Hermitian operators have real eigenvalues and orthogonal eigenvectors

Derivation · the atom

Hermitian operators have real eigenvalues and orthogonal eigenvectors

D-018 Home PU-104 Threads symmetry Depends on — verified

Statement

If Â^† = Â, then every eigenvalue of Â is real, and eigenvectors belonging to distinct eigenvalues are orthogonal.

Why it matters

Two theorems, four lines, and they carry an enormous amount of physics. Reality of eigenvalues is why observables are represented by Hermitian operators — measurement outcomes are real numbers. Orthogonality is why any state can be expanded in eigenstates with unambiguous coefficients, which is what makes the probability interpretation coherent. The same theorem, applied to real symmetric matrices, is what guarantees that the normal modes of D-003 exist and are independent.

Assumptions

Â is Hermitian: ⟨f|Âg⟩ = ⟨Âf|g⟩ for all f, g in the domain. For infinite-dimensional spaces the domain condition is not a technicality — a symmetric operator need not be self-adjoint, and the distinction determines whether the spectral theorem applies at all.

Eigenvectors are normalisable. Fails for continuous spectra: position and momentum eigenstates are not in the Hilbert space, and the theorem needs the rigged-Hilbert-space extension.

Derivation — reality of eigenvalues

Â|a⟩ = a|a⟩, ⟨a|a⟩ = 1

Take a normalised eigenvector with eigenvalue a, complex a priori. A

⟨a|Â|a⟩ = a⟨a|a⟩ = a

Act to the right. A

⟨a|Â|a⟩ = ⟨Âa|a⟩ = a*⟨a|a⟩ = a*

Act to the left instead, using Hermiticity to move the operator across. The eigenvalue emerges conjugated because the inner product is antilinear in its first argument. B

a = a* ⟹ a ∈ ℝ

The same quantity computed two ways. The only assumption used was Hermiticity. C

Derivation — orthogonality

Â|a⟩ = a|a⟩, Â|b⟩ = b|b⟩, a ≠ b

Two eigenvectors with distinct eigenvalues, both real by the first result. A

⟨b|Â|a⟩ = a⟨b|a⟩ and ⟨b|Â|a⟩ = b⟨b|a⟩

Evaluate the same matrix element in both directions, using Hermiticity and the reality of b. B

(a − b)⟨b|a⟩ = 0 ⟹ ⟨b|a⟩ = 0

Subtract. Since a ≠ b by hypothesis, the inner product must vanish. C

Result

Â^† = Â ⟹ a_n ∈ ℝ and ⟨a_m|a_n⟩ = δ_mn

Reading. The eigenvectors form an orthonormal basis. Any state can be expanded as |ψ⟩ = Σ c_n|a_n⟩ with c_n = ⟨a_n|ψ⟩ uniquely determined, and Σ|c_n|² = 1. The probability interpretation depends entirely on this.

Units check. Both statements are dimensionless relations among numbers: an eigenvalue equals its own conjugate (same units as the observable, purely real), and an inner product of normalised states is a pure number equal to δ_mn. No dimensional mismatch is introduced by either step.

Limiting cases

Real symmetric matrix (Â^T = Â, real entries) — the finite-dimensional real case; guarantees the normal modes of D-003 exist and are independent.
Distinct eigenvalues — orthogonality is automatic; the full spectral basis follows with no further choice.

Tier 3 · watch out

The degenerate case. When two eigenvalues coincide the argument at step 7 fails — and it should, because the eigenvectors are then genuinely not unique. Any linear combination within the degenerate subspace is also an eigenvector. Orthogonality can always be imposed by Gram–Schmidt, but it is a choice, not a consequence. Which basis is natural is then decided by an additional commuting observable — exactly why the complete-set-of-commuting-observables machinery exists in PU-301.

Breaks when

The operator is symmetric but not self-adjoint — domain issues that matter for momentum on a half-line, and for the Hamiltonian of a particle in a box with certain boundary conditions.
The spectrum is continuous — eigenstates are not normalisable and orthogonality becomes a delta function rather than a Kronecker delta.
Non-Hermitian systems, including the effective Hamiltonians of open and PT-symmetric systems, where eigenvalues may be complex and eigenvectors non-orthogonal.

