Eigenvalues and Eigenstates

Ask a quantum system a question and you get a number. Measure the energy of a hydrogen atom: −13.6 eV, or −3.4 eV, or one of a known ladder of values — never −7.2 eV. Who wrote the menu? The answer is an equation that looks almost too simple to matter: $\hat A|a\rangle = a|a\rangle$ . Feed an operator a very particular state and it hands the state back untouched, merely multiplied by a number. Those special states are the eigenstates, the numbers are the eigenvalues, and the eigenvalues are the menu — the complete list of readings any measurement of $\hat A$ can ever return. You have met the equation in wave-mechanics clothes: Time-Independent Schrödinger Equation read $\hat H\psi = E\psi$ as "operator in, number out," and Operators and Observables proved that an eigenstate answers with its eigenvalue every time. The Formalism chapter adds the machinery to solve it in any basis — the characteristic polynomial, worked here on the two-level system whose levels repel rather than cross — and the structure that makes solutions usable: functions of an operator inherit its eigenstates, and a degenerate eigenvalue owns a whole subspace. Mastering this buys the computational heart of the subject: diagonalize the Hamiltonian and dynamics becomes trivial; diagonalize any observable and its statistics fall into your lap. Everything downstream — bases, projectors, the Born rule, the postulates — is built on this equation and the vocabulary installed here.

Key takeaways

An eigenstate of $\hat A$ is a nonzero vector that $\hat A$ merely rescales: $\hat A|a\rangle = a|a\rangle$ . The scalar $a$ is its eigenvalue; the set of all eigenvalues is the operator's spectrum.
The spectrum is the measurement menu: a measurement of the observable $A$ can return an eigenvalue and nothing else. Eigenstates are the states of certainty — prepared in $|a\rangle$ , the apparatus reads $a$ with zero spread, every time.
In finite dimensions the eigenvalues are the roots of the characteristic equation $\det(\hat A - \lambda\hat 1) = 0$ . For the two-level Hamiltonian with coupling $g$ , the levels are $\bar\varepsilon \pm \sqrt{(\Delta/2)^2 + g^2}$ — they repel, never crossing while $g \neq 0$ .
Functions of an operator inherit its eigenstates: $f(\hat A)|a\rangle = f(a)|a\rangle$ . This one line is why $e^{-i\hat Ht/\hbar}$ is easy in the energy eigenbasis.
A degenerate eigenvalue owns a subspace, not a state: every linear combination of its eigenvectors is again an eigenvector, and any orthonormal basis of the eigenspace is equally valid.
Key idea: the eigenvalue equation $\hat A|a\rangle = a|a\rangle$ singles out the states of definite value; its eigenvalues are the only possible outcomes of a measurement, and solving it — diagonalization — is the central computation of quantum mechanics.

The core insight

The eigenvalue equation $\hat A|a\rangle = a|a\rangle$ identifies the states of definite value: for a Hermitian observable the eigenvalues are the possible measurement outcomes, and the eigenvalue problem — finding the menu and the states that pin each entry — is the master calculation of the theory.

Where this lives today

Level repulsion is a daily diagnostic: superconducting-qubit labs measure coupling strengths by tuning two circuits through resonance and reading the minimum gap off the spectroscopy chart — this article's $2g$ is literally a fit parameter in published figures, $g/2\pi$ quoted in megahertz. The same anticrossing physics decides where molecular potential-energy surfaces exchange character, how photons and excitons hybridize into polaritons, and why a Landau–Zener sweep through the gap can flip a qubit. Computationally, "diagonalize the Hamiltonian" may be the most-executed instruction in physical science: quantum-chemistry packages spend most of their cycles in eigensolvers, band structures are eigenvalue menus plotted against crystal momentum, and Lanczos-descended sparse solvers hunt eigenstates in $10^{10}$ -dimensional model magnets. Even quantum computers pay tribute — variational quantum eigensolvers exist because the ground-state entry of the menu is the number chemistry wants most.

