Hermitian Operators: Real Eigenvalues and Orthogonal Eigenstates

Derivation

Hermitian Operators: Real Eigenvalues and Orthogonal Eigenstates

D-140 Home PU-202 Threads chance · symmetry Depends on inner-product-space-axioms

Statement

Let Â be a Hermitian operator on a complex inner-product space, meaning ⟨φ|Âψ⟩ = ⟨Âφ|ψ⟩ for all vectors |φ⟩, |ψ⟩ in its domain. Then (i) every eigenvalue of Â is real, and (ii) eigenvectors belonging to distinct eigenvalues are orthogonal. For a Hermitian operator whose eigenvectors span the space, these orthogonal eigenstates form a complete orthonormal basis in which any state may be expanded, |ψ⟩ = Σ_n c_n|n⟩.

Why it matters

The measurable quantities of quantum mechanics — energy, position, momentum, angular momentum, spin — are represented by Hermitian operators precisely because a measurement must return a real number, and the possible outcomes are the operator's eigenvalues. Reality of the spectrum is therefore not an optional nicety but the structural guarantee that the theory can be compared with laboratory readings at all.

Orthogonality and completeness of the eigenstates are what make the Born rule coherent: an arbitrary state resolves uniquely into a superposition over measurement outcomes, the expansion coefficients are recovered by a single inner product, and the squared moduli of those coefficients sum to one, so they can be read as probabilities. This one theorem is the mathematical backbone shared by the chance thread (probabilities of outcomes) and the symmetry thread (conserved observables generate the symmetries of the dynamics).

Assumptions

The inner product is Hermitian-symmetric and positive-definite.Without conjugate symmetry ⟨φ|ψ⟩ = ⟨ψ|φ⟩* the reality argument collapses, since it hinges on comparing a number with its own complex conjugate; without positive-definiteness ⟨ψ|ψ⟩ = 0 could hold for |ψ⟩ ≠ 0 and we could not divide by the norm.

The operator satisfies ⟨φ|Âψ⟩ = ⟨Âφ|ψ⟩ on the relevant domain.If Â is merely symmetric on a domain smaller than that of its adjoint (formally Hermitian but not self-adjoint), eigenvalues can fail to be real or the eigenfunctions can fail to be complete; the spectral theorem needs genuine self-adjointness.

The vectors considered are genuine eigenvectors, i.e. normalizable elements of the space.For operators with continuous spectrum (position, momentum) the "eigenstates" are non-normalizable distributions; the finite-sum argument still runs formally, but completeness must be restated as an integral resolution of the identity and orthogonality becomes a Dirac delta rather than a Kronecker delta.

Derivation

Â|ψ⟩ = λ|ψ⟩, |ψ⟩ ≠ 0

Take an eigenvalue–eigenvector pair as the starting hypothesis; we prove λ is real. A

⟨ψ|Â|ψ⟩ = ⟨ψ|(λψ)⟩ = λ⟨ψ|ψ⟩

Form the expectation value by taking the inner product of the eigenvalue equation with |ψ⟩ on the left; linearity of the inner product in its second slot pulls the scalar λ out. A

⟨ψ|Â|ψ⟩ = ⟨Âψ|ψ⟩ = ⟨(λψ)|ψ⟩ = λ*⟨ψ|ψ⟩

Use the Hermitian property to move Â onto the left (bra) vector, then substitute the eigenvalue equation there; conjugate-linearity of the inner product in its first slot brings out λ*, not λ. B

λ⟨ψ|ψ⟩ = λ*⟨ψ|ψ⟩ ⟹ (λ − λ*)⟨ψ|ψ⟩ = 0

Steps 2 and 3 are two evaluations of the same number ⟨ψ|Â|ψ⟩, so they are equal; subtract. A

⟨ψ|ψ⟩ > 0 ⟹ λ = λ* ⟹ λ ∈ ℝ

Positive-definiteness makes the norm nonzero for |ψ⟩ ≠ 0, so the bracketed factor must vanish; a number equal to its own conjugate is real. This proves part (i). A

Â|m⟩ = λ_m|m⟩, Â|n⟩ = λ_n|n⟩, λ_m ≠ λ_n

Now take two eigenvectors with distinct eigenvalues to prove orthogonality (part ii). A

⟨m|Â|n⟩ = λ_n⟨m|n⟩

Let Â act to the right on |n⟩ and pull the scalar out. A

⟨m|Â|n⟩ = ⟨Âm|n⟩ = λ_m*⟨m|n⟩ = λ_m⟨m|n⟩

Let Â act to the left using the Hermitian property; the eigenvalue there emerges conjugated, but by part (i) λ_m* = λ_m, so the eigenvalue is real. B