Failure modes

Assuming eigenvectors of a Hermitian operator are always orthogonal. Only guaranteed for distinct eigenvalues.
Confusing Hermitian with unitary. Unitary operators satisfy Û^†Û = 1, have eigenvalues on the unit circle, and represent transformations rather than observables.
Forgetting the antilinearity of the inner product in the first slot — the source of nearly every sign or conjugation error in an introductory course.

Worked number

Take the Hermitian matrix Â = [[2, i], [−i, 2]], with Â^† = Â. Its eigenvalues solve (2 − λ)² − 1 = 0, giving λ = 3 and λ = 1 — both real, as promised. The eigenvectors are (1, −i)/√2 and (1, i)/√2; their inner product ⟨a|b⟩ = (1·1 + (i)(i))/2 = (1 − 1)/2 = 0 — orthogonal.

Run the check →

Discussion

The two claims of this theorem look modest but they are precisely the pair of facts that lets quantum mechanics attach numbers to reality. Every physical measurement returns a real number, so any operator that is to represent an observable must have a real spectrum — and Hermiticity is exactly the condition that guarantees it. Orthogonality is the second half of the bargain: because eigenvectors belonging to distinct eigenvalues are perpendicular, an arbitrary state resolves into eigenstates uniquely, with coefficients c_n = ⟨a_n|ψ⟩ that can be read off by a single projection. Reality makes the outcomes meaningful; orthonormality makes the probabilities |c_n|² add to one. Take away either and the Born rule collapses.

The proof is worth admiring for what it does not use. Reality of the eigenvalue is nothing more than computing ⟨a|Â|a⟩ two ways and demanding the answers agree; orthogonality is the same trick applied to the off-diagonal matrix element ⟨b|Â|a⟩. No basis is chosen, no dimension is assumed, and no property of Â is invoked except Â^† = Â together with the antilinearity of the inner product in its first slot. That economy is why the result is structured as a clean implication rather than a computation: it is a statement about the geometry that Hermiticity forces on Hilbert space, not about any particular operator.

Seen from higher ground, this is the finite-dimensional shadow of the spectral theorem. The genuinely general object is a normal operator, one satisfying Â^†Â = ÂÂ^†; Hermitian operators are the normal operators whose spectrum happens to lie on the real line, and unitary operators are the normal operators whose spectrum lies on the unit circle. The very same algebra, transcribed from bra–ket notation into the integral inner product ⟨f|g⟩ = ∫f*g dx, is Sturm–Liouville theory — which is why the energy eigenfunctions of a well-posed Hamiltonian are automatically real-energied and mutually orthogonal, and why Fourier series exist at all. Restricted to real symmetric matrices it is the principal-axis theorem that guarantees the normal modes of D-003. And when eigenvalues coincide the argument at step 7 correctly abstains: within a degenerate subspace orthogonality must be imposed by Gram–Schmidt or, more physically, selected by a commuting observable, which is the entire motivation for the complete-set-of-commuting-observables construction of PU-301.

Common misconceptions. Three errors recur. First, students assume all eigenvectors of a Hermitian operator are orthogonal; the theorem only promises this for distinct eigenvalues, and in a degenerate subspace orthogonality is a choice. Second, Hermitian is confused with unitary: a unitary operator preserves norms and represents a transformation (its eigenvalues sit on the unit circle), whereas a Hermitian operator represents an observable (its eigenvalues sit on the real line). Third — and this is the source of nearly every sign error in a first course — the inner product is antilinear in its first argument, so pulling Â to the left conjugates the eigenvalue; forgetting that conjugation makes the reality proof appear to prove nothing.

Worked examples

Example 1 — A 2×2 Hermitian matrix: real eigenvalues, orthogonal eigenvectors, and why antilinearity matters. Consider the observable represented in some orthonormal basis by

Â = ( 2 i
−i 2 )

which is Hermitian because the off-diagonal entries are complex conjugates, A₂₁ = A₁₂*. We find its spectrum and eigenvectors and test the theorem directly.

det(Â − λ𝕀) = (2 − λ)² − (i)(−i) = (2 − λ)² − 1 = 0

Write the characteristic equation. Note (i)(−i) = −i² = +1, so the imaginary parts cancel.