The physical picture

Watch what a linear operator does to vectors: almost every input comes back stretched and reoriented. A few privileged directions come back pointing exactly where they pointed — only their length (or phase) changed. Those are the eigenvectors; the rescaling factor is the eigenvalue. The operator, complicated in general, is utterly simple along its own axes.

Figure

Most vectors turn — eigenvectors only stretch

Why should physics care about a linear-algebra curiosity? Because Operators and Observables proved the dictionary — an eigenstate of $\hat Q$ answers a $Q$ -measurement with its eigenvalue, with certainty — and Standard Deviation and Quantum Uncertainty proved the converse: zero spread happens only in eigenstates. Definite values and eigenstates are the same thing, exactly. The eigenvalue problem is not a warm-up before the physics; it is the physics. "What energies can this molecule have?" asks for the spectrum of $\hat H$ ; "what states have them sharply?" asks for its eigenstates. The second picture is the menu. An operator's spectrum is the complete list of readings its measurement can produce: the ladder $E_n = n^2 E_1$ for Infinite Square Well, the even rungs of Quantum Harmonic Oscillator: Analytic Solution, a continuum for position. A general state is not on the menu — it is a superposition of menu items, and which item a measurement serves is statistical. But the menu is fixed by the operator alone, before any state is chosen. That division of labor — operator fixes the possible answers, state fixes their probabilities — is the deepest structural fact in the subject, and the probability half is written two serials ahead, in Born Rule and Measurement Probabilities.

The figure previews the flagship calculation. Couple two bare levels $\varepsilon_1$ , $\varepsilon_2$ by anything of strength $g$ — tunnelling, a field, a perturbation — and the eigenvalues bend away from the bare lines: the levels repel, closing no nearer than $2g$ . Every avoided crossing in molecular spectra or on a qubit tuning chart is this two-by-two problem wearing laboratory clothes.

The eigenvalue equation, read as physics

Here is the chapter's central equation, in the notation Dirac Bra-Ket Notation installed:

\hat A\,|a\rangle = a\,|a\rangle \qquad (1)

where: $\hat A$ — a linear operator (an observable when Hermitian); $|a\rangle$ — an eigenket, required nonzero; $a$ — its eigenvalue, a scalar. Read it as a promise: along this one direction the operator collapses to ordinary multiplication. We name the ket by the value it carries: " $|a\rangle$ " is the state on which $\hat A$ reads $a$ . Three facts arrive with equation (1). If $|a\rangle$ is an eigenket then so is $c|a\rangle$ , with the same eigenvalue — eigenstates are rays, per Hilbert Space and State Vectors. The certainty theorem (a system in $|a\rangle$ returns $a$ on every measurement) was proved by Operators and Observables, and its converse — zero spread forces an eigenstate — by Standard Deviation and Quantum Uncertainty; we lean on the pair without re-proving it. And for Hermitian $\hat A$ , eigenvalues are real and eigenkets of distinct eigenvalues orthogonal — the three-theorems package of Hermitian Operators and Observables, quoted here as settled law. What this article owns is the question those theorems leave open: how do you actually find the eigenvalues and eigenstates, and what structure do the solutions have — degeneracy, functions of operators, the anatomy of the spectrum?

Suppose the space is finite-dimensional — a spin, a two-level atom, a truncated model — so $\hat A$ is a matrix in a basis (the representation machinery of Linear Operators on Hilbert Space). Move everything in (1) to one side:

\big(\hat A - \lambda\,\hat 1\big)\,|a\rangle = 0 \qquad (2)

where: $\hat 1$ — the identity operator; $\lambda$ — a candidate eigenvalue. This homogeneous system always has the boring solution $|a\rangle = 0$ ; it has a nonzero one precisely when the matrix fails to be invertible, and non-invertibility has a scalar detector — the determinant:

\det\big(\hat A - \lambda\,\hat 1\big) = 0 \qquad (3)

where: the left side, expanded, is a degree- $N$ polynomial in $\lambda$ — the characteristic polynomial. Its roots are the eigenvalues, all of them, with multiplicity: an $N$ -dimensional system has an $N$ -entry menu counted with repeats. Equation (3) turns "find the spectrum" into "find polynomial roots" — why two-level problems reduce to quadratics, and why your linear-algebra course was secretly a measurement course.

