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Hermitian Operators & Real Eigenvalues

Also known as: Self-adjoint operators · Observable reality theorem

Measured quantities are always real numbers — you never read '3+2i volts' off a meter. Quantum mechanics guarantees this by demanding every observable be a Hermitian (self-adjoint) operator. Hermiticity forces the eigenvalues (the possible measurement outcomes) to be real and the eigenstates to be orthogonal, so different outcomes are perfectly distinguishable. It's the algebraic condition that makes operators safe to call 'observables.'

A^=A^†  ⇒  a∈R,  ⟨ϕm∣ϕn⟩=δmn\hat{A}=\hat{A}^{\dagger}\;\Rightarrow\; a\in\mathbb{R},\;\langle\phi_m|\phi_n\rangle=\delta_{mn}
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A real eigenvalue keeps an eigenstate's phase spinning on the real axis; a non-Hermitian part would make it grow or decay.

Equivalent forms

⟨ϕ∣A^ψ⟩=⟨A^ϕ∣ψ⟩\langle\phi|\hat{A}\psi\rangle=\langle\hat{A}\phi|\psi\rangle
One symmetry condition,  = †, simultaneously guarantees real outcomes and a clean orthonormal basis to measure in.