2 − λ = ±1 ⟹ λ₁ = 3, λ₂ = 1

Both eigenvalues are real, as the theorem requires — and this happened because the off-diagonal product turned real, which is the matrix-level fingerprint of Hermiticity.

(2 − 3)x + i y = 0 ⟹ x = i y ⟹ |a₁⟩ = (1/√2)(i, 1)^T

Eigenvector for λ₁ = 3, normalised: |i|² + |1|² = 2.

(2 − 1)x + i y = 0 ⟹ x = −i y ⟹ |a₂⟩ = (1/√2)(−i, 1)^T

Eigenvector for λ₂ = 1, normalised.

⟨a₁|a₂⟩ = ½[ (i)*(−i) + (1)*(1) ] = ½[ (−i)(−i) + 1 ] = ½[ −1 + 1 ] = 0

Here the antilinearity is decisive: the first component is conjugated, (i)* = −i. Had we forgotten to conjugate we would have got ½[(i)(−i)+1] = 1 ≠ 0 and falsely concluded the eigenvectors were not orthogonal.

λ₁ = 3, λ₂ = 1 ∈ ℝ, ⟨a₁|a₂⟩ = 0

Answer. Both eigenvalues are real and the eigenvectors are orthogonal, exactly as D-018 predicts. The check also exposes the one place a student can go wrong: orthogonality holds only when the inner product is taken antilinearly in the first slot.

Example 2 — The infinite square well: Hermiticity, real energies, and orthonormal eigenfunctions. Take the Hamiltonian Ĥ = −(ℏ²/2m) d²/dx² on 0 ≤ x ≤ L with the Dirichlet conditions ψ(0) = ψ(L) = 0. We show it is Hermitian on that domain, then confirm the theorem's two conclusions on its explicit eigenfunctions.

⟨f|Ĥg⟩ − ⟨Ĥf|g⟩ = −(ℏ²/2m) ∫₀^L (f* g″ − f*″ g) dx

Difference of the two matrix elements; the potential term (zero inside) cancels identically.

= −(ℏ²/2m) [ f* g′ − f*′ g ]₀^L

Integrate by parts twice; the interior integrals cancel, leaving only the Wronskian boundary term.

f(0) = f(L) = g(0) = g(L) = 0 ⟹ boundary term = 0 ⟹ ⟨f|Ĥg⟩ = ⟨Ĥf|g⟩

The Dirichlet conditions kill the boundary term, so Ĥ is Hermitian on this domain. (Change the boundary conditions and the boundary term can survive — this is the domain subtlety of the assumptions.)

ψ_n(x) = √(2/L) sin(nπx/L), E_n = n²π²ℏ²/(2mL²)

The normalised eigenfunctions and their energies. Every E_n is real — guaranteed in advance by step 3, before any of them is computed.

⟨ψ_m|ψ_n⟩ = (2/L) ∫₀^L sin(mπx/L) sin(nπx/L) dx = δ_mn

Using 2 sinα sinβ = cos(α−β) − cos(α+β); for m ≠ n both cosine integrals vanish over the well, for m = n the result is 1.

E_n ∈ ℝ, ⟨ψ_m|ψ_n⟩ = δ_mn

Answer. The well's energies are real and its stationary states orthonormal — the same theorem, now in an infinite-dimensional space with a discrete spectrum. The orthonormality is exactly what makes the Fourier-sine expansion of an arbitrary ψ(x) legitimate, with expansion coefficients c_n = ⟨ψ_n|ψ⟩ determined uniquely.