Derivation AThe characteristic equation and the avoided crossing

We solve the most important eigenvalue problem in quantum practice: the coupled two-level system. The goal is threefold — run the characteristic-equation machine end to end on a real case, derive the level-repulsion law behind every anticrossing in spectroscopy and qubit engineering, and extract the mixing angle that says how eigenstates interpolate between the uncoupled basis states. The route: write the matrix, expand the determinant, solve the quadratic, feed each root back for its eigenvector. Watch the discriminant — the physics of "levels cannot cross" lives inside that square root.

Step 1. Set up the matrix. In the basis $\{|1\rangle, |2\rangle\}$ of two uncoupled states with energies $\varepsilon_1$ and $\varepsilon_2$ , a real coupling $g$ connects them:

\hat H = \begin{pmatrix} \varepsilon_1 & g \\ g & \varepsilon_2 \end{pmatrix}

Why this matrix is everywhere: two wells joined by tunnelling, two levels dressed by a field, two qubit states near an anticrossing — any pair with one coupling channel gives exactly this Hermitian form.

Step 2. Expand the characteristic determinant:

\det\big(\hat H - \lambda\hat 1\big) = (\varepsilon_1 - \lambda)(\varepsilon_2 - \lambda) - g^2 = \lambda^2 - (\varepsilon_1 + \varepsilon_2)\lambda + \varepsilon_1\varepsilon_2 - g^2 = 0

Why the $-g^2$ matters: with $g = 0$ the roots are just $\varepsilon_1$ and $\varepsilon_2$ — the bare menu. The coupling term is what bends it.

Step 3. Solve the quadratic. With the mean $\bar\varepsilon = (\varepsilon_1 + \varepsilon_2)/2$ and the detuning $\Delta = \varepsilon_2 - \varepsilon_1$ :

E_\pm = \bar\varepsilon \pm \sqrt{\left(\frac{\Delta}{2}\right)^2 + g^2}

The gap between the two eigenvalues is $E_+ - E_- = \sqrt{\Delta^2 + 4g^2}$ . At fixed $g$ , sweep the detuning: the gap is smallest at $\Delta = 0$ , where it equals $2g$ — and it can never reach zero while $g \neq 0$ . That is level repulsion: coupled levels avoid each other, and the size of the avoided crossing measures the coupling.

Step 4. Feed each root back into (2) to get the eigenvectors. The algebra condenses into one angle, defined by $\tan 2\theta = 2g/\Delta$ :

|E_-\rangle = \cos\theta\,|1\rangle - \sin\theta\,|2\rangle, \qquad |E_+\rangle = \sin\theta\,|1\rangle + \cos\theta\,|2\rangle

Why the angle is the right variable: at $\Delta \gg g$ , $\theta \to 0$ — essentially bare states; at $\Delta = 0$ , $\theta = 45°$ — fifty-fifty mixtures. The eigenstates rotate smoothly from uncoupled to maximally hybridized as the levels approach. *Verified: sympy expands the determinant to the quadratic above and solves it to $E_\pm = \bar\varepsilon \pm \sqrt{(\Delta/2)^2 + g^2}$ exactly; numpy diagonalization at $\varepsilon_1 = 1.0$ , $\varepsilon_2 = 2.0$ , $g = 0.3$ eV returns 0.917 and 2.083 eV with orthonormal eigenvectors matching the $\theta = 15.5°$ forms to $10^{-12}$ ; a 401-point detuning sweep confirms the minimum gap is exactly $2g$ .*

The numbers arrive in Worked example 1. First, two structural theorems that make eigenvalue problems compose.

