houseboat on the Nile while recovering from rheumatic fever), Rayleigh posed the working musician's question in mathematician's dress: a string, a membrane, an organ pipe is almost uniform — how do its frequencies shift? His §90 answer is, to the letter, this article's method: expand the vibration problem about the solved uniform case, in powers of the non-uniformity; the first-order frequency shift is the perturbation averaged against the unperturbed mode shape — $\langle n^{(0)}|\hat{H}'|n^{(0)}\rangle$ with nineteenth-century typography — and the mode-shape correction is a sum over the other modes with resonance denominators. When Schrödinger built wave mechanics' perturbation theory in his third 1926 paper (computing the Stark effect of hydrogen — that payoff belongs to serials ahead), he knew exactly whose machinery he was porting and said so in the text, crediting the method to Rayleigh's Theory of Sound — the citation that welded the double name. Rayleigh's own second act — the meticulous density measurements that unmasked argon and won the 1904 Nobel — showed the same signature habit: treat every small discrepancy as a series to be pushed one order further.

History

1877

— Rayleigh's Theory of Sound derives frequency shifts of nearly uniform vibrating systems: the method is born acoustic

1926

— Schrödinger's third paper ports Rayleigh's expansion into wave mechanics, crediting him by name

1929

— von Neumann and Wigner prove the no-crossing theorem — the repulsion this article's laboratory exhibits exactly

1948

— Feynman, Schwinger and Tomonaga build QED's renormalized perturbation series on the same order-by-order grammar

1952

— Dyson argues QED's series is divergent-asymptotic — the honesty card, elevated to field theory

Going further

Griffiths & Schroeter Ch. 7 — the canonical first pass (Eqs. 7.9, 7.13, 7.15 are this article's tower projected).
Sakurai & Napolitano §5.1 — the same machinery at graduate register, with the two-level model as touchstone.
Shankar Ch. 17 — careful on normalization conventions and the degenerate danger.
Kato, Perturbation Theory for Linear Operators — where the analyticity questions of the honesty card live rigorously.

Provenance note: Equations should be checked against Griffiths & Schroeter Ch. 7 (Eqs. 7.9, 7.13, 7.15), Sakurai & Napolitano §5.1, and Shankar Ch. 17. Verifier: confirm the Rayleigh–Schrödinger first/second/third-order formulas, the sign of the ground-state second-order shift, and the two-level square-root expansion. Standard numbers cited: hydrogen ground-state polarizability $\alpha=\tfrac92 a_0^3$ (Gaussian units, Bethe-Salpeter / Griffiths) and the van der Waals $-C_6/R^6$ scaling. No nonstandard numerical values are introduced. — The two-level square-root expansion, the vanishing odd orders, the branch-point radius, and the convergence/divergence behavior are executed in this article's verify script (symbolic + numeric error table). The first-order formula's derivation and interpretation are owned BY NAME by First-Order Energy and State Corrections; the second-order formula, its ground-state sign theorem, and third-order structure by Second-Order Energy Correction (both consume this article's tower by citation). The provenance's quoted standard numbers — the hydrogen polarizability $\tfrac92a_0^3$ and the van der Waals $C_6$ scaling — are applications deferred BY NAME to Quadratic Stark Effect and Polarizability (which owns polarizability per the chapter map) and to the atomic-interactions serials; they are stated here for the record and neither derived nor verified in this article. Edition note: the base's level-repulsion derivation is retained here (the two-level laboratory); its general second-order form reappears, with the pairwise-push theorem, in Second-Order Energy Correction, cited both ways.

Is λ ever physical, and does setting λ = 1 at the end lose anything?

Sometimes physical (an applied field's strength in the Stark/Zeeman serials is a genuine dial), usually scaffolding. Nothing is lost at $\lambda=1$ provided the physical smallness (3) holds there — the series in λ is really a series in those ratios, and λ just counts their powers. When people say "second-order perturbation theory," they mean the $\lambda^2$ term evaluated at $\lambda=1$ : the dial's job was to sort terms, not to be small itself.

Why expand both the energy AND the state — couldn't I get energies alone more cheaply?

At the lowest orders you almost can: the energy at order $k$ needs the state only to order $k-1$ (and, by a theorem stated in the third-order context of the second-order serial, order $\lfloor k/2\rfloor$ suffices — the "2n+1 rule"). But the state corrections are not optional bookkeeping: transition rates, expectation values, and selection-rule breakdowns all live in the mixed wavefunction. The tower delivers both because physics needs both.

What actually goes wrong, mechanically, if two levels are degenerate?

The order-one equation demands $(\hat{H}_0-E^{(0)})|n^{(1)}\rangle=(E^{(1)}-\hat{H}')|n^{(0)}\rangle$ have a solution. Project onto the degenerate partner $|m^{(0)}\rangle$ (same $E^{(0)}$ ): the left side vanishes identically, forcing $H'_{mn}=0$ — a condition on the perturbation, generally false. The naive ansatz has assumed the perturbed state tends to $|n^{(0)}\rangle$ itself, when really it tends to some combination within the degenerate subspace that diagonalizes $\hat{H}'$ there. Choosing that combination first is the entire fix, and the next serial's subject.

How do I know how many orders to keep in practice?

For convergent series: until the next term is below your accuracy target — and the two-level radius warns that "next term smaller" must hold with margin. For asymptotic series (the generic case): terms first shrink, pass a minimum near order $\sim1/(\text{smallness})$ , then grow; truncate at the smallest term for exponentially good accuracy, and stop believing the series beyond. Practical QM applications (fine structure, Stark, Zeeman ahead) sit at such small coupling that first or second order is already beyond experimental resolution — the luxury regime this chapter mostly inhabits.

Does perturbation theory preserve orthogonality between different perturbed eigenstates?

Yes, automatically, order by order: eigenvectors of a hermitian operator with distinct eigenvalues are orthogonal, and the series construction respects it — one can verify $\langle m|n\rangle=\mathcal{O}(\lambda^2)$ cancellations happen exactly (a good exercise against the two-level model, where exact eigenvectors are available). What is not preserved by intermediate normalization is unit norm of each state separately — the budgeted cost of Step 2's gauge.

Is there a variational safety net under perturbation theory?

A partial one, and the chapter ahead builds it: the variational principle (its own serial) gives certified upper bounds on the ground-state energy, against which perturbative estimates can be checked — and indeed first-order perturbation theory is a variational evaluation with the unperturbed state as trial function, which is why $E^{(0)}+\lambda E^{(1)}$ always overestimates the true ground energy (the second-order term, negative for the ground state by the sign theorem ahead, is the first repair). When series and bound disagree beyond budget, believe the bound.