Problems

Show that the matrix Â = ( ( 4, 3i ), ( −3i, 4 ) ) is Hermitian and find its eigenvalues; confirm they are real.
Solution
Hermitian: the diagonal entries are real and A₂₁ = −3i = (3i)* = A₁₂*. Characteristic equation: (4 − λ)² − (3i)(−3i) = (4 − λ)² − 9 = 0, since (3i)(−3i) = 9. Hence 4 − λ = ±3, giving λ = 7 and λ = 1, both real. (The eigenvectors are (1/√2)(i, 1) for λ = 7 and (1/√2)(−i, 1) for λ = 1, and one checks ⟨·|·⟩ = ½[(−i)(−i) + 1] = 0.)
Prove that the expectation value ⟨Â⟩ = ⟨ψ|Â|ψ⟩ of a Hermitian operator is real for every normalised state, not only for eigenstates. Where exactly is Hermiticity used?
Solution
Compute the complex conjugate: ⟨Â⟩* = ⟨ψ|Â|ψ⟩* = ⟨Âψ|ψ⟩, because conjugating an inner product swaps its two arguments, ⟨u|v⟩* = ⟨v|u⟩. Now use Hermiticity, ⟨Âψ|ψ⟩ = ⟨ψ|Âψ⟩ = ⟨Â⟩. So ⟨Â⟩* = ⟨Â⟩, i.e. ⟨Â⟩ ∈ ℝ. Hermiticity is used in the single step that moves Â from the ket side to the bra side; conjugation-swap alone would not close the argument. This is the more general statement of which the reality of eigenvalues is the special case |ψ⟩ = |a⟩.
The eigenvalue λ = 2 of some Hermitian operator is doubly degenerate, with (non-orthogonal) eigenvectors |u⟩ = (1, 0, 0)^T and |v⟩ = (1, 1, 0)^T. Construct an orthonormal basis for the degenerate subspace and comment on why the theorem alone did not deliver one.
Solution
Gram–Schmidt. Take |e₁⟩ = |u⟩ = (1,0,0)^T (already normalised). Then remove its component from |v⟩: ⟨e₁|v⟩ = 1, so |v′⟩ = |v⟩ − 1·|e₁⟩ = (0,1,0)^T, which is already unit length, giving |e₂⟩ = (0,1,0)^T. Both are eigenvectors with eigenvalue 2 (any combination in the subspace is), and now ⟨e₁|e₂⟩ = 0. D-018 could not supply this because step 7 requires distinct eigenvalues; within one eigenvalue the eigenvectors are not pinned down, and orthogonality must be imposed by hand (or selected physically by a second commuting observable, per PU-301).
Let Â and Â ′ be Hermitian. (a) Show the product ÂÂ′ is Hermitian if and only if [Â, Â′] = 0. (b) Show that i[Â, Â′] is Hermitian even when they do not commute.
Solution
Use the reversal rule (ÂÂ′)^† = Â′^†Â^† = Â′Â, since both are Hermitian. (a) ÂÂ′ is Hermitian ⇔ (ÂÂ′)^† = ÂÂ′ ⇔ Â′Â = ÂÂ′ ⇔ [Â, Â′] = 0. So a product of observables is itself an observable only when they are compatible. (b) Let Č = [Â, Â′] = ÂÂ′ − Â′Â. Then Č^† = Â′Â − ÂÂ′ = −Č (anti-Hermitian). Hence (iČ)^† = −i·Č^† = −i(−Č) = iČ, so i[Â, Â′] is Hermitian. This is why the canonical commutator is written [x̂, p̂] = iℏ — the i is what keeps the right-hand side a legitimate (real-spectrum) object.
Given a Hermitian Hamiltonian Ĥ, define the time-evolution operator Û(t) = exp(−iĤt/ℏ). (a) Show Û is unitary. (b) If Ĥ|n⟩ = E_n|n⟩, find the eigenvalues of Û and show they lie on the unit circle. (c) In one sentence, contrast the roles of Ĥ and Û.
Solution
(a) Since Ĥ^† = Ĥ, taking the adjoint of the series gives Û^† = exp(+iĤ^†t/ℏ) = exp(+iĤt/ℏ). Because Ĥ commutes with itself, exponents add: Û^†Û = exp(+iĤt/ℏ) exp(−iĤt/ℏ) = exp(0) = 𝕀. So Û is unitary and preserves norms (probability). (b) Acting on an energy eigenstate, Û(t)|n⟩ = exp(−iE_nt/ℏ)|n⟩, so the eigenvalues are the phases u_n = exp(−iE_nt/ℏ). Since E_n is real (D-018), |u_n| = 1: the eigenvalues lie exactly on the unit circle, and this is where the reality of the spectrum pays off — a complex E_n would make |u_n| ≠ 1 and probability would leak. (c) Ĥ is Hermitian and represents the observable energy (real eigenvalues); Û is unitary and represents the transformation of the state in time (unit-modulus eigenvalues) — the two faces of a normal operator kept carefully distinct.