Functions of an operator inherit its eigenstates

You will constantly need not just $\hat A$ but operators built from it: $\hat A^2$ for variances, $\hat A^{-1}$ for resolvents, and above all $e^{-i\hat Ht/\hbar}$ for dynamics. The beautiful fact is that the eigenwork is already done.

Go deeper · Derivation B · 1 minFunctions of an operator keep the same eigenstates

Derivation BFunctions of an operator keep the same eigenstates

We show that if $\hat A|a\rangle = a|a\rangle$ , then any function of $\hat A$ definable by powers satisfies $f(\hat A)|a\rangle = f(a)|a\rangle$ — same eigenstates, transformed menu. The strategy is the only one available for "a function of an operator": define $f(\hat A)$ through the power series of $f$ , act term by term, and watch the operator series reassemble into a numerical one. The payoff outruns the effort — this argument is why the energy eigenbasis trivializes time evolution, and it quietly powers half the Formalism chapter.

Step 1. Powers first. Apply $\hat A$ twice to an eigenket:

\hat A^2|a\rangle = \hat A\big(a|a\rangle\big) = a\,\hat A|a\rangle = a^2|a\rangle

and by induction $\hat A^n|a\rangle = a^n|a\rangle$ for every $n$ . Why linearity is doing the work: the scalar $a$ slides out past the operator, so each application just deposits another factor of $a$ .

Step 2. Series next. For $f$ with a power series $f(z) = \sum_n f_n z^n$ , define $f(\hat A) \equiv \sum_n f_n \hat A^n$ . Acting on the eigenket:

f(\hat A)\,|a\rangle = \sum_n f_n \hat A^n|a\rangle = \Big(\sum_n f_n a^n\Big)|a\rangle = f(a)\,|a\rangle

The operator series, evaluated on an eigenstate, collapses to the ordinary numerical series. Why this is the whole content of "diagonalize once, compute forever": in its own eigenbasis, any function of $\hat A$ is just the function applied entry by entry to the menu.

Step 3. Cash the flagship case. Take $\hat H$ with eigenvalues $E_n$ and $f(z) = e^{-izt/\hbar}$ :

e^{-i\hat Ht/\hbar}\,|E_n\rangle = e^{-iE_nt/\hbar}\,|E_n\rangle

Each energy eigenstate simply spins a phase — the stationary-state clock of Stationary States and Separation of Variables, now derived in one line from operator structure rather than from the differential equation. Inverses ride along too: $\hat A^{-1}$ exists precisely when no eigenvalue is zero, and its menu is $\{1/a\}$ . Verified: numpy/scipy at $\varepsilon_1 = 1.0$ , $\varepsilon_2 = 2.0$ , $g = 0.3$ eV — $e^{-i\hat Ht/\hbar}$ is unitary to $10^{-12}$ and acts on each eigenvector as the pure phase $e^{-iE_\pm t/\hbar}$ ; $\hat H^2$ has eigenvalues exactly $E_\pm^2$ and $\hat H^{-1}$ exactly $1/E_\pm$ .

Nothing in equation (3) forbids repeated roots. When an eigenvalue $a$ appears $g_a$ times, the physics acquires a new layer of structure — and a new freedom.

Go deeper · Derivation C · 1 minA degenerate eigenvalue owns a subspace, not a state

Derivation CA degenerate eigenvalue owns a subspace, not a state

We prove that the eigenvectors belonging to one eigenvalue form a vector space of their own — the eigenspace — so a $g_a$ -fold degenerate eigenvalue comes with a $g_a$ -dimensional plane of equally valid eigenstates, not a single direction. The proof is two lines of linearity, but the consequence is a real physical freedom: no measurement of $\hat A$ can distinguish one frame of the eigenspace from another. Watch what the argument does not deliver: orthogonality inside the eigenspace is a choice you impose, not a fact you inherit.