(λ_n − λ_m)⟨m|n⟩ = 0 ⟹ ⟨m|n⟩ = 0

Subtract the two evaluations of ⟨m|Â|n⟩. Because the eigenvalues are distinct the prefactor is nonzero, forcing the overlap to vanish: the eigenvectors are orthogonal. A

λ_m = λ_n (degenerate) ⟹ Gram–Schmidt within the eigenspace

If two eigenvectors share an eigenvalue the subtraction gives 0 = 0 and no orthogonality is forced; but any linear combination inside a degenerate eigenspace is still an eigenvector, so Gram–Schmidt produces an orthonormal set spanning that eigenspace. Combined with step 9 the full eigenbasis can always be chosen orthonormal. C

Result

λ_n ∈ ℝ, ⟨m|n⟩ = δ_mn, Σ_n|n⟩⟨n| = Î ⟹ |ψ⟩ = Σ_n c_n|n⟩, c_n = ⟨n|ψ⟩

Reading. A Hermitian observable has real measurement outcomes λ_n. Its eigenstates can be chosen orthonormal and, when complete, resolve the identity, so any physical state expands uniquely as a superposition whose coefficients are recovered by a single projection c_n = ⟨n|ψ⟩. The squared moduli |c_n|² are then the Born probabilities of the outcomes.

Units check. The relations are dimensionless statements about vectors and their overlaps: δ_mn is a pure number, the projectors |n⟩⟨n| are dimensionless, and each eigenvalue λ_n carries exactly the physical units of the observable Â (joules for energy, kg·m·s⁻¹ for momentum), inherited unchanged from Â|n⟩ = λ_n|n⟩ since |n⟩ is dimensionless.

Limiting cases

Real symmetric matrix. Over ℝ, Hermitian reduces to A^T = A; the same proof gives the real spectral theorem — real eigenvalues, orthogonal eigenvectors, orthogonal diagonalization A = QΛQ^T.
Diagonal operator. If Â is already diagonal in an orthonormal basis, the eigenvalues are the diagonal entries (manifestly real) and the eigenvectors are the basis vectors (manifestly orthogonal); the theorem is trivially satisfied.
Non-degenerate spectrum. When all λ_n are distinct, orthogonality is automatic from step 9 and no Gram–Schmidt is needed; the eigenbasis is unique up to phases.
Two-dimensional spin. For Â = ½ħ(n̂·σ) the eigenvalues are ±½ħ (real) and the two spinors are orthogonal, recovering the qubit measurement basis.

Breaks when

The operator is only symmetric, not self-adjoint. On a finite interval the momentum operator p̂ = −iħ d/dx is symmetric on functions vanishing at the ends, but its adjoint acts on a larger domain. Depending on the boundary conditions it may have no real eigenvalues, complex eigenvalues, or a whole family of self-adjoint extensions with different real spectra — reality is not guaranteed by the formal expression alone.
Continuous spectrum. Position and momentum have no normalizable eigenstates: ⟨x|x′⟩ = δ(x − x′) is not a finite number, so ⟨ψ|ψ⟩ > 0 in step 5 cannot be applied to the "eigenvector" itself. The completeness sum becomes an integral ∫|x⟩⟨x| dx = Î and orthonormality is a delta, requiring the rigged-Hilbert-space (Gel'fand triple) framework rather than the elementary argument here.
Non-Hermitian (e.g. open or dissipative) systems. Operators lacking the Hermitian property — such as the non-Hermitian effective Hamiltonians of decaying states or PT-symmetric models — generically have complex eigenvalues and non-orthogonal (skew) eigenvectors; the left and right eigenvectors differ and one must use a biorthogonal expansion instead.

Failure modes

Forgetting conjugate-linearity in the first slot. Writing ⟨Âψ|ψ⟩ = λ⟨ψ|ψ⟩ instead of λ*⟨ψ|ψ⟩ silently assumes what is to be proved and makes the theorem vacuous; the whole content lives in that conjugate.
Claiming degenerate eigenvectors are automatically orthogonal. Step 9 only forces orthogonality for distinct eigenvalues; within a degenerate eigenspace one must impose orthogonality by Gram–Schmidt — it is a choice, not a consequence.
Confusing symmetric with self-adjoint. Assuming any operator with a symmetric-looking definition has a real, complete spectrum; boundary/domain issues can break this entirely for unbounded operators.
Treating |x⟩ as a normalizable state. Manipulating position/momentum eigenkets as if ⟨x|x⟩ = 1, then being surprised by infinities; these live in a rigged Hilbert space.
Dropping the phase freedom. Believing the orthonormal eigenbasis is unique; each |n⟩ is defined only up to a phase e^iθ (and up to unitary mixing within a degenerate subspace).

