Linear Operators

In quantum mechanics, states live in a Hilbert space, but the physics — measurement, time evolution, symmetry — happens when something acts on those states. That "something" is a linear operator. Operators are the verbs of the theory: they turn one state vector into another while respecting the linear structure that makes superposition possible.

What an operator is

An operator $A$ on a complex vector space $V$ is a rule that assigns to every vector $|\psi\rangle \in V$ another vector $A|\psi\rangle \in V$ . We write the action as juxtaposition: $A|\psi\rangle$ means " $A$ acting on $|\psi\rangle$ ."

We are almost never interested in arbitrary rules. The operators of quantum mechanics are linear, meaning they commute with the two ways of combining vectors — addition and scalar multiplication.

The linearity conditions

An operator $A$ is linear if for all vectors $|\psi\rangle, |\phi\rangle \in V$ and all scalars $\alpha, \beta \in \mathbb{C}$ ,

A\big(\alpha|\psi\rangle + \beta|\phi\rangle\big) = \alpha\, A|\psi\rangle + \beta\, A|\phi\rangle.

This single equation packs in two requirements:

Additivity: $A(|\psi\rangle + |\phi\rangle) = A|\psi\rangle + A|\phi\rangle$ .
Homogeneity: $A(\alpha|\psi\rangle) = \alpha\, A|\psi\rangle$ .

A consequence worth remembering: every linear operator sends the zero vector to the zero vector, $A\,|0\rangle_{\text{vec}} = 0$ , because $A(0 \cdot |\psi\rangle) = 0 \cdot A|\psi\rangle = 0$ . (Here $|0\rangle_{\text{vec}}$ is the zero of the vector space, not the computational basis state.)

Operators are determined by their action on a basis

Suppose $\{|e_1\rangle, \dots, |e_n\rangle\}$ is a basis of $V$ . Any vector expands uniquely as $|\psi\rangle = \sum_j c_j |e_j\rangle$ . By linearity,

A|\psi\rangle = A\sum_j c_j |e_j\rangle = \sum_j c_j\, A|e_j\rangle.

So the operator is completely fixed once we know the $n$ vectors $A|e_j\rangle$ . Each of those can be expanded back in the basis, $A|e_j\rangle = \sum_i A_{ij} |e_i\rangle$ , and the numbers $A_{ij}$ are the matrix elements of $A$ in this basis — the subject of the next lesson. In a finite-dimensional space, choosing a basis turns every linear operator into a matrix, and operator algebra becomes matrix algebra.

Examples and non-examples

The identity operator $\mathbb{1}$ leaves every vector alone: $\mathbb{1}|\psi\rangle = |\psi\rangle$ . It is linear.
A scalar multiple $cA$ , defined by $(cA)|\psi\rangle = c\,(A|\psi\rangle)$ , is linear whenever $A$ is.
The Pauli operator $X$ on a qubit, defined by $X|0\rangle = |1\rangle$ and $X|1\rangle = |0\rangle$ , extends linearly to $X(\alpha|0\rangle + \beta|1\rangle) = \alpha|1\rangle + \beta|0\rangle$ . Linear.
The rule "take the complex conjugate of every amplitude" is not linear over $\mathbb{C}$ : it satisfies additivity but fails homogeneity, since conjugating $\alpha|\psi\rangle$ gives $\bar{\alpha}\,\overline{|\psi\rangle}$ , not $\alpha\,\overline{|\psi\rangle}$ . Such a map is called antilinear, and it appears for time reversal, but it is not an operator in the linear sense used here.

Composition and the algebra of operators

Operators can be added, scaled, and composed. The product $AB$ means "first apply $B$ , then $A$ ":

(AB)|\psi\rangle = A\big(B|\psi\rangle\big).

The composition of two linear operators is again linear. Operator multiplication is associative, $(AB)C = A(BC)$ , but in general not commutative: $AB \neq BA$ . The failure of commutativity is not a technicality — it is the mathematical heart of the uncertainty principle, and we will return to it when we study commuting operators. For now, the takeaway is that the linear operators on a Hilbert space form an algebra: a set you can add, scale, and multiply within, with the identity as the multiplicative unit.

Why this matters

Every physical process in quantum mechanics is encoded by an operator. Observables are Hermitian operators (their eigenvalues are the possible measurement outcomes); time evolution is a unitary operator; symmetries and projections are operators too. The rest of this module unpacks these special families one at a time. Each is, first and foremost, a linear operator — so everything we say rests on the single defining property above.