Step 1. Closure under superposition. Let $\hat A|u\rangle = a|u\rangle$ and $\hat A|v\rangle = a|v\rangle$ share one eigenvalue. For any scalars $\alpha, \beta$ :

\hat A\big(\alpha|u\rangle + \beta|v\rangle\big) = \alpha\,a|u\rangle + \beta\,a|v\rangle = a\big(\alpha|u\rangle + \beta|v\rangle\big)

Every superposition is again an eigenvector with the same $a$ . Together with the zero vector, the $a$ -eigenvectors form a subspace; its dimension $g_a$ is the degeneracy (for a Hermitian matrix, equal to the multiplicity of the root $a$ in the characteristic polynomial). Why this fails across different eigenvalues: mixing $|a\rangle$ and $|a'\rangle$ with $a \neq a'$ gives $\hat A(\alpha|a\rangle + \beta|a'\rangle) = \alpha a|a\rangle + \beta a'|a'\rangle$ , which is not a multiple of the input — superpositions across the menu are not eigenstates, and that is precisely why measurement statistics exist.

Step 2. The freedom, exhibited. Take the $4$ -eigenspace of the matrix in Worked example 2, spanned by $|u_1\rangle = \tfrac{1}{\sqrt2}(1,1,0)$ and $|u_2\rangle = (0,0,1)$ . The remixed pair $|w_\pm\rangle = \tfrac{1}{\sqrt2}(|u_1\rangle \pm |u_2\rangle)$ is also an orthonormal basis of the same plane, and every member is still an exact eigenvector with eigenvalue 4. Nature expresses no preference; any orthonormal frame of the eigenspace serves.

Step 3. The repair, and its owner. Orthogonality is guaranteed only across distinct eigenvalues; inside an eigenspace it is manufactured — Gram–Schmidt, systematized in next serial's Complete Orthonormal Bases — or imposed physically by a second commuting observable that splits the degeneracy, the doctrine deferred by name to Simultaneous Diagonalization and Complete Sets of Commuting Observables. Verified: numpy — for $A$ with eigenvalues $\{2, 4, 4\}$ , five random complex combinations $\alpha|u_1\rangle + \beta|u_2\rangle$ all satisfy $A x = 4x$ to $10^{-10}$ ; the remixed pair $|w_\pm\rangle$ is orthonormal and still eigenvectors to $10^{-12}$ .

Where the spectrum lives: assembled forms and the continuous frontier

With eigenvalues and orthonormal eigenstates in hand, a Hermitian operator reassembles from its menu — the spectral form

\hat A = \sum_n a_n\,|a_n\rangle\langle a_n| \qquad (4)

where: $|a_n\rangle\langle a_n|$ — the outer-product machinery of Dirac Bra-Ket Notation, which already wielded this form for spin. The licensing theorems live ahead: the eigenbasis's completeness with Complete Orthonormal Bases, the assembly derivation with Completeness Relation and Resolution of Identity. Read (4) as the destination: an observable is its menu of outcomes, each glued to the state that pins it. One honesty clause. Position and momentum have gapless menus — continuous spectra — and their "eigenstates" ( $\delta$ -spikes, plane waves; Position Operator, Momentum Operator) are not normalizable vectors at all. The eigenvalue language survives, but sums become integrals and the rigor bill is paid in Continuous Spectra and Delta Normalization and Rigged Hilbert Space. This article is the discrete half of the story.