Discussion

The theorem is the quantum expression of a very old idea: the "principal axes" of a symmetric object are real and mutually perpendicular. In classical mechanics the moment-of-inertia tensor is real symmetric, so a rigid body has three real principal moments along orthogonal axes; the same linear algebra, promoted to a complex Hilbert space, tells us that an observable has real eigenvalues along orthogonal eigen-directions. Diagonalizing a Hermitian operator is nothing but finding the measurement basis in which the observable is "already sorted".

Completeness — the resolution of the identity Σ_n|n⟩⟨n| = Î — is the load-bearing extra input for physics and does not follow from Hermiticity alone in infinite dimensions; it is guaranteed by the spectral theorem for bounded self-adjoint operators and, with care, for unbounded ones. Once completeness holds, the expansion |ψ⟩ = Σ_n c_n|n⟩ with Σ_n|c_n|² = ⟨ψ|ψ⟩ = 1 makes the Born interpretation self-consistent: probabilities are non-negative, sum to one, and are basis-covariant. This is where the chance thread is anchored.

The symmetry thread enters because commuting Hermitian operators share a common eigenbasis: if [Â,B̂] = 0 then Â and B̂ can be simultaneously diagonalized, and their common eigenvalues are the good quantum numbers labelling states. A continuous symmetry generated by a Hermitian charge Ĝ (via the unitary e^−iĜα) that commutes with the Hamiltonian yields a conserved, real-valued observable — the quantum content of Noether's theorem. Reality of the spectrum is thus what lets a symmetry generator double as a measurable conserved quantity.

At the deepest level the requirement is self-adjointness, strictly stronger than symmetry. The distinction is invisible in finite dimensions but decisive for unbounded operators: von Neumann's deficiency-index analysis classifies when a symmetric operator admits self-adjoint extensions and how many, and different extensions correspond to genuinely different physics (for a particle on a half-line, different boundary conditions encode different wall interactions). The elementary two-line reality proof above is exact, but its hypothesis ⟨φ|Âψ⟩ = ⟨Âφ|ψ⟩ must be read as holding on a domain where Â equals its adjoint, not merely where the integration-by-parts boundary terms happen to be written down and ignored.

Common misconceptions. "Hermitian means the matrix looks symmetric" — no; it means A_ij = A_ji*, transpose and complex-conjugate, so diagonal entries must be real. "Real eigenvalues imply Hermitian" — false; non-Hermitian PT-symmetric operators can have entirely real spectra. "Orthogonal eigenstates are unique" — they are fixed only up to phases and up to rotations within degenerate subspaces.

Worked examples

Ĥ = ε₀ [[1, i],[−i, 1]] (check Hermiticity, find spectrum)

Take a two-level Hamiltonian. Hermiticity requires H₁₂ = H₂₁*: here H₁₂ = iε₀ and H₂₁* = (−iε₀)* = iε₀ ✓, and the diagonal entries ε₀ are real ✓. A

det(Ĥ − λÎ) = (ε₀ − λ)² − (iε₀)(−iε₀) = (ε₀ − λ)² − ε₀²

Characteristic equation; note (iε₀)(−iε₀) = ε₀², a real, positive off-diagonal product as Hermiticity guarantees. A

(ε₀ − λ)² = ε₀² ⟹ λ = ε₀ ± ε₀ ⟹ λ₊ = 2ε₀, λ₋ = 0

Solve; both roots real, as the theorem promised. A

|+⟩ = (1/√2)(1, i)^T, |−⟩ = (1/√2)(1, −i)^T

Eigenvectors from (Ĥ − λÎ)v = 0; normalized so ⟨±|±⟩ = ½(1·1 + (∓i)*(±i)) = ½(1+1) = 1. B

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

Overlap uses conjugation of the first vector's components: (1,i)* = (1,−i). Distinct eigenvalues ⟹ orthogonal, confirming part (ii). B

λ₊ = 2ε₀, λ₋ = 0 (real); ⟨+|−⟩ = 0

Reading. With ε₀ = 1.0 eV the energies are 2.0 eV and 0 eV, both real and measurable, and the two eigenstates are orthogonal — a concrete two-level system illustrating the theorem end to end.

σ_y = [[0, −i],[i, 0]], S_y = ½ħ σ_y (spin-½ along y)

The Pauli-y observable. Hermitian: (σ_y)₁₂ = −i = ((σ_y)₂₁)* = (i)* ✓. A

det(σ_y − λÎ) = λ² − (−i)(i) = λ² − 1 = 0 ⟹ λ = ±1

Characteristic equation; (−i)(i) = 1. Real eigenvalues ±1, so S_y = ±½ħ. A

|↑_y⟩ = (1/√2)(1, i)^T, |↓_y⟩ = (1/√2)(1, −i)^T

Solve σ_yv = ±v; normalized. B

|ψ⟩ = |↑_z⟩ = (1, 0)^T; c_↑y = ⟨↑_y|ψ⟩ = (1/√2)(1·1 + (i)*·0) = 1/√2

Expand a spin-up-along-z state in the y-basis using c_n = ⟨n|ψ⟩; conjugate the eigenvector components. Similarly c_↓y = 1/√2. B

