Observables are Hermitian operators
In quantum mechanics, every measurable quantity — position, momentum, energy, angular momentum — is represented by a linear operator \hat A acting on states. But not just any operator will do: a genuine observable must be Hermitian (self-adjoint), meaning \hat A^{\dagger}=\hat A. The special states on which \hat A acts by simple rescaling are its eigenstates.
The eigenvalue equation: |a\rangle is an eigenstate of \hat A with eigenvalue a.
Two theorems make Hermiticity exactly the right requirement for physics. First, the eigenvalues of a Hermitian operator are real — so the possible measured values are ordinary real numbers, as they must be. Second, eigenstates belonging to distinct eigenvalues are orthogonal, and together they form a complete basis — so any state can be expanded in the eigenstates of any observable.
Hermiticity guarantees real eigenvalues and an orthonormal eigenbasis — the two properties measurement demands.
The measurement postulate and the Born rule
The measurement postulate ties the formalism to the lab. A measurement of \hat A can only yield one of its eigenvalues a. If the system is in |\psi\rangle, first expand |\psi\rangle=\sum_a c_a|a\rangle; then the probability of the outcome a is |c_a|^2=|\langle a|\psi\rangle|^2 — the Born rule once more. Immediately after, the state `collapses` to the eigenstate |a\rangle that was found (wavefunction collapse).
The Born rule as a projection; the probabilities sum to one precisely because \langle\psi|\psi\rangle=1 and the eigenbasis is complete.
Repeat the same measurement on many identically prepared copies and average the outcomes: you get the expectation value, a probability-weighted mean of the eigenvalues. It has a beautifully compact form as a `sandwich` of the operator between the state and its bra.
The expectation value: the mean of many measurements on identically prepared systems.
Commutators and compatibility
For operators, order matters: in general \hat A\hat B\neq\hat B\hat A. The failure to commute is measured by the commutator [\hat A,\hat B]=\hat A\hat B-\hat B\hat A. When it vanishes, \hat A and \hat B share a common eigenbasis and can be measured simultaneously to definite values — they are compatible. When it does not vanish, no state can be a sharp eigenstate of both at once.
The commutator measures how badly two observables fail to be simultaneously measurable.
The canonical commutation relation — position and momentum are fundamentally incompatible.
The relation [\hat x,\hat p]=i\hbar is arguably the single equation from which all quantum strangeness flows. Check it in one line with \hat p=-i\hbar\,d/dx acting on a test function f: (\hat x\hat p-\hat p\hat x)f=-i\hbar\,x f'+i\hbar\,(xf)'=i\hbar f. The result is nonzero, so x and p can never both be perfectly sharp.
The uncertainty principle
Non-commuting observables obey a rigorous inequality — the Robertson relation. The product of the standard deviations of any two observables is bounded below by the expectation of their commutator. This is a theorem about the structure of states, not a limit of any apparatus.
The general (Robertson) uncertainty relation, valid for any pair of observables.
The Heisenberg position-momentum relation, obtained by putting [\hat x,\hat p]=i\hbar into Robertson's inequality.