Limiting cases

case	result	meaning
coupling off ( $g \to 0$ )	$E_\pm \to \varepsilon_{1,2}$ , $\theta \to 0$	bare menu recovered; the machine reduces to reading the diagonal
zero detuning ( $\Delta = 0$ )	gap $= 2g$ , $\theta = 45°$	maximal hybridization: eigenstates are fifty-fifty mixtures; the anticrossing bottom
large detuning ( $\Delta \gg g$ )	$E_\pm \approx \varepsilon_{2,1} \pm g^2/\Delta$	levels shift only at second order in $g$ — the seed of perturbation theory
degenerate root ( $g_a > 1$ )	eigenspace of dimension $g_a$	a plane of valid eigenstates; orthonormal frame is a choice, symmetry is the cause
$f(\hat A)$ for any power series	same eigenstates, menu $f(a_n)$	diagonalize once, compute every function of the operator forever
eigenvalue $0$ present	$\hat A^{-1}$ does not exist	the determinant vanishes; the operator annihilates a direction

Worked examples

Example 1 — the avoided crossing with numbers (confidence builder). Two coupled states have bare energies $\varepsilon_1 = 1.0$ eV and $\varepsilon_2 = 2.0$ eV, coupling $g = 0.3$ eV. Find the measurable energies, the eigenstates, and the statistics if the system is prepared in the bare state $|1\rangle$ .

Step 1. The menu. Mean $\bar\varepsilon = 1.5$ eV, detuning $\Delta = 1.0$ eV, radical $\sqrt{(0.5)^2 + (0.3)^2} = 0.583$ eV:

E_- = 1.5 - 0.583 = \textbf{0.917 eV}, \qquad E_+ = 1.5 + 0.583 = \textbf{2.083 eV}

The gap is $1.166$ eV — wider than the bare splitting of $1.0$ eV. Repulsion in action.

Step 2. The eigenstates. $\tan 2\theta = 2g/\Delta = 0.6$ , so $\theta = \textbf{15.5°}$ : $|E_-\rangle = \cos\theta|1\rangle - \sin\theta|2\rangle$ , $|E_+\rangle = \sin\theta|1\rangle + \cos\theta|2\rangle$ — mostly bare, with a 27%-amplitude admixture.

Step 3. Statistics of the bare state. Prepared in $|1\rangle$ , a measurement returns a menu value, never 1.0 eV: $P(E_-) = \cos^2\theta = \textbf{0.929}$ , $P(E_+) = \sin^2\theta = \textbf{0.071}$ (amplitude → probability is the Born rule, quantified in Born Rule and Measurement Probabilities).

Step 4. Two consistency dividends. The probability-weighted mean is $0.929 \times 0.917 + 0.071 \times 2.083 = \textbf{1.000 eV}$ — exactly $\varepsilon_1$ , as the diagonal matrix element demands. And the spread is $\sigma_E = \textbf{0.3 eV} = g$ exactly ( $\sigma_E^2 = (H^2)_{11} - H_{11}^2 = g^2$ ): the coupling is the energy uncertainty of a bare state.

Example 2 — a degenerate menu, organized (exam level). For the observable $\hat A = \begin{pmatrix} 3 & 1 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}$ (some energy-like quantity in eV on a three-state system), find the spectrum, exhibit the degeneracy freedom, and give the menu of $\hat A^2$ .

Step 1. Characteristic polynomial. The block structure factorizes it: the upper $2 \times 2$ block gives $(3-\lambda)^2 - 1 = 0$ , i.e. $\lambda = 2$ or $4$ ; the corner gives $\lambda = 4$ . Spectrum: $\{\textbf{2, 4, 4}\}$ — a two-entry menu with the 4 doubly degenerate.

Step 2. Eigenvectors. For $\lambda = 2$ : $\tfrac{1}{\sqrt2}(1, -1, 0)$ , unique up to phase. For $\lambda = 4$ : both $\tfrac{1}{\sqrt2}(1, 1, 0)$ and $(0, 0, 1)$ work — and so does every combination of them. The 4-eigenspace is a plane.

Step 3. Exhibit the freedom. The remixed pair $\tfrac{1}{\sqrt2}\big(\tfrac{1}{\sqrt2}(1,1,0) \pm (0,0,1)\big)$ is an equally valid orthonormal frame of the plane — verified as exact eigenvectors. A reading of 4 pins the state to the plane; the frame is bookkeeping.