P(S_y = +½ħ) = |c_↑y|² = ½, P(S_y = −½ħ) = ½, Σ = 1

Born rule on the orthonormal complete basis; probabilities sum to one exactly because Σ|n⟩⟨n| = Î. A

S_y ∈ {+½ħ, −½ħ}, P₊ = P₋ = ½

Reading. Measuring the y-spin of a z-up electron gives +½ħ = +5.27×10⁻³⁵ J·s or −½ħ, each with probability one-half. The real eigenvalues are the outcomes; the orthonormal complete eigenbasis is what lets the z-up state be resolved into equal parts and the probabilities to add to unity.

Problems

Show directly that the diagonal entries of any Hermitian matrix are real, and that A_ij and A_ji are complex conjugates.
Solution
Hermitian means A = A^†, i.e. A_ij = (A_ji)* for all i,j. Setting i = j: A_ii = (A_ii)*, so each diagonal entry equals its own conjugate and is therefore real. For i ≠ j the condition A_ij = (A_ji)* is exactly the statement that the two off-diagonal partners are complex conjugates, e.g. if A₁₂ = a + ib then A₂₁ = a − ib.
The trace and determinant of a Hermitian operator are real. Prove it for a 2×2 Hermitian matrix and connect to its eigenvalues.
Solution
Write A = [[a, w],[w*, d]] with a,d ∈ ℝ. Then tr A = a + d ∈ ℝ and det A = ad − w·w* = ad − |w|² ∈ ℝ. Since tr A = λ₁ + λ₂ and det A = λ₁λ₂, and both are real while the eigenvalues are real (by the theorem), this is consistent. Concretely the eigenvalues are λ_± = ½(a+d) ± √[¼(a−d)² + |w|²], manifestly real because the discriminant is a sum of non-negative reals.
Let Â be Hermitian with normalized eigenstate |n⟩, eigenvalue λ_n. Show the expectation value ⟨Â⟩ = ⟨ψ|Â|ψ⟩ in any normalized state is real, and lies between the smallest and largest eigenvalues.
Solution
Expand |ψ⟩ = Σ_n c_n|n⟩ with Σ|c_n|² = 1. Then ⟨Â⟩ = Σ_m,n c_m* c_n λ_n⟨m|n⟩ = Σ_n |c_n|² λ_n using orthonormality. Each λ_n is real (theorem) and |c_n|² ≥ 0, so ⟨Â⟩ is a convex combination of real numbers — real, and bounded by λ_min ≤ ⟨Â⟩ ≤ λ_max since Σ|c_n|² = 1.
Two Hermitian operators satisfy [Â,B̂] = 0 and Â has non-degenerate spectrum. Show they share eigenvectors.
Solution
Let Â|n⟩ = λ_n|n⟩. Apply Â to B̂|n⟩: Â(B̂|n⟩) = B̂Â|n⟩ = λ_n(B̂|n⟩), using ÂB̂ = B̂Â. So B̂|n⟩ is an eigenvector of Â with the same eigenvalue λ_n. Because that eigenvalue is non-degenerate, its eigenspace is one-dimensional, so B̂|n⟩ must be a scalar multiple of |n⟩: B̂|n⟩ = μ_n|n⟩. Hence |n⟩ is a simultaneous eigenvector, and μ_n is real since B̂ is Hermitian.
Consider Â = [[2, 1−i],[1+i, 3]] (with entries in eV). Verify it is Hermitian, find its eigenvalues, and confirm the eigenvectors are orthogonal.
Solution
Hermitian check: A₁₂ = 1−i = (A₂₁)* = (1+i)* ✓, diagonals 2,3 real ✓. Characteristic equation: (2−λ)(3−λ) − (1−i)(1+i) = 0. Now (1−i)(1+i) = 1 − i² = 2, so λ² − 5λ + 6 − 2 = λ² − 5λ + 4 = 0, giving λ = (5 ± √(25−16))/2 = (5 ± 3)/2, i.e. λ₊ = 4 eV, λ₋ = 1 eV — both real. Eigenvector for λ₊=4: (2−4)x + (1−i)y = 0 ⟹ y = 2x/(1−i) = x(1+i), so v₊ ∝ (1, 1+i). For λ₋=1: (2−1)x + (1−i)y = 0 ⟹ y = −x/(1−i) = −x(1+i)/2, so v₋ ∝ (2, −(1+i)). Overlap ⟨v₊|v₋⟩ = (1)*(2) + (1+i)*·(−(1+i)) = 2 − (1−i)(1+i) = 2 − 2 = 0 ✓ orthogonal, as required for distinct eigenvalues.