Observables as Operators

In classical mechanics every measurable quantity — position, momentum, energy — is just a number (or a function of numbers). You can, in principle, read off all of them simultaneously without disturbing anything. Quantum mechanics shatters that picture. To see why, we need to elevate each measurable quantity to a linear operator acting on the state space.

What is a linear operator?

A linear operator $\hat{A}$ is a rule that maps every state $|\psi\rangle$ to another state $\hat{A}|\psi\rangle$ , with the linearity condition

$\hat{A}\bigl(\alpha|\psi\rangle + \beta|\phi\rangle\bigr) = \alpha\,\hat{A}|\psi\rangle + \beta\,\hat{A}|\phi\rangle$

for any complex numbers $\alpha, \beta$ . The hat notation $\hat{A}$ is conventional for operators to distinguish them from ordinary numbers.

In a finite-dimensional Hilbert space (the setting you will encounter for qubits and spin-1/2 systems) every linear operator can be written as a matrix. For instance, the two-by-two matrix

$\hat{\sigma}_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

is the Pauli- $z$ matrix; the spin- $z$ observable for a spin-1/2 particle is $\hat{S}_z = \tfrac{\hbar}{2}\hat{\sigma}_z$ . Acting on the spin-up state $|0\rangle = \begin{pmatrix}1\\0\end{pmatrix}$ it gives $+1\cdot|0\rangle$ ; acting on the spin-down state $|1\rangle = \begin{pmatrix}0\\1\end{pmatrix}$ it gives $-1\cdot|1\rangle$ . Those eigenvalues $\pm 1$ are precisely the two possible outcomes when you measure the $z$ -component of spin, in units of $\hbar/2$ — i.e. the measured values $\pm\hbar/2$ .

The operator postulate

One of the central postulates of quantum mechanics can be stated as:

To every observable physical quantity there corresponds a linear, Hermitian operator on the Hilbert space of the system. The only possible outcomes of a measurement of that observable are the eigenvalues of the corresponding operator.

"Hermitian" here means the operator equals its own conjugate transpose: $\hat{A} = \hat{A}^\dagger$ . This constraint will be explored in the next lesson; for now the key consequence is that all eigenvalues of a Hermitian operator are real numbers — a prerequisite for them to represent physical measurement results.

The position and momentum operators

For a particle moving in one dimension the state is described by a wave function $\psi(x)$ , a complex-valued function of position. The two fundamental operators are:

Position: $\hat{x}\,\psi(x) = x\,\psi(x)$ — it simply multiplies the wave function by the coordinate $x$ .
Momentum: $\hat{p}\,\psi(x) = -i\hbar\dfrac{d}{dx}\psi(x)$ — it differentiates the wave function (and multiplies by $-i\hbar$ , where $\hbar = h/(2\pi)$ ).

Both are Hermitian on the appropriate function space, so their eigenvalues — the measurable values of position and momentum — are real, as they must be.

The Hamiltonian

The observable corresponding to total energy is the Hamiltonian $\hat{H}$ . For a particle in a potential $V(x)$ it is built directly from $\hat{x}$ and $\hat{p}$ :

$\hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x).$

The eigenvalue equation $\hat{H}|\psi\rangle = E|\psi\rangle$ is the time-independent Schrödinger equation: its solutions $|\psi\rangle$ are the stationary states and its eigenvalues $E$ are the allowed energies. Every discrete energy level you computed in the infinite square well or the harmonic oscillator is an eigenvalue of that system's Hamiltonian.

Why operators, not numbers?

Replacing numbers with operators is not mere mathematical decoration. It is what allows quantum mechanics to encode the fact that measuring one observable can irreversibly disturb another. The position operator $\hat{x}$ and the momentum operator $\hat{p}$ satisfy the canonical commutation relation

$[\hat{x}, \hat{p}] \equiv \hat{x}\hat{p} - \hat{p}\hat{x} = i\hbar\,\hat{I},$

where $\hat{I}$ is the identity operator. Because $\hat{x}$ and $\hat{p}$ do not commute — $\hat{x}\hat{p} \neq \hat{p}\hat{x}$ — they cannot both have definite values simultaneously. This non-commutativity is the algebraic root of the Heisenberg uncertainty principle $\sigma_x \sigma_p \geq \hbar/2$ .