Step 4. Functions ride along. $\hat A^2$ has the same eigenvectors and menu $\{2^2, 4^2, 4^2\} = \{\textbf{4, 16, 16}\}$ — no new diagonalization needed, by the inheritance derivation. (Every number in both examples — eigenvalues, angle, probabilities, $\sigma_E = g$ , the remix check, the squared menu — is recomputed in this article's verification script.)

Origins · 1 minJames Joseph Sylvester and how this was discovered

James Joseph Sylvester

1814–1897 · London, Baltimore, Oxford

The vocabulary of this article is largely Sylvester's. In 1850 he coined the word matrix — the "womb" out of which determinants are born, he said, characteristically flamboyant — and in his 1883 lectures he named the roots of the characteristic equation the latent roots: values hidden inside the array, becoming "patent" only when the right vector draws them out. Quantum mechanics could not ask for a better metaphor — an observable's eigenvalues are latent in exactly this sense, invisible until a measurement on the right state turns one into an instrument reading. His life had latent phases too: barred from a Cambridge degree as a Jew, he worked as an actuary and lawyer (tutoring Florence Nightingale on the side) before mathematics reclaimed him; at 62 he founded America's first mathematics research school at Johns Hopkins, and at 69 became Oxford's Savilian Professor. His friend Arthur Cayley — whose 1858 Memoir built the matrix algebra itself, a story told with Linear Operators on Hilbert Space — supplied the machinery; Sylvester supplied the names physics still uses every day.

History

1829

— Cauchy proves the eigenvalues of a real symmetric matrix are real, decades before physics needs the fact.

1850

— Sylvester coins "matrix"; the 1858 Cayley Memoir turns arrays into an algebra with multiplication and inverses.

1883

— Sylvester names the characteristic roots "latent roots" — values hidden in the operator until the right state exposes them.

1925

— Heisenberg, Born and Jordan cast mechanics as matrix algebra; observable quantities become arrays to diagonalize.

1926

— Schrödinger titles his four-paper series "Quantisierung als Eigenwertproblem" — quantization as eigenvalue problem — and shows wave mechanics computes the same menus.

1929

— von Neumann's spectral theorem extends eigenvalue theory to unbounded operators, giving the continuous spectrum a rigorous home.

Going further

References. Griffiths, Introduction to Quantum Mechanics, 3rd ed., Ch. 3 (Formalism) · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 1 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 1 & 4 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. 1, Ch. II · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 1–2

Provenance. Upgraded to full canon at Step-2 from the AllQFT base article (QUANT.FORM.EIGEN.0870). Base provenance, kept verbatim: "All equations and numbers are standard textbook results. A verifier should recompute: (1) the reality/orthogonality proof from $A=A^{\dagger}$ (Griffiths §3.3, Sakurai §1.3); (2) the $S_x$ eigenvalues $\pm\hbar/2$ and eigenstates $\tfrac1{\sqrt2}(|{+}\rangle\pm|{-}\rangle)$ by diagonalizing $\tfrac{\hbar}{2}\sigma_x$ (Sakurai §1.4, Townsend Ch. 1); (3) the harmonic-oscillator spectrum $E_n=\hbar\omega(n+1/2)$ (Griffiths Ch. 2); (4) the spectral decomposition and completeness relations (Shankar Ch. 1, Cohen-Tannoudji Ch. II). No nonstandard numbers are introduced." Per the chapter ownership map, the reality/orthogonality theorems are owned by Hermitian Operators and Observables and recalled here without re-proof; the base's spin-½ diagonalization exhibit lives with Operators and Observables and Dirac Bra-Ket Notation, replaced here by the avoided-crossing calculation; completeness and the spectral assembly are owned ahead by Complete Orthonormal Bases and Completeness Relation and Resolution of Identity. For this edition every number was machine-verified (sympy/numpy/scipy): the symbolic characteristic polynomial and $E_\pm$ , the numeric eigenpair (0.917/2.083 eV), the mixing angle 15.5°, the $2g$ minimum gap over a detuning sweep, $P = 0.929/0.071$ with mean 1.000 eV and $\sigma_E = g$ exactly, unitarity and eigenphases of $e^{-i\hat Ht/\hbar}$ , the $\{2,4,4\}$ spectrum with eigenspace remixing to $10^{-12}$ , and the $\hat A^2$ menu $\{4,16,16\}$ .

