In quantum mechanics, measurable quantities such as position, momentum and energy are represented by Hermitian operators acting on quantum states. Hermiticity ensures that the operator's eigenvalues—the possible ideal measurement values—are real.

For a one-dimensional wavefunction, the position operator acts by multiplication: $$\hat x\psi=x\psi,$$ while the momentum operator is $$\hat p=-i\hbar\frac{\partial}{\partial x}.$$

Eigenstates and measurement values

If a state satisfies $$\hat A\psi=a\psi,$$ then $\psi$ is an eigenstate of the observable $A$ with eigenvalue $a$. A measurement of $A$ in that state returns $a$ with certainty in the idealized model.

For a general state, several eigenvalues can be possible, with probabilities determined by the state's components in the corresponding eigenstates.

Expectation value

The average result over many identically prepared systems is the expectation value. For a normalized wavefunction, $$\langle A\rangle=\int\psi^*(x),\hat A\psi(x),dx.$$

Operators need not commute

For two operators, $$[\hat A,\hat B]=\hat A\hat B-\hat B\hat A$$ measures whether their order matters. Position and momentum obey $$[\hat x,\hat p]=i\hbar.$$

Noncommuting observables cannot generally be assigned simultaneous sharp values in the same state; this algebra leads to quantum uncertainty relations.

Operators connect the abstract quantum state to experimental quantities, while their eigenstructure and commutation relations encode which values can be measured and which observables are mutually compatible.