Observable

An observable is a physical quantity that can be measured. In quantum mechanics, observables are represented by Hermitian linear operators acting on Hilbert space.

The key dictionary is:

Physics idea	Quantum mathematical object
State	Vector $
Observable	Hermitian operator $\hat{A}$
Possible measurement results	Eigenvalues of $\hat{A}$
Sharp-value states	Eigenvectors of $\hat{A}$

For a normalised state, the expectation value is

$$⟨A⟩=⟨\psi |\hat{A}|\psi ⟩.$$

This is not necessarily the result of one measurement. It is the average predicted over many identically prepared systems.

Noncommuting observables

If two operators do not commute,

$$[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}\ne 0,$$

then the order of operations matters and simultaneous sharp values may be impossible. This leads directly to the Uncertainty principle.

Classical bridge

In Hamiltonian mechanics, quantities on Phase space evolve via the Poisson bracket. Quantum mechanics replaces that algebra with operator commutators.

Common pitfalls

• Hermitian does not mean “visibly symmetric” unless you are in an orthonormal basis and include complex conjugation.
• Expectation value is an ensemble average, not a promise about a single trial.
• Measurement changes the state in ideal projective-measurement models.

Source trail: Susskind The Theoretical Minimum index; reference book: Quantum Mechanics: The Theoretical Minimum by Leonard Susskind and Art Friedman.