Builds on: Hilbert Space and State Vectors · Linear Operators on Hilbert Space · Hermitian Operators and Observables · Dirac Bra-Ket Notation · Inner Product and Norm of States

Leads to: Complete Orthonormal Bases · Completeness Relation and Resolution of Identity · Born Rule and Measurement Probabilities · Measurement and Wavefunction Collapse · Expectation Values and Variance · Projection Operators

Is an eigenstate a special state of the system, or a special state of the operator?

Of the operator. "Eigenstate" describes a relationship between a state and one operator — that the operator merely rescales it. The same ket can be an eigenstate of $\hat H$ , not of $\hat x$ , and of $\hat H^2$ , all at once. Whenever you say "eigenstate," finish the sentence: eigenstate of what.

If the eigenvalues are the only possible outcomes, where do all the in-between numbers go when I measure a superposition?

They are simply never produced. A state $\alpha|E_1\rangle + \beta|E_2\rangle$ has an average energy between $E_1$ and $E_2$ , but each individual measurement returns $E_1$ or $E_2$ ; the average is realized only across an ensemble. The menu is fixed by the operator — the state only sets the frequencies, exactly the histogram picture of Operators and Observables.

Why does the characteristic equation stop working for position and momentum?

Because $\det(\hat A - \lambda\hat 1)$ is a finite-matrix construction, and $\hat x$ , $\hat p$ act on an infinite-dimensional space with continuous spectra. There is no polynomial whose roots list a continuum, and the "eigenvectors" ( $\delta$ -spikes, plane waves) are not normalizable states. The eigenvalue concept survives — that is the miracle — but the machinery changes: delta normalization and spectral integrals, owned by Continuous Spectra and Delta Normalization.

Do the two coupled levels ever actually cross if I keep tuning?

Not while the coupling is nonzero: the gap $\sqrt{\Delta^2 + 4g^2}$ has a strictly positive minimum $2g$ . What looks like a crossing in real spectra is either genuine $g = 0$ — a symmetry forbids the coupling, so the states pass through each other — or an avoided crossing too narrow to resolve. Measuring the minimum gap is how labs measure $g$ .

What is the point of the mixing angle θ — is it physical or just algebra?

Physical, and measurable. $\theta$ says how much of each bare state lives in each eigenstate: the admixtures $\sin^2\theta$ , $\cos^2\theta$ appear directly in transition rates and in Example 1's statistics. Sweeping the detuning sweeps $\theta$ from $0$ to $45°$ at the anticrossing — the eigenstates visibly rotate in spectroscopy data.

Can a non-Hermitian operator have eigenvalues, and do they mean anything?

Certainly — eigenvalue theory is pure linear algebra. But the eigenvalues need not be real (a unitary operator's lie on the unit circle) and the eigenvectors need not be orthogonal, so they are not measurement menus in the postulate sense. Non-Hermitian spectra still carry physics — decay rates in open systems — but the measurement reading is reserved for Hermitian observables, per Hermitian Operators and Observables.

Eigenvalues and Eigenstates

Where this lives today

The physical picture

The eigenvalue equation, read as physics

Finding the menu: the characteristic equation

Functions of an operator inherit its eigenstates

Degeneracy: when one menu entry owns a subspace

Where the spectrum lives: assembled forms and the continuous frontier

Limiting cases

Worked examples

Going further

Before this

Where this leads