13.2 Adjoints, self-adjointness, and observables

Not every operator can represent a physical quantity. A measured value is a real number, and a special class of operators — the self-adjoint ones — are exactly those whose measurable outputs are guaranteed real, with a complete set of states in which the quantity has a definite value. This lesson defines the adjoint, identifies the self-adjoint operators, and explains why every observable in quantum mechanics is one.

The adjoint

For a matrix, the spectral-theorem lesson introduced the conjugate transpose $A^\dagger$ , defined by the property $\langle \mathbf u, A\mathbf v\rangle = \langle A^\dagger\mathbf u, \mathbf v\rangle$ . The same defining property carries over to operators verbatim. The adjoint $\hat A^\dagger$ of an operator $\hat A$ is the operator satisfying

\boxed{\;\langle f,\, \hat A g\rangle \;=\; \langle \hat A^\dagger f,\, g\rangle\;}

for all functions $f, g$ in the space, with the inner product $\langle f,g\rangle = \int f^* g\,dx$ . In words: moving $\hat A$ from acting on the right slot to acting on the left slot replaces it by $\hat A^\dagger$ .

where

$\hat A^\dagger$: adjoint of the operator A, defined by ⟨f, Ag⟩ = ⟨A†f, g⟩
$\langle f, g\rangle = \int f^* g\, dx$: the inner product against which the adjoint is defined

Self-adjoint operators

An operator is self-adjoint (physicists say Hermitian) when it is its own adjoint,

\hat A^\dagger = \hat A \quad\Longleftrightarrow\quad \langle f, \hat A g\rangle = \langle \hat A f, g\rangle \ \text{ for all } f,g.

The spectral theorem — proved for matrices in Chapter 4 and extending to operators — gives such operators three properties that are the reason they matter:

Real eigenvalues. If $\hat A|a\rangle = a|a\rangle$ and $\hat A$ is self-adjoint, then $a$ is real.
Orthogonal eigenfunctions. Eigenfunctions belonging to different eigenvalues are orthogonal.
Completeness. The eigenfunctions form a basis: any state expands as $|\psi\rangle = \sum_a c_a|a\rangle$ (or an integral, for a continuous spectrum), with $c_a = \langle a|\psi\rangle$ .

▶ Self-adjoint ⟹ real eigenvalues Derivation

Let $\hat A|a\rangle = a|a\rangle$ with $|a\rangle \neq 0$ . Take the inner product with $|a\rangle$ : $\langle a|\hat A|a\rangle = a\,\langle a|a\rangle$ . Now use self-adjointness to move $\hat A$ to the left slot, and the definition of the adjoint: $\langle a|\hat A|a\rangle = \langle \hat A a|a\rangle = \overline{\langle a|\hat A|a\rangle}$ — the quantity equals its own complex conjugate, hence is real. Since $\langle a|a\rangle > 0$ is real and positive, $a = \langle a|\hat A|a\rangle/\langle a|a\rangle$ is real. ∎

The same one-line manoeuvre shows the expectation value $\langle\hat A\rangle = \langle\psi|\hat A|\psi\rangle$ is real for any state $\psi$ when $\hat A$ is self-adjoint — which is precisely what a physical measurement must return.

Why observables are self-adjoint

This is the physical payoff. A measurement yields one of the possible values of the quantity, and those values are real numbers. In the operator formalism the possible outcomes of measuring $\hat A$ are its eigenvalues, and the state collapses to the corresponding eigenfunction. For that to be consistent — real outcomes, and a complete set of definite-value states to collapse into — the operator must have real eigenvalues and a complete orthonormal eigenbasis. Self-adjoint operators are exactly the operators with those two properties. Hence the postulate: every observable is represented by a self-adjoint operator. Position, momentum, energy, angular momentum are all self-adjoint, and their reality and completeness are not assumptions but theorems about such operators.

▶ x̂ and p̂ are self-adjoint Derivation

Position. $(\hat x f)(x) = x f(x)$ with $x$ real. Then $\langle f, \hat x g\rangle = \int f^*\, x\, g\, dx = \int (x f)^* g\, dx = \langle \hat x f, g\rangle$ , using $x = x^*$ . So $\hat x^\dagger = \hat x$ : multiplication by a real function is self-adjoint.

Momentum. $\hat p = -i\hbar\,d/dx$ . Move it across the inner product by integration by parts:

\langle f, \hat p g\rangle = \int f^*(-i\hbar\, g')\, dx = \underbrace{\big[-i\hbar f^* g\big]_{-\infty}^{\infty}}_{=\,0} + \int (-i\hbar f')^* g\, dx = \langle \hat p f, g\rangle.

The boundary term vanishes for functions that decay at infinity. The crucial detail is the factor $-i$ : without it, $d/dx$ alone is anti-self-adjoint ( $\hat D^\dagger = -\hat D$ , the extra minus sign coming from the integration by parts), and the $-i$ flips that sign to make $\hat p$ genuinely self-adjoint. This is why momentum carries the $i$ — reality of measured momentum requires it.

Discrete and continuous spectra

The set of eigenvalues of an operator is its spectrum, and it comes in two flavours:

Discrete spectrum. For a particle bound in a well — the harmonic oscillator of 13.4, a box, an atom — the energy operator has isolated eigenvalues $E_0, E_1, E_2, \dots$ with normalisable eigenfunctions. Quantisation is discreteness of a spectrum; the “quantum” in quantum mechanics is here.
Continuous spectrum. Position and momentum have a continuum of eigenvalues. Their eigenfunctions — $\hat x$ has $\delta(x-x_0)$ , $\hat p$ has $e^{ikx}$ — are not square-integrable and live just outside the Hilbert space (they are handled as limits or “generalised” eigenvectors), but the expansion still works: it is the Fourier transform (Ch 7) that resolves a state into momentum eigenfunctions.

Many operators have both: a Hamiltonian can carry discrete bound states below a continuum of scattering states. Either way the spectral theorem provides a complete basis, and the expansion of a state in eigenfunctions of a self-adjoint operator is the single computation that underlies almost all of quantum mechanics.

▶ Self-adjoint differential operators: the Sturm–Liouville connection Worked Example

The self-adjoint operators built from derivatives are exactly the Sturm–Liouville operators $\hat L = -\frac{d}{dx}\!\left(p(x)\frac{d}{dx}\right) + q(x)$ met in Foundations 6.5. Their real eigenvalues and complete orthonormal eigenfunctions are why separation of variables works and why mode expansions (Fourier series, the normal modes of a string, the spherical harmonics) form bases. The operator formalism and the mode expansions of the PDE chapter are the same spectral theorem, seen from two sides.

With the observables identified as self-adjoint operators, the next lesson turns to the structure that distinguishes quantum from classical: the fact that two observables generally cannot be measured together, encoded in their commutator.