A quantum state is represented by a vector $|\psi\rangle$ in a complex Hilbert space. Unlike classical states, quantum states can superpose: if $|\psi_1\rangle$ and $|\psi_2\rangle$ are allowed states, then $a|\psi_1\rangle+b|\psi_2\rangle$ is also allowed for complex coefficients $a$ and $b$. This linearity is central to interference.

Observable quantities correspond to Hermitian operators such as $\hat{x}$, $\hat{p}$, and $\hat{H}$. Measurement outcomes are eigenvalues of these operators, and the state is projected onto the associated eigenstate or subspace according to the measurement postulate.

If a system is in state $|\psi\rangle$ and we measure an observable with eigenstate $|a\rangle$, the probability of obtaining result $a$ is $P(a)=|\langle a|\psi\rangle|^2$. Probabilities are computed from complex amplitudes, so phases matter even when they are not directly observed.